tag:blogger.com,1999:blog-89483053591185412432024-03-14T10:09:40.843+08:00OMG! MathsFor clarification purposes, this blog is about Math and only Math.
If you have arithmophobia, please abandon this page immediately for your own mental sanity.4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.comBlogger25125tag:blogger.com,1999:blog-8948305359118541243.post-87481764758203395162011-07-28T23:20:00.000+08:002011-07-28T23:20:07.320+08:00Mathematics Quizzes<div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">1. </span><span style="font-size: 12pt; line-height: 18px;">Find the place value of 9 in 348.719.</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">a. </span><span style="font-size: 12pt; line-height: 18px;">9 hundredths</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">b. </span><span style="font-size: 12pt; line-height: 18px;">9 thousandths</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">c. </span><span style="font-size: 12pt; line-height: 18px;">9 hundreds</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">d. </span><span style="font-size: 12pt; line-height: 18px;">9 thousands</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;"><br />
</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">2. </span><span style="font-size: 12pt; line-height: 18px;">Which is the greatest 2 digit prime number?</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">a. </span><span style="font-size: 12pt; line-height: 18px;">99</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">b. </span><span style="font-size: 12pt; line-height: 18px;">97</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">c. </span><span style="font-size: 12pt; line-height: 18px;">91</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">d. </span></strong></span></span></span><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;"> </span><span style="font-size: 12pt; line-height: 18px;">83</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;"><br />
</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">3. </span><span style="font-size: 12pt; line-height: 18px;">30 cents+2 dimes=________</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">a. </span><span style="font-size: 12pt; line-height: 18px;">2 quarters</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">b. </span><span style="font-size: 12pt; line-height: 18px;">4 nickels</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">c. </span><span style="font-size: 12pt; line-height: 18px;">10 dimes</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">d. </span><span style="font-size: 12pt; line-height: 18px;">100 cents</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;"><br />
</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">4. </span><span style="font-size: 12pt; line-height: 18px;">The</span></strong></span></span></span><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;"> <span class="IL_AD" id="IL_AD5" style="background-attachment: scroll !important; background-clip: initial !important; background-color: transparent !important; background-image: none !important; background-origin: initial !important; background-position: 0% 50%; background-repeat: repeat repeat !important; border-bottom-color: rgb(255, 0, 0) !important; border-bottom-style: solid !important; border-bottom-width: 1px !important; color: rgb(255, 0, 0) !important; cursor: pointer !important; display: inline !important; float: none !important; font-family: 'Trebuchet MS', sans-serif !important; font-size: 16px !important; font-style: normal !important; font-weight: bold !important; padding-bottom: 1px !important; padding-left: 0px !important; padding-right: 0px !important; padding-top: 0px !important; position: static; text-decoration: underline !important;">Math test</span> scores of Grade 4 students are given below:</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in;"><span style="color: blue; font-family: 'Trebuchet MS', sans-serif; font-size: 12pt; line-height: 18px;"><strong>90, 56, 36, 55, 99, 55, 67, 42, 51, 63 and 65.</strong></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in;"><span style="color: blue; font-family: 'Trebuchet MS', sans-serif; font-size: 12pt; line-height: 18px;"><strong>Find the range of the above data.</strong></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: x-small;"><strong><span style="font-size: 12pt; line-height: 18px;">a. </span><span style="font-size: 12pt; line-height: 18px;">99</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">b. </span><span style="font-size: 12pt; line-height: 18px;">36</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">c. </span><span style="font-size: 12pt; line-height: 18px;">63</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">d. </span><span style="font-size: 12pt; line-height: 18px;">55</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;"><br />
</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">5. </span><span style="font-size: 12pt; line-height: 18px;">A shelf contains 7 boxes. There are 12 pencils in a box. How many pencils are there in total?</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">a. </span><span style="font-size: 12pt; line-height: 18px;">19</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">b. </span><span style="font-size: 12pt; line-height: 18px;">5</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">c. </span><span style="font-size: 12pt; line-height: 18px;">84</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">d. </span><span style="font-size: 12pt; line-height: 18px;">14</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;"><br />
</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">6. </span><span style="font-size: 12pt; line-height: 18px;">Estimate the area of a picture in the given grid:</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue; font-family: 'Trebuchet MS', sans-serif; font-size: 12pt; line-height: 18px;"><strong> <img alt="Estimate the area" src="http://i570.photobucket.com/albums/ss141/athiappan/46.jpg" style="-webkit-box-shadow: rgba(0, 0, 0, 0.0976563) 1px 1px 5px; background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial initial; background-repeat: initial initial; border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(238, 238, 238); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(238, 238, 238); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(238, 238, 238); border-top-style: solid; border-top-width: 1px; box-shadow: rgba(0, 0, 0, 0.0976563) 1px 1px 5px; padding-bottom: 5px; padding-left: 5px; padding-right: 5px; padding-top: 5px;" /></strong></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">a. </span><span style="font-size: 12pt; line-height: 18px;">18 sq. units</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">b. </span><span style="font-size: 12pt; line-height: 18px;">14 sq. units</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">c. </span><span style="font-size: 12pt; line-height: 18px;">12 sq. units</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">d. </span><span style="font-size: 12pt; line-height: 18px;">8 sq. units</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;"><br />
</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">7. </span><span style="font-size: 12pt; line-height: 18px;">5 ft 3 in + 2 ft 5 in is equal to ______</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">a. </span><span style="font-size: 12pt; line-height: 18px;">7 ft 8 in</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">b. </span><span style="font-size: 12pt; line-height: 18px;">8 ft 7 in</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">c. </span><span style="font-size: 12pt; line-height: 18px;">3 ft 2 in</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">d. </span><span style="font-size: 12pt; line-height: 18px;">0 ft 2 in</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;"><br />
</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">8. </span><span style="font-size: 12pt; line-height: 18px;">2 cm=___ mm</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">a. </span><span style="font-size: 12pt; line-height: 18px;">10</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">b. </span><span style="font-size: 12pt; line-height: 18px;">20</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">c. </span><span style="font-size: 12pt; line-height: 18px;">100</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">d. </span><span style="font-size: 12pt; line-height: 18px;">200</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;"><br />
</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">9. </span><span style="font-size: 12pt; line-height: 18px;">There are 8000 grams of rice in a container. Dawn wanted to make 1 kg of rice in each bag. How many bags are required to empty the container?</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">a. </span><span style="font-size: 12pt; line-height: 18px;">8000</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">b. </span><span style="font-size: 12pt; line-height: 18px;">800</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">c. </span><span style="font-size: 12pt; line-height: 18px;">80</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">d. </span><span style="font-size: 12pt; line-height: 18px;">8</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;"><br />
</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">10.</span> </strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue; font-family: 'Trebuchet MS', sans-serif;"><strong> <img alt="Type of triangle" src="http://i570.photobucket.com/albums/ss141/athiappan/49.jpg" style="-webkit-box-shadow: rgba(0, 0, 0, 0.0976563) 1px 1px 5px; background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial initial; background-repeat: initial initial; border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(238, 238, 238); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(238, 238, 238); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(238, 238, 238); border-top-style: solid; border-top-width: 1px; box-shadow: rgba(0, 0, 0, 0.0976563) 1px 1px 5px; padding-bottom: 5px; padding-left: 5px; padding-right: 5px; padding-top: 5px;" /></strong></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in;"><span style="color: blue; font-family: 'Trebuchet MS', sans-serif; font-size: 12pt; line-height: 18px;"><strong>The given triangle is _________</strong></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">a. </span><span style="font-size: 12pt; line-height: 18px;">Equilateral triangle</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">b. </span><span style="font-size: 12pt; line-height: 18px;">Isosceles triangle</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">c. </span><span style="font-size: 12pt; line-height: 18px;">Right triangle</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: x-small;"><strong><span style="font-size: 12pt; line-height: 18px;">d. </span><span style="font-size: 12pt; line-height: 18px;">Scalene triangle</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: x-small;"><strong><span style="font-size: 12pt; line-height: 18px;"><br />
</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">11. </span><span style="font-size: 12pt; line-height: 18px;">In a rectangle, which of the following statement is not true?</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">a. </span><span style="font-size: 12pt; line-height: 18px;">Sides are equal</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">b. </span><span style="font-size: 12pt; line-height: 18px;">The angles are 90 degree</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">c. </span><span style="font-size: 14pt; line-height: 21px;">Diagonals bisect each other</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">d. </span><span style="font-size: 12pt; line-height: 18px;">Diagonals are equal</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;"><br />
</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">12. </span><span style="font-size: 12pt; line-height: 18px;">If a ray makes an angle 50 degree with a straight line, the angle formed is _____</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">a. </span><span style="font-size: 12pt; line-height: 18px;">Acute angle</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">b. </span><span style="font-size: 12pt; line-height: 18px;">Obtuse angle</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">c. </span><span style="font-size: 12pt; line-height: 18px;">Right angle</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">d. </span><span style="font-size: 12pt; line-height: 18px;">Straight angle</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;"><br />
</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="font-size: 12pt; line-height: 18px;"><span style="color: blue; font-family: 'Trebuchet MS', sans-serif; font-size: 12pt;"><strong>13. </strong></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="font-size: 12pt; line-height: 18px;"><span style="color: blue; font-family: 'Trebuchet MS', sans-serif; font-size: 12pt;"><strong> <img alt="Triangle based on angles" src="http://i570.photobucket.com/albums/ss141/athiappan/412.jpg" style="-webkit-box-shadow: rgba(0, 0, 0, 0.0976563) 1px 1px 5px; background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial initial; background-repeat: initial initial; border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(238, 238, 238); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(238, 238, 238); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(238, 238, 238); border-top-style: solid; border-top-width: 1px; box-shadow: rgba(0, 0, 0, 0.0976563) 1px 1px 5px; padding-bottom: 5px; padding-left: 5px; padding-right: 5px; padding-top: 5px;" /></strong></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in;"><span style="color: blue; font-family: 'Trebuchet MS', sans-serif; font-size: 12pt; line-height: 18px;"><strong>The given triangle is ________triangle</strong></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">a. </span><span style="font-size: 12pt; line-height: 18px;">Acute</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">b. </span><span style="font-size: 12pt; line-height: 18px;">Obtuse</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">c. </span><span style="font-size: 12pt; line-height: 18px;">Right</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">d. </span><span style="font-size: 12pt; line-height: 18px;">Straight</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;"><br />
</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">14. </span><span style="font-size: 12pt; line-height: 18px;">Fill in the missing number in the given number sequence:</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in;"><span style="color: blue; font-family: 'Trebuchet MS', sans-serif; font-size: 12pt; line-height: 18px;"><strong>1, 4, 10, 13, 16, ___, 22, 25.</strong></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: x-small;"><strong><span style="font-size: 12pt; line-height: 18px;">a. </span><span style="font-size: 12pt; line-height: 18px;">17</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">b. </span><span style="font-size: 12pt; line-height: 18px;">18</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">c. </span><span style="font-size: 12pt; line-height: 18px;">19</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">d. </span><span style="font-size: 12pt; line-height: 18px;">20</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;"><br />
</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">15. </span><span style="font-size: 12pt; line-height: 18px;">Choose the correct expression for the given statement:</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in;"><span style="color: blue; font-family: 'Trebuchet MS', sans-serif; font-size: 12pt; line-height: 18px;"><strong>Cathy has n number of apples. He gave away 3 of the apples to his friend.</strong></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">a. </span><span style="font-size: 12pt; line-height: 18px;">n+3</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">b. </span><span style="font-size: 12pt; line-height: 18px;">3n</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">c. </span><span style="font-size: 12pt; line-height: 18px;">n-3</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">d. </span><span style="font-size: 12pt; line-height: 18px;">3</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;"><br />
</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">16. </span><span style="font-size: 12pt; line-height: 18px;">Jason filled 6 cups of water in the first can. 12 cups of water in the second can and 18 cups of water in the third can. If the pattern continues, which can is filled with 36 cups of water?</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: x-small;"><strong><span style="font-size: 12pt; line-height: 18px;">a. </span><span style="font-size: 12pt; line-height: 18px;">4</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">b. </span><span style="font-size: 12pt; line-height: 18px;">5</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">c. </span><span style="font-size: 12pt; line-height: 18px;">6</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">d. </span><span style="font-size: 12pt; line-height: 18px;">7</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">17. </span><span style="font-size: 12pt; line-height: 18px;">If y=3+x and x takes the value as 4, then the value of y is ______</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">a. </span><span style="font-size: 12pt; line-height: 18px;">7</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">b. </span><span style="font-size: 12pt; line-height: 18px;">6</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">c. </span><span style="font-size: 12pt; line-height: 18px;">5</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">d. </span><span style="font-size: 12pt; line-height: 18px;">4</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;"><br />
</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">18.</span><span style="font-size: 12pt; line-height: 18px;">There are 5 green color beads and 6 yellow color beads in a box. Find the probability of selecting a green bead from the box.</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">a. </span><span style="font-size: 12pt; line-height: 18px;">5/6</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">b. </span><span style="font-size: 12pt; line-height: 18px;">1/11</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">c. </span><span style="font-size: 12pt; line-height: 18px;">6/5</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">d. </span><span style="font-size: 12pt; line-height: 18px;">1/5</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">19. </span><span style="font-size: 12pt; line-height: 18px;">Find the median for the given data:</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in;"><span style="color: blue; font-family: 'Trebuchet MS', sans-serif; font-size: 12pt; line-height: 18px;"><strong>45, 22, 76, 90, 35, 78,92</strong></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">a. </span><span style="font-size: 12pt; line-height: 18px;">45</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">b. </span><span style="font-size: 12pt; line-height: 18px;">76</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">c. </span><span style="font-size: 12pt; line-height: 18px;">78</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">d. </span><span style="font-size: 12pt; line-height: 18px;">90</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;"><br />
</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">20. </span><span style="font-size: 12pt; line-height: 18px;"><span class="IL_AD" id="IL_AD9" style="background-attachment: scroll !important; background-clip: initial !important; background-color: transparent !important; background-image: none !important; background-origin: initial !important; background-position: 0% 50%; background-repeat: repeat repeat !important; border-bottom-color: rgb(255, 0, 0) !important; border-bottom-style: solid !important; border-bottom-width: 1px !important; color: rgb(255, 0, 0) !important; cursor: pointer !important; display: inline !important; float: none !important; font-family: 'Trebuchet MS', sans-serif !important; font-size: 16px !important; font-style: normal !important; font-weight: bold !important; padding-bottom: 1px !important; padding-left: 0px !important; padding-right: 0px !important; padding-top: 0px !important; position: static; text-decoration: underline !important;">The bar</span> graph given below shows the Math scores of grade 4 classmates. Who is in the second highest position?</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.5in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue; font-family: 'Trebuchet MS', sans-serif; font-size: 12pt; line-height: 18px;"><strong> <img alt="Bar Graph" src="http://i570.photobucket.com/albums/ss141/athiappan/420.jpg" style="-webkit-box-shadow: rgba(0, 0, 0, 0.0976563) 1px 1px 5px; background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial initial; background-repeat: initial initial; border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(238, 238, 238); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(238, 238, 238); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(238, 238, 238); border-top-style: solid; border-top-width: 1px; box-shadow: rgba(0, 0, 0, 0.0976563) 1px 1px 5px; padding-bottom: 5px; padding-left: 5px; padding-right: 5px; padding-top: 5px;" /></strong></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">a. </span><span style="font-size: 12pt; line-height: 18px;">Jack</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">b. </span><span style="font-size: 12pt; line-height: 18px;">Tony</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 0pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">c. </span><span style="font-size: 12pt; line-height: 18px;">Lara</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 10pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;">d. </span><span style="font-size: 12pt; line-height: 18px;">Eric</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 10pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;"><br />
</span></strong></span></span></span></div><div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 10pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="color: blue;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;"><br />
</span></strong></span></span></span></div><div style="font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 10pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;"><span class="Apple-style-span" style="color: #e06666;">Nurain Nazuraa</span></span></strong></span></span></div><div style="font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14px; line-height: 19px; margin-bottom: 10pt; margin-left: 0.75in; margin-right: 0in; margin-top: 0in; text-indent: -0.25in;"><span style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: 12pt;"><strong><span style="font-size: 12pt; line-height: 18px;"><span class="Apple-style-span" style="color: #e06666;">4sc</span></span></strong></span></span></div>4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com0tag:blogger.com,1999:blog-8948305359118541243.post-37507606692109680142011-07-27T22:34:00.000+08:002011-07-27T22:34:03.180+08:00<i><b><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;">If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is. ~John Louis von Neumann</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;">Arithmetic is where the answer is right and everything is nice and you can look out of the window and see the blue sky - or the answer is wrong and you have to start over and try again and see how it comes out this time. ~Carl Sandburg</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;">Pure mathematics is, in its way, the poetry of logical ideas. ~Albert Einstein</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;">Mathematics are well and good but nature keeps dragging us around by the nose. ~Albert Einstein</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;">Black holes result from God dividing the universe by zero. ~Author Unknown</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;">Mathematics - the unshaken Foundation of Sciences, and the plentiful Fountain of Advantage to human affairs. ~Isaac Barrow</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;">I never did very well in math - I could never seem to persuade the teacher that I hadn't meant my answers literally. ~Calvin Trillin</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;">I don't agree with mathematics; the sum total of zeros is a frightening figure. ~Stanislaw J. Lec,</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"> </span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;">More Unkempt Thoughts</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;">If you think dogs can't count, try putting three dog biscuits in your pocket and then giving Fido only two of them. ~Phil Pastoret</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;">[A mathematician is a] scientist who can figure out anything except such simple things as squaring the circle and trisecting an angle. ~Evan Esar,</span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"> </span></b></i><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><i><b>Esar's Comic Di</b>ctionary</i></span><span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><br />
</span><br />
<span class="Apple-style-span" style="border-collapse: collapse; color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><i><br />
</i></span><br />
<span class="Apple-style-span" style="color: #330000; font-family: georgia, 'bookman old style', 'palatino linotype', 'book antiqua', palatino, 'trebuchet ms', helvetica, garamond, sans-serif, arial, verdana, 'avante garde', 'century gothic', 'comic sans ms', times, 'times new roman', serif;"><span class="Apple-style-span" style="border-collapse: collapse;"><i>Done By, Jannani :)</i></span></span>4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com0tag:blogger.com,1999:blog-8948305359118541243.post-88820425948922161462011-07-26T22:35:00.005+08:002011-07-26T23:46:07.300+08:00Probability I<div class="MsoNormal"><span class="Apple-style-span" style="line-height: 18px;"></span><br />
<div class="MsoNormal" style="font-family: inherit;"><span style="font-size: small;"><b><span lang="EN-US" style="line-height: 115%;">7.1 Sample Space<o:p></o:p></span></b></span></div><div class="MsoNormal" style="font-family: inherit;"><br />
<span lang="EN-US" style="font-size: small; line-height: 115%;">An experiment is a process or an operation with an outcome.<o:p></o:p></span></div><div class="MsoNormal" style="font-family: inherit;"><br />
<span style="font-size: small;"><b><span lang="EN-US" style="line-height: 115%;">E.g. 1<o:p></o:p></span></b></span></div><div style="border: medium none; font-family: inherit; padding: 0cm 0cm 1pt;"><div align="center" class="MsoNormal" style="border: medium none; padding: 0cm; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><span style="font-size: small;"><a href="http://4.bp.blogspot.com/-iIxC40vnnyo/Ti7SWgXKNJI/AAAAAAAAAA0/ijX_yr1G1dU/s1600/123.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="143" src="http://4.bp.blogspot.com/-iIxC40vnnyo/Ti7SWgXKNJI/AAAAAAAAAA0/ijX_yr1G1dU/s320/123.jpg" width="320" /></a></span></div><span lang="EN-US" style="font-size: small;"><span class="Apple-style-span" style="line-height: 18px;"><br />
</span><span class="Apple-style-span"><span class="Apple-style-span" style="line-height: 115%;"> Toss a balanced dice once and observe its uppermost surface<o:p></o:p></span></span></span></div></div><div class="MsoNormal" style="font-family: inherit;"><span style="font-size: small;"><b><span lang="EN-US"><span class="Apple-style-span"><span class="Apple-style-span" style="line-height: 115%;">E.g.</span></span><span style="line-height: 115%;"> </span><span class="Apple-style-span"><span class="Apple-style-span" style="line-height: 115%;">2</span></span><span class="Apple-style-span" style="line-height: 18px;"><br />
</span></span></b><a href="http://4.bp.blogspot.com/-QRjgk4ej0oI/Ti7Ss8YKZsI/AAAAAAAAAA4/WN_LUl7olKY/s1600/124.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-QRjgk4ej0oI/Ti7Ss8YKZsI/AAAAAAAAAA4/WN_LUl7olKY/s1600/124.png" /></a></span><span class="Apple-style-span" style="font-size: small; line-height: 18px;"> </span><span class="Apple-style-span" style="font-size: small; line-height: 18px;">When a coin is tossed, we can only get 2 results :</span></div><div class="MsoListParagraphCxSpFirst" style="font-family: inherit; margin-left: 234pt; text-indent: -18pt;"><span lang="EN-US" style="font-size: small; line-height: 115%;">Ø<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-size: small; line-height: 115%;">Head <o:p></o:p></span></div><div class="MsoListParagraphCxSpLast" style="font-family: inherit; margin-left: 234pt; text-indent: -18pt;"><span lang="EN-US" style="font-size: small; line-height: 115%;">Ø<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-size: small; line-height: 115%;">Tail <o:p></o:p></span></div><div style="border: medium none; font-family: inherit; padding: 0cm 0cm 1pt;"><div class="MsoNormal" style="border: medium none; padding: 0cm;"><span style="font-size: small;"><br />
</span></div></div><div class="MsoNormal" style="font-family: inherit; margin-left: 36pt; text-indent: -36pt;"><span lang="EN-US" style="font-size: small; line-height: 115%;">The set of all possible outcomes of an experiment is called a sample space. It is usually denoted by S.<o:p></o:p></span></div><div class="MsoNormal" style="font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div class="MsoNormal" style="font-family: inherit;"><span style="font-size: small;"><b><span lang="EN-US" style="line-height: 115%;">E.g. 3<o:p></o:p></span></b></span></div><div class="MsoNormal" style="font-family: inherit;"><span lang="EN-US" style="font-size: small; line-height: 115%;">En. Adam has a fruit stall. He sells bananas, apples, watermelons, papayas and durians. Students of 4KP were asked to select their favourite fruit from the variety of fruits sold at En. Adam’s stall.<o:p></o:p></span></div><div class="MsoNormal" style="font-family: inherit;"><span lang="EN-US" style="font-size: small; line-height: 115%;">S = { bananas, apples, watermelons, papayas, durians }<o:p></o:p></span></div><div class="MsoNormal" style="font-family: inherit;"><br />
<span style="font-size: small;"><b><span lang="EN-US" style="line-height: 115%;">E.g. 4<o:p></o:p></span></b></span></div><div class="MsoNormal" style="font-family: inherit;"><span lang="EN-US" style="font-size: small; line-height: 115%;">A month is randomly selected from a year. Describe the sample space of this experiment by using set notation.<o:p></o:p></span></div><div style="border: medium none; font-family: inherit; padding: 0cm 0cm 1pt;"><div class="MsoNormal" style="border: medium none; padding: 0cm;"><span lang="EN-US" style="font-size: small; line-height: 115%;">S = { January, February, March, April, June, July, August, September, October, November, December }<br />
<br />
<o:p></o:p></span></div></div><div class="MsoNormal" style="font-family: inherit;"><span style="font-size: small;"><b><span lang="EN-US" style="line-height: 115%;">7.2 Event</span></b><b><span lang="EN-US" style="line-height: 115%;"><o:p></o:p></span></b></span></div><div class="MsoNormal" style="font-family: inherit;"><span style="font-size: small;"><b><span lang="EN-US" style="line-height: 115%;"><br />
- </span></b></span><span lang="EN-US" style="font-size: small; line-height: 115%;">Is a subset of the sample space.<o:p></o:p></span></div><div class="MsoNormal" style="font-family: inherit;">-<span lang="EN-US" style="font-size: small; line-height: 115%;"> Is an outcome or a set of outcomes that satisfies certain conditions.<o:p></o:p> </span><br />
<span lang="EN-US" style="font-size: small; line-height: 115%;">- Denoted by a capital letter.<o:p></o:p></span></div><div class="MsoNormal" style="font-family: inherit;"><span lang="EN-US" style="font-size: small; line-height: 115%;"><br />
<b>E.g. 1 <o:p></o:p></b></span></div><div class="MsoNormal" style="font-family: inherit;"><span lang="EN-US" style="font-size: small; line-height: 115%;">A box contains five cards written with 1, 2, 3, 4 and 5 respectively. A card is picked randomly from a box.<o:p></o:p></span></div><div class="MsoNormal" style="font-family: inherit;"><span lang="EN-US" style="font-size: small; line-height: 115%;">S = { 1, 2, 3, 4, 5 }<o:p></o:p></span></div><div class="MsoNormal" style="font-family: inherit;"><span lang="EN-US" style="font-size: small; line-height: 115%;">If we define just “the card with even numbers”, the outcome of <b>J</b> in set notation will be<b> <br />
J = { 2, 4 }<o:p></o:p></b></span></div><div class="MsoNormal" style="font-family: inherit;"><span lang="EN-US" style="font-size: small; line-height: 115%;"><br />
<b>E.g. 2<o:p></o:p></b></span></div><div class="MsoNormal" style="font-family: inherit;"><span lang="EN-US" style="font-size: small; line-height: 115%;">A letter is randomly selected from the word “COMPUTER”. Determine the number of the possible outcomes of the event that the selected letter is.<o:p></o:p></span></div><div class="MsoListParagraphCxSpFirst" style="font-family: inherit; text-indent: -36pt;"><span lang="EN-US" style="font-size: small; line-height: 115%;"><span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;"> </span>i.<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-size: small; line-height: 115%;">A vowel<o:p></o:p></span></div><div class="MsoListParagraphCxSpLast" style="font-family: inherit; text-indent: -36pt;"><span lang="EN-US" style="font-size: small; line-height: 115%;"><span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;"> </span>ii.<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-size: small; line-height: 115%;">A consonant<o:p></o:p></span></div><div class="MsoNormal" style="font-family: inherit;"><span style="font-size: small;"><b><span lang="EN-US" style="line-height: 115%;">Solution : <o:p></o:p></span></b></span></div><div class="MsoNormal" style="font-family: inherit;"><span lang="EN-US" style="font-size: small; line-height: 115%;">Let A = event that the selected letter is a vowel = { O, U, E }<br />
Therefore, n (A) = 3<o:p></o:p></span></div><div style="border: medium none; font-family: inherit; padding: 0cm 0cm 1pt;"><div class="MsoNormal" style="border: medium none; padding: 0cm;"><span lang="EN-US" style="font-size: small; line-height: 115%;">Let B = event that the selected letter is a consonant = { C, M, P, T, R }<br />
Therefore, n (A) = 5<br />
<br />
<o:p></o:p></span></div></div><div class="MsoNormal" style="font-family: inherit;"><span lang="EN-US" style="font-size: small; line-height: 115%;"><br />
</span><span style="font-size: small;"><b><span lang="EN-US" style="line-height: 115%;">7.3 Probability of an Event</span></b><b><span lang="EN-US" style="line-height: 115%;"><o:p></o:p></span></b></span></div><div class="MsoNormal" style="font-family: inherit;"><span style="font-size: small;"><b><span lang="EN-US" style="line-height: 115%;"><br />
</span></b></span><span lang="EN-US" style="font-size: small; line-height: 115%;">Probability of an event E, <o:p></o:p></span></div><div class="MsoNormal" style="font-family: inherit;">-<span lang="EN-US" style="font-size: small; line-height: 115%;"> P (E) = ___<u>number of outcomes of the event___</u><br />
number of outcomes of the sample space<o:p></o:p></span></div><div class="MsoNormal" style="font-family: inherit;"><span lang="EN-US" style="font-size: small; line-height: 115%;">- P (E) = <u>n (E)<br />
</u> n (S)<o:p></o:p></span></div><div class="MsoNormal" style="font-family: inherit;"><span lang="EN-US" style="font-size: small; line-height: 115%;">- 0 </span><span lang="EN-US" style="font-size: small; line-height: 115%;">≤ P (E) ≤ 1</span><span lang="EN-US" style="font-size: small; line-height: 115%;"><o:p></o:p></span></div><div class="MsoNormal" style="font-family: inherit;"><span lang="EN-US" style="font-size: small;">- </span><span lang="EN-US" style="font-size: small; line-height: 115%;">P (E) = 0 means that it is impossible for the event to happen.<o:p></o:p></span></div><div class="MsoNormal" style="font-family: inherit;"><span lang="EN-US" style="font-size: small; line-height: 115%;">- P (E) = 1 means that the event is certain to happen.<o:p></o:p></span></div><div class="MsoNormal" style="font-family: inherit;"><span lang="EN-US" style="font-size: small; line-height: 115%;">- The closer the probability of a given event is to 1, the more likely it is to happen.<o:p></o:p></span></div><div class="MsoNormal" style="font-family: inherit;"><br />
<span style="font-size: small;"><b><span lang="EN-US" style="line-height: 115%;">E.g. 1<o:p></o:p></span></b></span></div><div class="MsoNormal" style="font-family: inherit;"><span lang="EN-US" style="font-size: small; line-height: 115%;">A bag contains 3 red balls and 4 white ones. If Rashid puts his hand in the bag and picks a ball, what is the probability that the ball he picked is white?<o:p></o:p></span></div><div class="MsoNormal" style="font-family: inherit;"><span style="font-size: small;"><b><span lang="EN-US" style="line-height: 115%;">Solution : <o:p></o:p></span></b></span></div><div class="MsoNormal" style="font-family: inherit;"><span lang="EN-US" style="font-size: small; line-height: 115%;">S = { R1, R2, R3, W1, W2, W3, W4 } <br />
n (S) = 7<o:p></o:p></span></div><div class="MsoNormal" style="font-family: inherit;"><span lang="EN-US" style="font-size: small; line-height: 115%;">Let E is the event of drawing a white ball<br />
E = { W1, W2, W3, W4 }<br />
n (E) = 4<o:p></o:p></span></div><div class="MsoNormal" style="font-family: inherit;"><span lang="EN-US" style="font-size: small; line-height: 115%;">Therefore, the probability of drawing a white ball is : <u>4<br />
</u> 7<o:p></o:p></span></div><div class="MsoNormal" style="font-family: inherit;"><span lang="EN-US" style="font-size: small; line-height: 115%;"><br />
</span></div><div class="MsoNormal" style="font-family: inherit;"><span lang="EN-US" style="font-size: small; line-height: 115%;">By Melissa Teh</span></div></div>4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com0tag:blogger.com,1999:blog-8948305359118541243.post-92050358256359243172011-07-24T23:07:00.003+08:002011-07-26T16:01:38.742+08:004 Significant Figures<div style="color: black; font-family: inherit;"><span class="Apple-style-span" style="font-size: small; line-height: 17px;">1. Any digit that is not zero is significant. 1234.56 6 significant figures </span></div><blockquote style="color: black; font-family: inherit; line-height: 17px;"><span style="font-size: small;">1234.56 6 significant figures</span></blockquote><div style="color: black; font-family: inherit;"><span style="font-size: small;">2. Zeros between non-zero digits are significant.</span></div><blockquote style="color: black; font-family: inherit; line-height: 17px;"><span style="font-size: small;">1002.5 5 significant figures</span></blockquote><div style="color: black; font-family: inherit;"><span style="font-size: small;">3. Zeros to the left of the first non-zero digit are not significant.</span></div><blockquote style="color: black; font-family: inherit; line-height: 17px;"><span style="font-size: small;">000456 3 significant figures</span></blockquote><blockquote style="color: black; font-family: inherit; line-height: 17px;"><span style="font-size: small;">0.0056 2 significant figures</span></blockquote><div style="color: black; font-family: inherit;"><span style="font-size: small;">4. If the number is greater than one (1), then all zeros to the right of the decimal point are significant.</span></div><blockquote style="color: black; font-family: inherit; line-height: 17px;"><span style="font-size: small;">457.12 5 significant figures</span></blockquote><blockquote style="color: black; font-family: inherit; line-height: 17px;"><span style="font-size: small;">400.00 5 significant figures</span></blockquote><div style="color: black; font-family: inherit;"><span style="font-size: small;">5. If the number is less than one, then only zeros that are at the end of the number and between non-zero digits are significant.</span></div><blockquote style="color: black; font-family: inherit; line-height: 17px;"><span style="font-size: small;">0.01020 4 significant figures</span></blockquote><div style="color: black; font-family: inherit;"><span style="font-size: small;">6. For numbers that do not contain decimal points, the trailing zeros may or may not be significant. In this course assume the digits are significant unless told otherwise.</span></div><blockquote style="color: black; font-family: inherit; line-height: 17px;"><span style="font-size: small;">1000 1, 2, 3, or 4 significant figures. UNCLEAR assume 4 in calculation</span></blockquote><blockquote style="color: black; font-family: inherit; line-height: 17px;"><span style="font-size: small;">0.0010 2 significant figures</span></blockquote><blockquote style="color: black; font-family: inherit; line-height: 17px;"><span style="font-size: small;">1.000 4 significant figures</span></blockquote><div style="color: black; font-family: inherit;"><span style="font-size: small;">7. Assume defined and counted quantities have an unlimited number of significant figures.</span></div><blockquote style="color: black; font-family: inherit; line-height: 17px;"><span style="font-size: small;">NOTE: It is much easier to count and keep track of significant figures if the number is written in scientific notation.</span></blockquote><div style="color: black; font-family: inherit;"><span style="font-size: small;">By Pravetha Nair</span></div>4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com0tag:blogger.com,1999:blog-8948305359118541243.post-4120001107758470232011-07-24T19:12:00.001+08:002011-07-26T15:56:45.181+08:00Mathematics as science<span class="Apple-style-span" style="font-family: sans-serif; font-size: 13px; line-height: 19px;"></span><br />
<div class="thumb tleft" style="background-color: transparent; clear: left; float: left; margin-bottom: 1.3em; margin-left: 0px; margin-right: 1.4em; margin-top: 0.5em; width: auto;"><div class="thumbinner" style="background-color: #f9f9f9; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(204, 204, 204); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(204, 204, 204); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(204, 204, 204); border-top-style: solid; border-top-width: 1px; font-size: 12px; min-width: 100px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 3px !important; padding-left: 3px !important; padding-right: 3px !important; padding-top: 3px !important; text-align: center; width: 172px;"><span class="Apple-style-span" style="color: #0645ad;"><span class="Apple-style-span"><img alt="" class="thumbimage" height="218" src="http://upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Carl_Friedrich_Gauss.jpg/170px-Carl_Friedrich_Gauss.jpg" style="background-color: white; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; border-color: initial; border-left-color: rgb(204, 204, 204); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(204, 204, 204); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(204, 204, 204); border-top-style: solid; border-top-width: 1px; border-width: initial; vertical-align: middle;" width="170" /></span></span><br />
<div class="thumbcaption" style="border-bottom-style: none; border-color: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; font-size: 11px; line-height: 1.4em; padding-bottom: 3px !important; padding-left: 3px !important; padding-right: 3px !important; padding-top: 3px !important; text-align: left;"><div class="magnify" style="background-attachment: initial !important; background-clip: initial !important; background-color: initial !important; background-image: none !important; background-origin: initial !important; background-position: initial initial !important; background-repeat: initial initial !important; border-bottom-style: none !important; border-color: initial !important; border-left-style: none !important; border-right-style: none !important; border-top-style: none !important; border-width: initial !important; float: right;"><span class="Apple-style-span" style="color: #0645ad;"><span class="Apple-style-span"></span></span></div><span class="Apple-style-span" style="color: #0645ad;"><span class="Apple-style-span">Carl Friedrich Gauss</span></span>, himself known as the "prince of mathematicians", referred to mathematics as "the Queen of the Sciences".</div></div></div><div style="color: black; font-family: inherit; line-height: 1.5em; margin: 0.4em 0px 0.5em;">Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences". In the original Latin Regina Scientiarum, as well as in German Königin der Wissenschaften, the word corresponding to science means a "field of knowledge", and this was the original meaning of "science" in English, also. Of course, mathematics is in this sense a field of knowledge. The specialization restricting the meaning of "science" to <span class="Apple-style-span"><span class="Apple-style-span">N</span></span>atural science follows the rise of Baconian science, which contrasted "natural science" to<span class="Apple-style-span"><span class="Apple-style-span"> scholasticism</span></span>, the Aristotelean method of inquiring from <span class="Apple-style-span"><span class="Apple-style-span">first principles</span></span>. Of course, the role of empirical experimentation and observation is negligible in mathematics, compared to natural sciences such as <span class="Apple-style-span"><span class="Apple-style-span">psychology</span></span>, <span class="Apple-style-span"><span class="Apple-style-span">biology</span></span>, or <span class="Apple-style-span"><span class="Apple-style-span">physics</span></span>. <span class="Apple-style-span"><span class="Apple-style-span">Albert Einstein</span></span> stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."</div><div style="color: black; font-family: inherit; line-height: 1.5em; margin: 0.4em 0px 0.5em;">Many philosophers believe that mathematics is not experimentally <span class="Apple-style-span"><span class="Apple-style-span">falsifiable</span></span>, and thus not a science according to the definition of <span class="Apple-style-span"><span class="Apple-style-span">Karl Popper</span></span>. However, in the 1930s <span class="Apple-style-span"><span class="Apple-style-span">Gödel's incompleteness theorems</span></span> convinced many mathematicians that mathematics cannot be reduced to logic alone, and Karl Popper concluded that "most mathematical theories are, like those of <span class="Apple-style-span"><span class="Apple-style-span">physics</span></span> and <span class="Apple-style-span"><span class="Apple-style-span">biology</span></span>, <span class="Apple-style-span"><span class="Apple-style-span">hypothetico</span></span>-<span class="Apple-style-span"><span class="Apple-style-span">deductive</span></span>: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently." Other thinkers, notably <span class="Apple-style-span"><span class="Apple-style-span">Imre Lakatos</span></span>, have applied a version of <span class="Apple-style-span"><span class="Apple-style-span">falsificationism</span></span> to mathematics itself.</div><div style="color: black; font-family: inherit; line-height: 1.5em; margin: 0.4em 0px 0.5em;">An alternative view is that certain scientific fields (such as <span class="Apple-style-span"><span class="Apple-style-span">theoretical physics</span></span>) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, <span class="Apple-style-span"><span class="Apple-style-span">J. M. Ziman</span></span>, proposed that science is public knowledge and thus includes mathematics. In any case, mathematics shares much in common with many fields in the physical sciences, notably the <span class="Apple-style-span"><span class="Apple-style-span">exploration of the logical consequences</span></span> of assumptions. <span class="Apple-style-span"><span class="Apple-style-span">Intuition</span></span> and <span class="Apple-style-span"><span class="Apple-style-span">experimentation</span></span> also play a role in the formulation of <span class="Apple-style-span"><span class="Apple-style-span">conjectures</span></span> in both mathematics and the (other) sciences. <span class="Apple-style-span"><span class="Apple-style-span">Experimental mathematics</span></span> continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics, weakening the objection that mathematics does not use the <span class="Apple-style-span"><span class="Apple-style-span">scientific method</span></span></div><div style="color: black; font-family: inherit; margin: 0.4em 0px 0.5em;"><span class="Apple-style-span" style="line-height: 1.5em;">The opinions of mathematicians on this matter are varied. Many mathematicians</span><span class="Apple-style-span" style="font-size: x-small;"><span class="Apple-style-span" style="line-height: 13px; white-space: nowrap;"> </span></span><span class="Apple-style-span" style="line-height: 1.5em;">feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven </span><span class="Apple-style-span" style="line-height: 1.5em;"><span class="Apple-style-span">liberal arts</span></span><span class="Apple-style-span" style="line-height: 1.5em;">; others</span><span class="Apple-style-span" style="font-size: x-small;"><span class="Apple-style-span" style="line-height: 13px; white-space: nowrap;"> </span></span><span class="Apple-style-span" style="line-height: 1.5em;">feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and </span><span class="Apple-style-span" style="line-height: 1.5em;"><span class="Apple-style-span">engineering</span></span><span class="Apple-style-span" style="line-height: 1.5em;"> has driven much development in mathematics. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is </span><span style="line-height: 1.5em;">created</span><span class="Apple-style-span" style="line-height: 1.5em;"> (as in art) or </span><span style="line-height: 1.5em;">discovered</span><span class="Apple-style-span" style="line-height: 1.5em;"> (as in science). It is common to see </span><span class="Apple-style-span" style="line-height: 1.5em;"><span class="Apple-style-span">universities</span></span><span class="Apple-style-span" style="line-height: 1.5em;"> divided into sections that include a division of </span><span style="line-height: 1.5em;">Science and Mathematics</span><span class="Apple-style-span" style="line-height: 1.5em;">, indicating that the fields are seen as being allied but that they do not coincide. In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the </span><span class="Apple-style-span" style="line-height: 1.5em;"><span class="Apple-style-span">philosophy of mathematics</span></span><span class="Apple-style-span" style="line-height: 1.5em;">.</span></div><div style="color: black; font-family: inherit; margin: 0.4em 0px 0.5em;"><span class="Apple-style-span" style="line-height: 1.5em;"><br />
</span></div><div style="color: black; font-family: inherit; margin: 0.4em 0px 0.5em;"><span class="Apple-style-span" style="line-height: 1.5em;">by Nadhirah</span></div>4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com0tag:blogger.com,1999:blog-8948305359118541243.post-22949207244480805912011-07-24T17:18:00.002+08:002011-07-26T15:59:42.291+08:00Statistics III<div style="font-family: Times,"Times New Roman",serif;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;">1.<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;"> </span>Data can be grouped into classes such as 1-5, 6-10, 11-15, ... </span></div><div class="MsoListParagraphCxSpMiddle" style="color: black; font-family: inherit; line-height: 150%; text-indent: -0.25in;"><span style="font-size: small;"> Each class is known as a class interval<br />
<br />
2.<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;"> </span>Range of the class interval is the difference between the smallest and the largest values.</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"> Range of class interval = Upper limit – Lower limit<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;"> <br />
</span></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;">3. </span>For a class interval such as 1-5, the number 1 is called the lower limit and the number 5 is called the upper limit. 0.5 is the lower boundary and 5.5 is the upper boundary.<br />
</span></div><div class="MsoListParagraphCxSpMiddle" style="color: black; font-family: inherit; line-height: 150%; text-indent: -0.25in;"><span style="font-size: small;">4.<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;"> 4. </span>Class size or size of the class interval = Upper boundary – Lower boundary<br />
</span></div><div class="MsoListParagraphCxSpMiddle" style="color: black; font-family: inherit; line-height: 150%; text-indent: -0.25in;"><span style="font-size: small;">5.<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;"> </span>5. Modal class is the class with highest frequency.<br />
</span></div><div class="MsoListParagraphCxSpLast" style="color: black; font-family: inherit; line-height: 150%; text-indent: -0.25in;"><span style="font-size: small;">6.<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;"> 6. </span>Midpoint of a class = <u>Lower limit + Upper limit</u><br />
</span><span style="font-size: small;"> 2</span></div><div class="MsoListParagraphCxSpLast" style="color: black; font-family: inherit; line-height: 150%; text-indent: -0.25in;"><span style="font-size: small;"> </span></div><div class="MsoListParagraph" style="color: black; font-family: inherit; line-height: 150%; text-indent: -0.25in;"><span style="font-size: small;">1.<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;"> 7. </span>Mean of a grouped data, </span><span style="font-size: small;"> <img alt="" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADMAAAA0CAIAAACoxpfFAAAA40lEQVRYhe2W0Q6EIAwE+f+fxhcDxkDZrU0xZOdJkbqjFu5K/Stlt8AUmfHIjEdmPDLjOc6smMxmZpi1SHuwnaaaIXkOoV7rKwOzI82Q1gHjX+WzU9TMhy03HFm+ywAz49GNpbq+7fBe+Nd0XHWaUfhaEHxtSfvZc2RWG2M2++jPyNfI8NjqBJ9ZAjLjOc7MXgFuAswSkBnPuWZ4R1ds6w8zq/AfijaSZwaGgUJ9/jclInibmZ1NddhdEifWDdhL4/lBSuts9RmfusGM3c+SzMDfgP1rMxCZ8ciMR2Y8MuP5r9kFR+5uHDsIQAoAAAAASUVORK5CYII=" /> where f is its frequency and x is the midpoint of a class. ∑ means ‘sum of’.</span></div><div class="MsoListParagraph" style="color: black; font-family: inherit; line-height: 150%; text-indent: -0.25in;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"> <span style="line-height: 115%;">8. Histogram is a graphical representation of a frequency distribution.</span><img alt="" src="data:image/png;base64,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" /></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"> </span></div><div class="MsoListParagraph" style="color: black; font-family: inherit; line-height: 150%; text-indent: -0.25in;"><span style="font-size: small;">1.<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;"> </span>Horizontal axis can be labeled with midpoints or class boundaries or class intervals.<br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"> <span style="line-height: 115%;">9. Frequency polygon is another graphical representation of a frequency distribution.</span><img alt="" src="data:image/png;base64,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" 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" /></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small; line-height: 115%;">10. The cumulative frequency of a data or a class interval in a frequency table is the sum of its frequency and the total frequency of the entire class interval before it.</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><img alt="" 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" /></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small; line-height: 115%;">11. An ogive or cumulative frequency curve is a graphical representation of a cumulative frequency table.</span><span style="font-size: small;"><img alt="" src="data:image/png;base64,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" /></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div class="MsoListParagraph" style="color: black; font-family: inherit; line-height: 150%; text-indent: -0.25in;"><span style="font-size: small;">1.<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;"> </span>12. For ungrouped data, Range = Largest value – Smallest value</span><br />
<span style="font-size: small;">For example: 2, 4, 1, 8, 10, 7, 13, 5</span><br />
<span style="font-size: small;">Range = 13 – 1 = 12</span><br />
<span style="font-size: small;">For grouped data, Range = Midpoint of the last class interval – Midpoint of the first class interval</span><br />
<span style="font-size: small;">For example:</span></div><table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: medium none; color: black; font-family: inherit;"><tbody>
<tr> <td style="border: 1pt solid windowtext; padding: 0in 5.4pt; width: 79.8pt;" valign="top" width="106"><div class="MsoNormal" style="line-height: 150%; margin-bottom: 0in;"><span style="font-size: small;">Mass (g)</span></div></td> <td style="border: 1pt solid windowtext; padding: 0in 5.4pt; width: 42.6pt;" valign="top" width="57"><div class="MsoNormal" style="line-height: 150%; margin-bottom: 0in;"><span style="font-size: small;"> 4 - 8</span></div></td> <td style="border: 1pt solid windowtext; padding: 0in 5.4pt; width: 40.5pt;" valign="top" width="54"><div class="MsoNormal" style="line-height: 150%; margin-bottom: 0in;"><span style="font-size: small;">9 - 13</span></div></td> <td style="border: 1pt solid windowtext; padding: 0in 5.4pt; width: 0.75in;" valign="top" width="72"><div class="MsoNormal" style="line-height: 150%; margin-bottom: 0in;"><span style="font-size: small;"> 14 - 18</span></div></td> <td style="border: 1pt solid windowtext; padding: 0in 5.4pt; width: 49.5pt;" valign="top" width="66"><div class="MsoNormal" style="line-height: 150%; margin-bottom: 0in;"><span style="font-size: small;">19 - 23</span></div></td> <td style="border: 1pt solid windowtext; padding: 0in 5.4pt; width: 0.75in;" valign="top" width="72"><div class="MsoNormal" style="line-height: 150%; margin-bottom: 0in;"><span style="font-size: small;">24 – 28</span></div></td> </tr>
<tr> <td style="border: 1pt solid windowtext; padding: 0in 5.4pt; width: 79.8pt;" valign="top" width="106"><div class="MsoNormal" style="line-height: 150%; margin-bottom: 0in;"><span style="font-size: small;">Frequency</span></div></td> <td style="border-color: -moz-use-text-color windowtext windowtext -moz-use-text-color; border-style: none solid solid none; border-width: medium 1pt 1pt medium; padding: 0in 5.4pt; width: 42.6pt;" valign="top" width="57"><div class="MsoNormal" style="line-height: 150%; margin-bottom: 0in;"><span style="font-size: small;"> 3</span></div></td> <td style="border-color: -moz-use-text-color windowtext windowtext -moz-use-text-color; border-style: none solid solid none; border-width: medium 1pt 1pt medium; padding: 0in 5.4pt; width: 40.5pt;" valign="top" width="54"><div class="MsoNormal" style="line-height: 150%; margin-bottom: 0in;"><span style="font-size: small;"> 6</span></div></td> <td style="border-color: -moz-use-text-color windowtext windowtext -moz-use-text-color; border-style: none solid solid none; border-width: medium 1pt 1pt medium; padding: 0in 5.4pt; width: 0.75in;" valign="top" width="72"><div class="MsoNormal" style="line-height: 150%; margin-bottom: 0in;"><span style="font-size: small;"> </span><span style="font-size: small;"> </span><span style="font-size: small;"> 12</span></div></td> <td style="border-color: -moz-use-text-color windowtext windowtext -moz-use-text-color; border-style: none solid solid none; border-width: medium 1pt 1pt medium; padding: 0in 5.4pt; width: 49.5pt;" valign="top" width="66"><div class="MsoNormal" style="line-height: 150%; margin-bottom: 0in;"><span style="font-size: small;"> 5</span></div></td> <td style="border-color: -moz-use-text-color windowtext windowtext -moz-use-text-color; border-style: none solid solid none; border-width: medium 1pt 1pt medium; padding: 0in 5.4pt; width: 0.75in;" valign="top" width="72"><div class="MsoNormal" style="line-height: 150%; margin-bottom: 0in;"><span style="font-size: small;"> 2</span></div></td> </tr>
</tbody></table><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: 150%;"><span style="font-size: small;">Range </span><span style="font-size: small;"> = <u>24 + 28</u></span><span style="font-size: small;"> _</span><span style="font-size: small;"> <u>4 + 8</u></span><br />
<span style="font-size: small;"> </span><span style="font-size: small;"> 2</span><span style="font-size: small;"> 2</span><br />
<span style="font-size: small;"> = 26 – 6</span><span style="font-size: small;"> <br />
= 20</span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: 150%;"><span style="font-size: small;">by: Mawar Ayu</span></div>4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com0tag:blogger.com,1999:blog-8948305359118541243.post-21552798346788321802011-07-24T14:08:00.003+08:002011-07-26T16:04:26.400+08:00Maths In Nature<div class="post-body entry-content" style="color: #555544; font-family: tahoma, 'Trebuchet MS', lucida, helvetica, sans-serif; font-size: 13px; line-height: 18px; margin-bottom: 1em; margin-top: 5px;"><div style="color: black; font-family: inherit; text-align: left;"><span style="font-size: small;">"<i>The laws of nature are but the mathematical thoughts of God</i>"</span></div><div style="color: black; font-family: inherit; text-align: left;"><span style="font-size: small;"> - Euclid</span></div><div style="color: black; font-family: inherit; text-align: left;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit; text-align: left;"><span style="font-size: small;">Mathematics is everywhere in this universe. We seldom note it. We enjoy nature and are not interested in going deep about what mathematical idea is in it. Here are a very few properties of mathematics that are depicted in nature.</span></div><div style="color: black; font-family: inherit; text-align: center;"><br />
<span style="font-size: small;"><u><b>SYMMETRY</b></u></span></div><div style="color: black; font-family: inherit; text-align: center;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;">Symmetry is everywhere you look in nature .</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><i>Symmetry is when a figure has two sides that are mirror images of one another. It would then be possible to draw a line through a picture of the object and along either side the image would look exactly the same. This line would be called a line of symmetry. </i></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><b>There are two kinds of symmetry.</b></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;">One is <b>bilateral symmetry</b> in which an object has two sides that are mirror images of each other.</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><i>The human body would be an excellent example of a living being that has bilateral symmetry.</i></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><a href="http://1.bp.blogspot.com/_NYaJQllvyA0/THv0gWgyUsI/AAAAAAAABvs/ibo4RnEmAtA/s1600/bilateral_symmetry.jpg" imageanchor="1" style="background-color: transparent; font-weight: bold; margin-left: 1em; margin-right: 1em; text-decoration: none;"><img border="0" height="320" src="http://1.bp.blogspot.com/_NYaJQllvyA0/THv0gWgyUsI/AAAAAAAABvs/ibo4RnEmAtA/s320/bilateral_symmetry.jpg" style="border-width: 0px; display: block;" width="210" /></a></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
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</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><b><i>Few more pictures in nature showing bilateral symmetry</i></b>.</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
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<span style="font-size: small;"><a href="http://3.bp.blogspot.com/_NYaJQllvyA0/THv0sQicKRI/AAAAAAAABv0/jBXUEed2FTY/s1600/butterfly.jpg" imageanchor="1" style="background-color: transparent; font-weight: bold; margin-left: 1em; margin-right: 1em; text-decoration: none;"><img border="0" src="http://3.bp.blogspot.com/_NYaJQllvyA0/THv0sQicKRI/AAAAAAAABv0/jBXUEed2FTY/s320/butterfly.jpg" style="border-width: 0px; display: block;" /></a></span><br />
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<span style="font-size: small;"><a href="http://4.bp.blogspot.com/_NYaJQllvyA0/THv2dj6o07I/AAAAAAAABwI/HeTOY1WMAGw/s1600/untitled.bmp" imageanchor="1" style="background-color: transparent; font-weight: bold; margin-left: 1em; margin-right: 1em; text-decoration: none;"><img border="0" ox="true" src="http://4.bp.blogspot.com/_NYaJQllvyA0/THv2dj6o07I/AAAAAAAABwI/HeTOY1WMAGw/s320/untitled.bmp" style="border-width: 0px; display: block;" /></a></span><br />
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<span style="font-size: small;"><a href="http://3.bp.blogspot.com/_NYaJQllvyA0/THv2PB6ZBKI/AAAAAAAABwA/2fFRfyX2tdc/s1600/220px-Leaf_1_web.jpg" imageanchor="1" style="background-color: transparent; font-weight: bold; margin-left: 1em; margin-right: 1em; text-decoration: none;"><img border="0" ox="true" src="http://3.bp.blogspot.com/_NYaJQllvyA0/THv2PB6ZBKI/AAAAAAAABwA/2fFRfyX2tdc/s320/220px-Leaf_1_web.jpg" style="border-width: 0px; display: block;" /></a></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;">The other kind of symmetry is <b>radial symmetry.</b> This is where there is a center point and numerous lines of symmetry could be drawn.</span></div><div class="separator" style="clear: both; color: black; font-family: inherit; text-align: center;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><i>The most obvious geometric example would be a circle.</i></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><a href="http://3.bp.blogspot.com/_NYaJQllvyA0/THv4nxuasNI/AAAAAAAABwU/gOXzhr4F1ow/s1600/circle%20lines%20of%20symmetry.jpg" imageanchor="1" style="background-color: transparent; font-weight: bold; margin-left: 1em; margin-right: 1em; text-decoration: none;"><img border="0" ox="true" src="http://3.bp.blogspot.com/_NYaJQllvyA0/THv4nxuasNI/AAAAAAAABwU/gOXzhr4F1ow/s320/circle%2520lines%2520of%2520symmetry.jpg" style="border-width: 0px; display: block;" /></a></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><b><i>Few more pictures in nature showing radial symmetry.</i></b></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
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</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><a href="http://3.bp.blogspot.com/_NYaJQllvyA0/THv6D4ty0AI/AAAAAAAABwg/Nd8HKJEcoVI/s1600/stock-photo-isolated-half-cut-orange-with-perfect-geometrical-shape-11920849.jpg" imageanchor="1" style="background-color: transparent; font-weight: bold; margin-left: 1em; margin-right: 1em; text-decoration: none;"><img border="0" ox="true" src="http://3.bp.blogspot.com/_NYaJQllvyA0/THv6D4ty0AI/AAAAAAAABwg/Nd8HKJEcoVI/s320/stock-photo-isolated-half-cut-orange-with-perfect-geometrical-shape-11920849.jpg" style="border-width: 0px; display: block;" /></a></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
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</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><a href="http://3.bp.blogspot.com/_NYaJQllvyA0/THv6Q_Cyh5I/AAAAAAAABww/DCth0MQHNUA/s1600/Starfish-1.jpg" imageanchor="1" style="background-color: transparent; clear: left; float: left; font-weight: bold; margin-bottom: 1em; margin-right: 1em; text-decoration: none;"><img border="0" ox="true" src="http://3.bp.blogspot.com/_NYaJQllvyA0/THv6Q_Cyh5I/AAAAAAAABww/DCth0MQHNUA/s320/Starfish-1.jpg" style="border-width: 0px; display: block;" /></a></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><a href="http://1.bp.blogspot.com/_NYaJQllvyA0/THv6MkCZfCI/AAAAAAAABwo/6mCov5do804/s1600/1466346615_7b0de441fd.jpg" imageanchor="1" style="background-color: transparent; clear: left; float: left; font-weight: bold; margin-bottom: 1em; margin-right: 1em; text-decoration: none;"><img border="0" ox="true" src="http://1.bp.blogspot.com/_NYaJQllvyA0/THv6MkCZfCI/AAAAAAAABwo/6mCov5do804/s320/1466346615_7b0de441fd.jpg" style="border-width: 0px; display: block;" /></a></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
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</span></div><div style="color: black; font-family: inherit; text-align: center;"><span style="font-size: small;"><b>SHAPES</b></span></div><div style="color: black; font-family: inherit; text-align: center;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;">Geometry is the branch of mathematics that describes shapes.</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><b>Sphere:</b></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><i>A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball.</i></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;">The shape of the Earth is very close to that of an oblate spheroid, a sphere flattened along the axis from pole to pole such that there is a bulge around the equator.</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
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</span></div><div class="separator" style="clear: both; color: black; font-family: inherit; text-align: center;"><span style="font-size: small;"><a href="http://3.bp.blogspot.com/_NYaJQllvyA0/THwCOBhg-7I/AAAAAAAABw8/jFEzm5mp838/s1600/blue-marble-1-trim.jpg" imageanchor="1" style="background-color: transparent; clear: left; float: left; font-weight: bold; margin-bottom: 1em; margin-right: 1em; text-decoration: none;"><img border="0" ox="true" src="http://3.bp.blogspot.com/_NYaJQllvyA0/THwCOBhg-7I/AAAAAAAABw8/jFEzm5mp838/s320/blue-marble-1-trim.jpg" style="border-width: 0px; display: block;" /></a></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
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</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><b>Hexagons:</b></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><i>Hexagons are six-sided polygons, closed, 2-dimensional, many-sided figures with straight edges.</i></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
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</div><div style="color: black; font-family: inherit;"><span style="font-size: small;">For a beehive, close packing is important to maximise the use of space. Hexagons fit most closely together without any gaps; so hexagonal wax cells are what bees create to store their eggs and larvae.</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
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</span></div><div class="separator" style="clear: both; color: black; font-family: inherit; text-align: center;"><span style="font-size: small;"><a href="http://1.bp.blogspot.com/_NYaJQllvyA0/THwCaTq504I/AAAAAAAABxE/6DvvdUw-f1w/s1600/4.jpg" imageanchor="1" style="background-color: transparent; clear: left; float: left; font-weight: bold; margin-bottom: 1em; margin-right: 1em; text-decoration: none;"><img border="0" ox="true" src="http://1.bp.blogspot.com/_NYaJQllvyA0/THwCaTq504I/AAAAAAAABxE/6DvvdUw-f1w/s320/4.jpg" style="border-width: 0px; display: block;" /></a></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
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</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><b>Cones:</b></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><i>A cone is a three-dimensional geometric shape that tapers smoothly from a flat, usually circular base to a point called the apex or vertex.</i></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;">Volcanoes form cones, the steepness and height of which depends on the runniness (viscosity) of the lava. Fast, runny lava forms flatter cones; thick, viscous lava forms steep-sided cones.</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div class="separator" style="clear: both; color: black; font-family: inherit; text-align: center;"><span style="font-size: small;"><a href="http://3.bp.blogspot.com/_NYaJQllvyA0/THwFARUnV1I/AAAAAAAABxQ/-rvRd39K4Ro/s1600/Amazing-Volcanoes-Screensaver_hfn2.jpg" imageanchor="1" style="background-color: transparent; clear: left; float: left; font-weight: bold; margin-bottom: 1em; margin-right: 1em; text-decoration: none;"><img border="0" ox="true" src="http://3.bp.blogspot.com/_NYaJQllvyA0/THwFARUnV1I/AAAAAAAABxQ/-rvRd39K4Ro/s320/Amazing-Volcanoes-Screensaver_hfn2.jpg" style="border-width: 0px; display: block;" /></a></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
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</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><b>Few more cones in nature:</b></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
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</span></div><div class="separator" style="clear: both; color: black; font-family: inherit; text-align: center;"><span style="font-size: small;"><a href="http://4.bp.blogspot.com/_NYaJQllvyA0/THwFUCRlVAI/AAAAAAAABxY/9PoAqPinjZo/s1600/chocolate_hills.jpg" imageanchor="1" style="background-color: transparent; clear: left; float: left; font-weight: bold; margin-bottom: 1em; margin-right: 1em; text-decoration: none;"><img border="0" ox="true" src="http://4.bp.blogspot.com/_NYaJQllvyA0/THwFUCRlVAI/AAAAAAAABxY/9PoAqPinjZo/s320/chocolate_hills.jpg" style="border-width: 0px; display: block;" /></a></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
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</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><b>Parallel lines:</b></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><i>In mathematics, parallel lines stretch to infinity, neither converging nor diverging.</i></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;">These parallel dunes in the Australian desert aren't perfect - the physical world rarely is.</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
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</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><b>Fibonacci spiral:</b></span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><i>If you construct a series of squares with lengths equal to the Fibonacci numbers (1,1,2,3,5, etc) and trace a line through the diagonals of each square, it forms a Fibonacci spiral</i>.</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;">Many examples of the Fibonacci spiral can be seen in nature, including in the chambers of a nautilus shell.</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
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</span></div><div style="color: black; font-family: inherit;"><span style="font-size: small;">by Iqffa Nadia</span></div><div style="clear: both;"></div></div><div class="post-footer" style="color: #555544; font-family: tahoma, 'Trebuchet MS', lucida, helvetica, sans-serif; font-size: 13px; font-style: italic;"></div>4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com0tag:blogger.com,1999:blog-8948305359118541243.post-85893335389024309772011-07-23T21:01:00.001+08:002011-07-26T16:06:33.771+08:00Probability<div class="MsoNormal" style="color: black; font-family: inherit;"><span style="font-size: small;"><b>Probability</b> is a way of expressing knowledge or belief that an <a href="http://en.wikipedia.org/wiki/Event_%28probability_theory%29" title="Event (probability theory)">event</a> will occur or has occurred. The concept has an exact <a href="http://en.wikipedia.org/wiki/Mathematics" title="Mathematics">mathematical</a> meaning in <a href="http://en.wikipedia.org/wiki/Probability_theory" title="Probability theory">probability theory</a>, which is used extensively in such <a href="http://en.wikipedia.org/wiki/Areas_of_study" title="Areas of study">areas of study</a> as <a href="http://en.wikipedia.org/wiki/Mathematics" title="Mathematics">mathematics</a>, <a href="http://en.wikipedia.org/wiki/Statistics" title="Statistics">statistics</a>, <a href="http://en.wikipedia.org/wiki/Finance" title="Finance">finance</a>, <a href="http://en.wikipedia.org/wiki/Gambling" title="Gambling">gambling</a>, <a href="http://en.wikipedia.org/wiki/Science" title="Science">science</a>, <a href="http://en.wikipedia.org/wiki/Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a>/<a href="http://en.wikipedia.org/wiki/Machine_learning" title="Machine learning">machine learning</a> and <a href="http://en.wikipedia.org/wiki/Philosophy" title="Philosophy">philosophy</a> to draw conclusions about the likelihood of potential events and the underlying mechanics of <a href="http://en.wikipedia.org/wiki/Complex_systems" title="Complex systems">complex systems</a>.</span></div><div class="MsoNormal" style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;"><b>SAMPLE SPACE <o:p></o:p></b></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;">An experiment is a process or an operation with an outcomes.<o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;">T<span style="font-size: small;">he set of all possible outcomes of an experiment is called the <b>sample space.</b> It usually denoted by S.</span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: 18pt; margin-bottom: 0in;"><span style="font-size: small;"><br />
</span><br />
<span class="Apple-style-span" style="font-size: small; line-height: normal;">When toss the coin, we can get only 2 results:</span></div><ul style="color: black; font-family: inherit;"><li><span class="Apple-style-span" style="font-size: small;"> Head</span></li>
<li><span class="Apple-style-span" style="font-size: small;"><span class="Apple-style-span"> </span></span><span class="Apple-style-span" style="font-size: small;">Tail </span></li>
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</span></div><div class="MsoListParagraphCxSpLast" style="color: black; font-family: inherit; line-height: normal;"><span style="font-size: small;"><o:p></o:p></span><span style="font-size: small;">Example 1:</span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: 13.3pt; margin-bottom: 0in;"><span style="font-size: small;">En. Adam has a fruit stall that sells bananas, apples,watermelons, papayas and durians. Students of class 4KP are asked to select their favorite fruit from the fruits at En. Adam’s stall.<o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: 13.3pt; margin-bottom: 0in;"><span style="font-size: small; letter-spacing: 4.2pt;">S</span><span style="font-size: small;"> = { banana, apple, watermelon, papaya, durian}<o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin: 0in 0in 0in 0.25in;"><span style="font-size: small;"><br />
</span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin: 7.5pt 7.5pt 15pt 0in;"><span style="font-size: small;">Example 2:</span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin: 7.5pt 7.5pt 15pt 0in;"><span class="Apple-style-span" style="font-size: small;">A month is randomly selected from a year. Describe the sample space of this experiment by using set notation.</span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin: 7.5pt 7.5pt 15pt 0in;"><span style="font-size: small;">S= { January, February, March, April, May, June, July, August, September, October, November, December</span><span style="font-size: small;">}</span><span style="font-size: small;"><o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal;"><span style="font-size: small;"><b>EVENT <o:p></o:p></b></span></div><ul style="color: black; font-family: inherit;"><li><span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-size-adjust: none; font-size: small; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;"><span class="Apple-style-span"><span class="Apple-style-span"> </span></span></span><span style="font-size: small;">Is a subset of the sample space.</span></li>
<li><span style="font-size: small;"><span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;"> </span></span><span style="font-size: small;">Is an outcome or a set of outcomes that satisfies certain condition.</span></li>
<li><span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-size-adjust: none; font-size: small; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;"><span class="Apple-style-span"><span class="Apple-style-span"> </span><span class="Apple-style-span"> </span></span></span><span style="font-size: small;">Denoted by a capital letter.</span></li>
</ul><div style="color: black; font-family: inherit;"><span style="font-size: small;"><br />
</span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;">Example 1: <o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: 14.9pt;"><span style="font-size: small;">A box contains five cards written with 1,2,3,4 and 5 respectively. A card is picked randomly from the box.<o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small; letter-spacing: 5.4pt;">S</span><span style="font-size: small;"> = {1, 2, 3, 4, 5}. <o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;">If we define J as ‘the card with J as ‘the card with an even number’ <span style="letter-spacing: 4.3pt;">,</span>the outcome<span style="letter-spacing: 4.3pt;"> </span>of J in set notation will be J = { 2, 4 }.<o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;">J is known as an event of the experiment.<o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;">The number of outcome of an event n(P)=2<o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;"><br />
</span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;">Example 2: <o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: 13.3pt; margin-bottom: 0in; text-align: justify;"><span style="font-size: small;">A letter is randomly selected from the word ‘COMPUTER’. Determine the number of possible outcomes of the event that the selected letter is <o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;">i. A vowel <o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;">ii. A consonant <o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;"><br />
</span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;">Solution :<o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;">Let A = event that the selected letter is vowel = {O, U, E} <o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;">Therefore n (A) = 3 <o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;"><br />
</span></div><div class="MsoNormal" style="color: black; font-family: inherit;"><span style="font-size: small; line-height: 115%;">Let B = event that the selected letter is consonant = {C, M</span><span style="font-size: small; line-height: 115%;"> </span><span style="font-size: small; line-height: 115%;">, P, T, R} Therefore n (B) = 5<o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;"><b><br />
</b></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;"><b>PROBABILITY OF AN EVENT <o:p></o:p></b></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;"><br />
</span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;">P(E) = number of outcomes of the event <o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;"> _________________________________<o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;"> number of outcomes of the sample space <o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;">P(E) = n (E) <o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;"> = n (S) <o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;"><br />
</span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;">P(E) = 0 means that it is impossible for the event to happen. <o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;">P(E) =1 means that the event is certain to happen. <o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;">The closer the probability of a given event is to 1, the more likely it is to happen.<o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;"><br />
</span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;">Example <o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; margin-bottom: 0in; text-align: justify;"><span style="font-size: small;">A bag contains 3 red balls and 4 white ones. If Rashid puts his hand in the bag and picks a ball, what is the probability that the ball he picked is white? <o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;">Solution: <o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;">S = {R1,R2,R3,W1,W2,W3,W4} <o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;">n(S)= 7 <o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;"><br />
</span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;">Let E is the event of drawing a white ball <o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;">E = {W1,W2,W3,W4} <o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal; margin-bottom: 0in;"><span style="font-size: small;">n(E)=4 <o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; line-height: normal;"><span style="font-size: small;">Therefore, the probability of drawing a white ball is 4/7<o:p></o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; page-break-after: avoid;"><span style="font-size: small;"><o:p><br />
</o:p></span></div><div class="MsoNormal" style="color: black; font-family: inherit; page-break-after: avoid;"><span style="font-size: small;"><o:p>by Jannathul </o:p></span></div>4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com0tag:blogger.com,1999:blog-8948305359118541243.post-81130856407808843632011-07-23T18:24:00.002+08:002011-07-26T16:12:37.570+08:00Pythagoras<div class="separator" style="clear: both; text-align: left;"><span class="Apple-style-span" style="font-family: inherit;">As you all may know, Pythagoras is one of the most recognized mathematicians in the world. He came up with the Pythagoras' Theorem, a formula we learnt in Form 2 and is still being used now. </span></div><div class="separator" style="clear: both; text-align: left;"><span class="Apple-style-span" style="font-family: inherit;"><br />
</span></div><div class="separator" style="clear: both; text-align: center;"><a href="http://upload.wikimedia.org/wikipedia/commons/1/1a/Kapitolinischer_Pythagoras_adjusted.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span class="Apple-style-span" style="font-family: inherit;"><img border="0" height="320" src="http://upload.wikimedia.org/wikipedia/commons/1/1a/Kapitolinischer_Pythagoras_adjusted.jpg" width="239" /></span></a></div><div class="separator" style="clear: both; text-align: center;"><span class="Apple-style-span" style="font-family: inherit; line-height: 16px;">Pythagoras (Πυθαγόρας)</span></div><div class="separator" style="clear: both; text-align: center;"><span class="Apple-style-span" style="font-family: inherit; line-height: 16px;"><br />
</span></div><div class="separator" style="clear: both; text-align: left;"><span class="Apple-style-span" style="line-height: 16px;"><span class="Apple-style-span" style="font-family: inherit;">By Seri</span></span></div>4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com0tag:blogger.com,1999:blog-8948305359118541243.post-8228821671416553912011-07-23T18:04:00.001+08:002011-07-23T18:04:50.366+08:00Female mathematicians<span class="Apple-style-span" style="line-height: 19px;"><span class="Apple-style-span" style="font-family: inherit;"></span></span><br />
<div style="margin-bottom: .0001pt; margin: 0in;"><span class="Apple-style-span" style="font-family: inherit;"><span class="apple-style-span"><span style="color: black;">The majority of mathematicians are male, although more women have entered mathematics since World War II.</span></span><span class="apple-converted-space"><span style="color: black;"> </span></span><span class="apple-style-span"><span style="color: black;">Women are still a small minority of notables in mathematics.</span></span><span style="color: black;"><o:p></o:p></span></span></div><span class="Apple-style-span" style="font-family: inherit;"> </span><br />
<div style="margin-bottom: .0001pt; margin: 0in;"><span class="Apple-style-span" style="font-family: inherit;"><br />
</span></div><div style="margin-bottom: .0001pt; margin: 0in;"><span class="Apple-style-span" style="font-family: inherit;"><span class="apple-style-span"><span style="color: black;"><span class="Apple-style-span" style="font-size: large;">Cecilia Krieger</span></span></span><span style="color: black;"><o:p></o:p></span></span></div><div class="separator" style="clear: both; text-align: center;"><span class="Apple-style-span" style="font-family: inherit;"><a href="http://2.bp.blogspot.com/-OAJdWOJVCik/Tiqby4BzTII/AAAAAAAAAAs/Jn2gsSNWkBY/s1600/images.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-OAJdWOJVCik/Tiqby4BzTII/AAAAAAAAAAs/Jn2gsSNWkBY/s1600/images.jpg" /></a></span></div><div style="margin-bottom: .0001pt; margin: 0in;"><span class="Apple-style-span" style="font-family: inherit;"><span class="apple-style-span"><span style="color: black;"><br />
</span></span></span></div><div style="margin-bottom: .0001pt; margin: 0in;"><span class="Apple-style-span" style="font-family: inherit;"><span class="apple-style-span"><span style="color: black;">Cypra Cecilia Krieger-Dunaij (April 9, 1894 - August 17, 1974) was an Austro-Hungarian</span></span><span class="apple-converted-space"><span style="color: black;"> </span></span><span class="apple-style-span"><span style="color: black;">(Galician)-born mathematician of Jewish ancestry who lived and worked in <st1:country-region w:st="on"><st1:place w:st="on">Canada</st1:place></st1:country-region>. She was the 3rd person (& 1st woman) to earn a Ph.D in mathematics from a university in <st1:country-region w:st="on">Canada</st1:country-region>, in 1930 as well as the 3rd woman to have been awarded a doctorate in any discipline in <st1:country-region w:st="on"><st1:place w:st="on">Canada</st1:place></st1:country-region>. Krieger is well-known for having translated two works of Sierpinski in general topology. The Krieger-Nelson Prize,</span></span><span class="apple-converted-space"><span style="color: black;"> </span></span><span class="apple-style-span"><span style="color: black;">awarded annually by the Canadian Mathematical Society since 1995 for outstanding research by a female mathematician, is named in honour of Krieger and Evelyn Nelson.</span></span><span style="color: black;"><o:p></o:p></span></span></div><div style="margin-bottom: .0001pt; margin: 0in;"><span class="Apple-style-span" style="font-family: inherit;"><span class="apple-style-span"><span style="color: black;">Krieger's thesis for her Ph.D, under the supervision of W.J. Webber, was entitled "On the summability of trigonometric series with localized parameters-on Fourier constants and convergence factors of double Fourier series."</span></span><span style="color: black;"><o:p></o:p></span></span></div><div style="margin-bottom: .0001pt; margin: 0in;"><span class="Apple-style-span" style="font-family: inherit;"><br />
</span></div><div style="margin-bottom: .0001pt; margin: 0in;"><span class="Apple-style-span" style="font-family: inherit;"><span class="apple-style-span"><span style="color: black;"><span class="Apple-style-span" style="font-size: large;">Emmy Noether</span></span></span><span style="color: black;"><o:p></o:p></span></span></div><div class="separator" style="clear: both; text-align: center;"><span class="Apple-style-span" style="font-family: inherit;"><a href="http://2.bp.blogspot.com/-Ij-lQ8Q9rs0/TiqbzEqOZEI/AAAAAAAAAAw/TYEXqC6fxQI/s1600/images+%25281%2529.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-Ij-lQ8Q9rs0/TiqbzEqOZEI/AAAAAAAAAAw/TYEXqC6fxQI/s1600/images+%25281%2529.jpg" /></a></span></div><div style="margin-bottom: .0001pt; margin: 0in;"><span class="Apple-style-span" style="font-family: inherit;"><span class="apple-style-span"><span style="color: black;"><br />
</span></span></span></div><div style="margin-bottom: .0001pt; margin: 0in;"><span class="Apple-style-span" style="font-family: inherit;"><span class="apple-style-span"><span style="color: black;">Amalie Emmy Noether, (23 March 1882 – 14 April 1935) was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of rings, fields and algebras. </span></span><span style="color: black;"><o:p></o:p></span></span></div><div style="margin-bottom: .0001pt; margin: 0in;"><span class="Apple-style-span" style="font-family: inherit;"><span class="apple-style-span"><span style="color: black;">After completing her dissertation in 1907 under the supervision of Paul Gordan, she worked at the Mathematical Institute of Erlangen without pay for 7 years. In 1915, she was invited by David Hilbert and Felix Klein to join the mathematics department at the <st1:place w:st="on"><st1:placetype w:st="on">University</st1:placetype> of <st1:placename w:st="on">Göttingen</st1:placename></st1:place>, a world-renowned center of mathematical research. The philosophical faculty objected, however, and she spent four years lecturing under Hilbert's name.</span></span><span style="color: black;"><o:p></o:p></span></span></div><div style="margin-bottom: .0001pt; margin: 0in;"><span class="Apple-style-span" style="font-family: inherit;"><span class="apple-style-span"><span style="color: black;">Her work was the foundation for the second volume of his influential 1931 textbook, <i>Moderne Algebra</i>. By the time of her plenary address at the 1932 International Congress of Mathematicians in Zürich, her algebraic acumen was recognized around the world. The following year, <st1:country-region w:st="on">Germany</st1:country-region>'s Nazi government dismissed Jews from university positions, and Noether moved to the <st1:country-region w:st="on">United States</st1:country-region> to take up a position at <st1:placename w:st="on">Bryn</st1:placename> <st1:placename w:st="on">Mawr</st1:placename> <st1:placetype w:st="on">College</st1:placetype> in <st1:state w:st="on"><st1:place w:st="on">Pennsylvania</st1:place></st1:state>. In 1935 she underwent surgery for an ovarian cyst and, despite signs of a recovery, died 4 days later at the age of 53.</span></span><span style="color: black;"><o:p></o:p></span></span></div><div style="margin-bottom: .0001pt; margin: 0in;"><span class="Apple-style-span" style="font-family: inherit;"><span class="apple-style-span"><span style="color: black;">Noether's mathematical work has been divided into three "</span></span><span class="apple-style-span">epochs". In the first (1908–1919), she made significant contributions to the theories of <span style="color: windowtext; text-decoration: none;">algebraic invariants</span> and <span style="color: windowtext; text-decoration: none;">number fields</span>. Her work on differential invariants in the <span style="color: windowtext; text-decoration: none;">calculus of variations</span>, <i><span style="color: windowtext; text-decoration: none;">Noether's theorem</span></i>, has been called "one of the most important mathematical theorems ever proved in guiding the development<span style="color: black;"> of modern physics". <o:p></o:p></span></span></span></div><div style="margin-bottom: .0001pt; margin: 0in;"><span class="Apple-style-span" style="font-family: inherit;"><span class="apple-style-span"><span style="color: black;">In the 2nd epoch, (1920–1926), she began work that "changed the face of [abstract] algebra". In her classic paper<i>Idealtheorie in Ringbereichen</i> (<i>Theory of Ideals in Ring Domains</i>, 1921) Noether developed the theory </span></span><span class="apple-style-span">of <span style="color: windowtext; text-decoration: none;">ideals</span> in <span style="color: windowtext; text-decoration: none;">commutative rings</span> into a powerful tool with wide-ranging applications. She made elegant use of the <span style="color: windowtext; text-decoration: none;">ascending chain condition</span>, and objects satisfying it are named <i><span style="color: windowtext; text-decoration: none;">Noetherian</span></i> in her honor. <o:p></o:p></span></span></div><div style="margin-bottom: .0001pt; margin: 0in;"><span class="Apple-style-span" style="font-family: inherit;"><span class="apple-style-span">In the 3rd epoch, (1927–1935), she published major works on <span style="color: windowtext; text-decoration: none;">noncommutative algebras</span> and <span style="color: windowtext; text-decoration: none;">hypercomplex numbers</span> and united the <span style="color: windowtext; text-decoration: none;">representation theory</span> of <span style="color: windowtext; text-decoration: none;">groups</span> with the theory of <span style="color: windowtext; text-decoration: none;">modules</span> and ideals. In addition to her own publications, Noether was generous with her ideas and is credited with several lines of research published by other mathematicians, even in fields far removed from her main work, such as <span style="color: windowtext; text-decoration: none;">algebraic topology</span>.</span><o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0in;"><span class="Apple-style-span" style="font-family: inherit;"><br />
</span></div><div style="margin-bottom: .0001pt; margin: 0in;"><span class="Apple-style-span" style="font-family: inherit;"><span class="apple-style-span"><span style="color: black;"><span class="Apple-style-span" style="font-size: large;">Sun-Yung <st1:city w:st="on"><st1:place w:st="on">Alice</st1:place></st1:city> Chang</span></span></span><span style="color: black;"><o:p></o:p></span></span></div><div class="separator" style="clear: both; text-align: center;"><span class="Apple-style-span" style="font-family: inherit;"><a href="http://4.bp.blogspot.com/-S_-GfSUO2ys/TiqbyaaGQmI/AAAAAAAAAAo/W71EYL6dQCc/s1600/download" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://4.bp.blogspot.com/-S_-GfSUO2ys/TiqbyaaGQmI/AAAAAAAAAAo/W71EYL6dQCc/s1600/download" /></a></span></div><span class="Apple-style-span" style="line-height: 19px;">Alice Chang is a <span style="color: windowtext; text-decoration: none;">Chinese American</span> <span style="color: windowtext; text-decoration: none;">mathematician</span> specializing in aspects of <span style="color: windowtext; text-decoration: none;">mathematical analysis</span> ranging from <span style="color: windowtext; text-decoration: none;">harmonic analysis</span> and <span style="color: windowtext; text-decoration: none;">partial differential equations</span> to <span style="color: windowtext; text-decoration: none;">differential geometry</span>. She is a professor of mathematics and chair of the department at <span style="color: windowtext; text-decoration: none;">Princeton University</span>.</span><span class="Apple-style-span" style="line-height: 19px;"><span class="Apple-style-span" style="font-family: inherit;"><div style="margin-bottom: .0001pt; margin: 0in;"><o:p></o:p></div><div style="margin-bottom: .0001pt; margin: 0in;">Chang was born in <span style="color: windowtext; text-decoration: none;">Xian, China</span> in 1948. She received her Bachelor of Science degree in 1970 from <span style="color: windowtext; text-decoration: none;">National Taiwan University</span>, and her doctorate in 1974 from the <span style="color: windowtext; text-decoration: none;">University of California, Berkeley</span>. At <st1:city w:st="on"><st1:place w:st="on">Berkeley</st1:place></st1:city>, Chang wrote her thesis on the study of bounded <span style="color: windowtext; text-decoration: none;">analytic functions</span>.<o:p></o:p></div><div style="margin-bottom: .0001pt; margin: 0in;">Working with her husband Paul Yang and others, they produced “deep contributions” to <span style="color: windowtext; text-decoration: none;">differential equations</span> in relationship to geometry and <span style="color: windowtext; text-decoration: none;">topology</span>. Chang became a full professor at UCLA in 1980. She was a speaker at the <span style="color: windowtext; text-decoration: none;">International Congress of Mathematicians</span>. She also worked to promote the careers of women in mathematics.<o:p></o:p></div><div style="margin-bottom: .0001pt; margin: 0in;">Chang’s research interests include the study of geometric types of <span style="color: windowtext; text-decoration: none;">nonlinear partial differential equations</span> and problems in isospectral geometry. She was awarded the Ruth Lyttle Satter Prize in Mathematics on January 1995 in <span style="color: windowtext; text-decoration: none;">San Francisco</span>. This prize is awarded to women who contributed in mathematics research in the past five years. Sun-Yung served as a vice president of the <span style="background-attachment: initial; background-clip: initial; background-color: initial; background-origin: initial; color: windowtext; text-decoration: none;">American Mathematical Society </span>for 3 years until 1991.</div><div style="margin-bottom: .0001pt; margin: 0in;"><br />
</div><div style="margin-bottom: .0001pt; margin: 0in;">By Eleanor :)</div><div style="margin-bottom: .0001pt; margin: 0in;">PS: I also created the Cbox.</div></span></span>4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com0tag:blogger.com,1999:blog-8948305359118541243.post-67969163185841625202011-07-23T15:20:00.001+08:002011-07-25T18:30:12.625+08:00Mathematics as a career<center style="color: #000066;"><h3><span class="Apple-style-span" style="font-family: Arial, Helvetica, 'Times New Roman'; font-size: small; font-weight: normal;"><i>Those who qualify in mathematics are in the fortunate position of having a wide range of career choices. The abilities</i></span></h3></center><br />
<ul style="color: #000066;"><li><span style="color: #000066; font-family: Arial, Helvetica, 'Times New Roman';"><i>to use logical thought,</i></span></li>
<li><span style="color: #000066; font-family: Arial, Helvetica, 'Times New Roman';"><i>to formulate a problem in a way which allows for computation and decision,</i></span></li>
<li><span style="color: #000066; font-family: Arial, Helvetica, 'Times New Roman';"><i>to make deductions from assumption,</i></span></li>
<li><span style="color: #000066; font-family: Arial, Helvetica, 'Times New Roman';"><i>to use advanced concepts,</i></span></li>
</ul><div style="color: #000066;"><span style="color: #000066; font-family: Arial, Helvetica, 'Times New Roman';"><i>are all enhanced by a mathematics degree course. It is for this reason that mathematician are increasingly in demand. With a mathematics degree, you should be able to turn your hand to finance, statistics, engineering, computers, teaching or accountancy with a success not possible to other graduates. This flexibility is even more important nowadays, with the considerable uncertainty as to which areas will be the best for employment in future years.</i></span></div><div style="color: #000066;"><span style="color: #000066; font-family: Arial, Helvetica, 'Times New Roman';"><i>[The most recent surveys show graduates in mathematicians and computer science at the top of the earning lists six years after graduation.]</i></span></div><div style="color: #000066;"><span style="color: #000066; font-family: Arial, Helvetica, 'Times New Roman';"><i>Computer science has a considerable mathematical component, which is becoming more important as the designers of software are required to prove that the software meets its specification. This kind of rigour is one of the basic techniques of mathematics, and can be learned only through a mathematics course.</i></span></div><div style="color: #000066;"><span style="color: #000066; font-family: Arial, Helvetica, 'Times New Roman';"><i><br />
</i></span></div><div style="color: #000066;"><span style="color: #000066; font-family: Arial, Helvetica, 'Times New Roman';"><i><br />
</i></span></div><div style="color: #000066;"><span style="color: #000066; font-family: Arial, Helvetica, 'Times New Roman';"><i>By Nur Shahirah Hider(shiro)</i></span></div><div style="color: #000066;"><span style="color: #000066; font-family: Arial, Helvetica, 'Times New Roman';"><i><br />
</i></span></div><div style="color: #000066;"><span class="Apple-style-span" style="font-family: Arial, Helvetica, 'Times New Roman';"><i><br />
</i></span></div><div style="color: #000066;"><span style="color: #000066; font-family: Arial, Helvetica, 'Times New Roman';"><i><br />
</i></span></div>4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com1tag:blogger.com,1999:blog-8948305359118541243.post-77142346151806849812011-07-23T15:18:00.004+08:002011-07-24T18:42:28.641+08:00Why study Mathematics???<center style="color: #000066;"><h2><span class="Apple-style-span" style="font-family: inherit; font-size: small; font-weight: normal;"><i>The main reason for studying mathematics to an advanced level is that it is interesting and enjoyable. People like its challenge, its clarity, and the fact that you know when you are right. The solution of a problem has an excitement and a satisfaction. You will find all these aspects in a university degree course.</i></span></h2></center><br />
<div style="color: #000066;"><span style="color: #000066; font-family: inherit;"><i>You should also be aware of the wide importance of Mathematics, and the way in which it is advancing at a spectacular rate. Mathematics is about pattern and structure; it is about logical analysis, deduction, calculation within these patterns and structures. When patterns are found, often in widely different areas of science and technology, the mathematics of these patterns can be used to explain and control natural happenings and situations. Mathematics has a pervasive influence on our everyday lives, and contributes to the wealth of the country.</i></span></div><br />
<br />
<center style="color: #000066;"><h3><span style="color: #000066; font-family: inherit; font-size: small;"><i>The importance of mathematics</i></span></h3></center><br />
<div style="color: #000066;"><span style="color: #000066; font-family: inherit;"><i>The everyday use of arithmetic and the display of information by means of graphs, are an everyday commonplace. These are the elementary aspects of mathematics. Advanced mathematics is widely used, but often in an unseen and unadvertised way.</i></span></div><ul style="color: #000066;"><li><span style="color: #000066; font-family: inherit;"><i>The mathematics of error-correcting codes is applied to CD players and to computers.</i></span></li>
<li><span style="color: #000066; font-family: inherit;"><i>The stunning pictures of far away planets sent by Voyager II could not have had their crispness and quality without such mathematics.</i></span></li>
<li><span style="color: #000066; font-family: inherit;"><i>Voyager's journey to the planets could not have been calculated without the mathematics of differential equations.</i></span></li>
<li><span style="color: #000066; font-family: inherit;"><i>Whenever it is said that advances are made with supercomputers, there has to be a mathematical theory which instructs the computer what is to be done, so allowing it to apply its capacity for speed and accuracy.</i></span></li>
<li><span style="color: #000066; font-family: inherit;"><i>The development of computers was initiated in this country by mathematicians and logicians, who continue to make important contributions to the theory of computer science.</i></span></li>
<li><span style="color: #000066; font-family: inherit;"><i>The next generation of software requires the latest methods from what is called category theory, a theory of mathematical structures which has given new perspectives on the foundations of mathematics and on logic.</i></span></li>
<li><span style="color: #000066; font-family: inherit;"><i>The physical sciences (chemistry, physics, oceanography, astronomy) require mathematics for the development of their theories.</i></span></li>
<li><span style="color: #000066; font-family: inherit;"><i>In ecology, mathematics is used when studying the laws of population change.</i></span></li>
<li><span style="color: #000066; font-family: inherit;"><i>Statistics provides the theory and methodology for the analysis of wide varieties of data.</i></span></li>
<li><span style="color: #000066; font-family: inherit;"><i>Statistics is also essential in medicine, for analysing data on the causes of illness and on the utility of new drugs. .</i></span></li>
<li><span style="color: #000066; font-family: inherit;"><i>Travel by aeroplane would not be possible without the mathematics of airflow and of control systems.</i></span></li>
<li><span style="color: #000066; font-family: inherit;"><i>Body scanners are the expression of subtle mathematics, discovered in the 19th century, which makes it possible to construct an image of the inside of an object from information on a number of single X-ray views of it. Thus mathematics is often involved in matters of life and death.</i></span></li>
</ul><div style="color: #000066;"><span style="color: #000066; font-family: inherit;"><i>These applications have often developed from the study of general ideas for their own sake: numbers, symmetry, area and volume, rate of change, shape, dimension, randomness and many others. Mathematics makes an especial contribution to the study of these ideas, namely the methods of</i></span></div><ul style="color: #000066;"><li><span style="color: #000066; font-family: inherit;"><i>precise definitions;</i></span></li>
<li><span style="color: #000066; font-family: inherit;"><i>careful and rigorous argument; representation of ideas by many methods, including symbols and formulae, pictures and graphics;</i></span></li>
<li><span style="color: #000066; font-family: inherit;"><i>means of calculation;</i></span></li>
<li><span style="color: #000066; font-family: inherit;"><i>and the obtaining of precise solutions to clearly stated problems, or clear statements of the limits of knowledge.</i></span></li>
</ul><div style="color: #000066;"><span style="color: #000066; font-family: inherit;"><i>These features allow mathematics to provide a solid foundation to many aspects of daily life, and to give a comprehension of the complexities inherent in apparently quite simple situations.</i></span></div><div style="color: #000066;"><span style="color: #000066; font-family: inherit;"><i>For these reasons, mathematics and calculation have been associated from earliest times. In modern times, the need to perform rapid mathematical calculations in war time, particularly in ballistics, and in decoding, was a strong stimulus to the development of the electronic computer. The existence of high speed computers has now helped mathematicians to calculate and to make situations visual as never before. Also this calculation has developed from numerical calculation, to symbolic calculation, and currently to calculation with the mathematical structures themselves. This last is very recent, and is likely to lead to a major transformation. These capacities change, not the nature of mathematics, but the power of the mathematican, which increases perhaps a millionfold the possibility to comprehend, to argue, to explore.</i></span></div><div style="color: #000066;"><span style="color: #000066; font-family: inherit;"><i>There is also a reverse interaction.</i></span></div><div style="color: #000066;"><span style="color: #000066; font-family: inherit;"><i>The notion of computing would not have made sense without Mathematics, and it was the analysis of the methods of Mathematics by mathematicians, philosophers, logicians and engineers which led to the concept of a programmable computer</i></span></div><div style="color: #000066;"><span style="color: #000066; font-family: inherit;"><i>. Indeed, two mathematicians, von Neumann in the USA and Turing in the UK, are known as the fathers of the modern computers. Analysis of computing, and attempts to make it as reliable as possible, needs deep Mathematics, and this need is likely to grow. A computer, unless it is programmed, is just a box made of metal, glass, silicon, etc. Programming expresses algorithms in a form suitable for the computer. Mathematics is needed as a language for specification, for determining what is to be done, how and when, and for the verification that the programs and algorithms work correctly. Mathematics is essential for the correct use of computers in most of their applications and the mathematical needs of computing have sparked off many new and exciting questions. Thus computers, while they have, fortunately, done away with the need for humans to carry out routine calculations, have also required from mathematicians a deeper analysis of the process and logic of computation, and its representation in a machine.</i></span></div><div style="color: #000066;"><span style="color: #000066; font-family: inherit;"><i>The imagination of mathematicians is also stirred by its rigorous nature, which forces them to follow through the logic of their ideas. There are many examples of mathematicians producing apparently strange and inapplicable theories, noting simply that this is the way the mathematics seems to go, only to find these vindicated perhaps decades later by surprising applications. A recent example is the theory of knots, which was developed as a part of pure mathematics since 1870. A wonderful advance in 1985 showed how the theory could be applied in physics in relation to quantum theory, and in biology in relation to the way DNA unknots itself before dividing. Similarly, modern notions of chaos andfractals were pioneered by mathematicians in the early years of this century. Now fractals are a practical tool for compressing data on computer discs.</i></span></div><div style="color: #000066;"><span style="color: #000066; font-family: inherit;"><i>The study of mathematics can satisfy a wide range of interests and abilities. It develops the imagination. It trains in clear and logical thought. It is a challenge, with varieties of difficult ideas and unsolved problems, because it deals with the questions arising from complicated structures. Yet it also has a continuing drive to simplification, to finding the right concepts and methods to make difficult things easy, to explaining why a situation must be as it is. In so doing, it develops a range of language and insights, which may then be applied to make a crucial contribution to our understanding and appreciation of the world, and our ability to find and make our way in it.</i></span><br />
<span style="color: #000066; font-family: inherit;"><i><br />
</i></span><br />
<span style="color: #000066; font-family: inherit;"><i>by Atiqah Ramzan</i></span></div>4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com0tag:blogger.com,1999:blog-8948305359118541243.post-57296282695433000622011-07-22T22:48:00.004+08:002011-07-23T18:28:38.848+08:00Standard Form<h1 style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; color: navy; text-align: left;"><span class="Apple-style-span" style="color: black; font-family: inherit; font-size: small; font-weight: normal;">Standard form is a way of writing down very large or very small numbers easily. 10³ = 1000, so 4 × 10³ = 4000 . So 4000 can be written as 4 × 10³ . This idea can be used to write even larger numbers down easily in standard form.</span></h1><div style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="font-family: inherit;">Small numbers can also be written in standard form. However, instead of the <a href="http://www.mathsrevision.net/gcse/pages.php?page=26" style="text-decoration: none;">index</a> being positive (in the above example, the index was 3), it will be negative.</span><br />
<span class="Apple-style-span" style="font-family: inherit;"><br />
</span><br />
<span class="Apple-style-span" style="font-family: inherit;">The rules when writing a number in standard form is that first you write down a number between 1 and 10, then you write × 10(to the power of a number).</span></div><h4 style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; color: navy; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span class="Apple-style-span" style="font-family: inherit;">Example</span></h4><div style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="font-family: inherit;">Write 81 900 000 000 000 in standard form:</span></div><div style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="font-family: inherit;">81 900 000 000 000 = 8.19 × 10<sup>13</sup></span></div><div style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i><span class="Apple-style-span" style="font-family: inherit;">It’s 10<sup>13</sup> because the decimal point has been moved 13 places to the left to get the number to be 8.19</span></i></div><h4 style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; color: navy; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span class="Apple-style-span" style="font-family: inherit;">Example</span></h4><div style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="font-family: inherit;">Write 0.000 001 2 in standard form:</span></div><div style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="font-family: inherit;">0.000 001 2 = 1.2 × 10<sup>-6</sup></span></div><div style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="font-family: inherit;">It’s 10<sup>-6</sup> because the decimal point has been moved 6 places to the right to get the number to be 1.2</span></div><div style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="font-family: inherit;">On a calculator, you usually enter a number in standard form as follows:</span><br />
<span class="Apple-style-span" style="font-family: inherit;">Type in the first number (the one between 1 and 10). Press EXP . Type in the power to which the 10 is risen.</span></div><h3 style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; color: navy;"><span class="Apple-style-span" style="font-family: inherit; font-size: small;">Manipulation in Standard Form</span></h3><div style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="font-family: inherit;">This is best explained with an example:</span></div><h4 style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; color: navy; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span class="Apple-style-span" style="font-family: inherit;">Example</span></h4><div style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="font-family: inherit;">The number p written in standard form is 8 × 10<sup>5</sup></span><br />
<span class="Apple-style-span" style="font-family: inherit;">The number q written in standard form is 5 × 10<sup>-2</sup></span><br />
<span class="Apple-style-span" style="font-family: inherit;">Calculate p × q. Give your answer in standard form.</span></div><div style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="font-family: inherit;">Multiply the two first bits of the numbers together and the two second bits together:</span><br />
<span class="Apple-style-span" style="font-family: inherit;">8 × 5 × 10<sup>5</sup> × 10<sup>-2</sup></span></div><div style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="font-family: inherit;">= 40 × 10<sup>3</sup> (Remember 10<sup>5</sup> × 10<sup>-2</sup> = 10<sup>3</sup>)</span></div><div style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="font-family: inherit;">The question asks for the answer in standard form, but this is not standard form because the first part (the 40) should be a number between 1 and 10.</span></div><div style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="font-family: inherit;">=<u> 4 × 10</u><sup>4</sup></span></div><div style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="font-family: inherit;">Calculate p ÷ q. Give your answer in standard form.</span></div><div style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="font-family: inherit;">This time, divide the two first bits of the standard forms. Divide the two second bits.</span></div><div style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="font-family: inherit;">(8 ÷ 5) × (10<sup>5</sup> ÷ 10<sup>-2</sup>)</span></div><div style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="font-family: inherit;">= <u>1.6 × 10</u><sup>7</sup></span></div><div style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><sup><span class="Apple-style-span" style="font-family: inherit; font-size: small;"><br />
</span></sup></div><div style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><sup><span class="Apple-style-span" style="font-family: inherit; font-size: small;">By : Amira :)</span></sup></div>4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com0tag:blogger.com,1999:blog-8948305359118541243.post-35510327695639651232011-07-21T22:10:00.001+08:002011-07-23T18:28:17.692+08:00Roman Numerals :)<div style="color: #330066; line-height: 22px;"><span class="Apple-style-span" style="font-family: inherit;">Roman numerals are expressed by letters of the alphabet:</span></div><div class="center" style="color: #330066; line-height: 22px; text-align: center;"><table border="1" class="sgmltable" id="A0908425" style="border-bottom-color: rgb(51, 51, 153); border-collapse: collapse; border-left-color: rgb(51, 51, 153); border-right-color: rgb(51, 51, 153); border-top-color: rgb(51, 51, 153);"><tbody>
<tr><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">I=1</span></td></tr>
<tr><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">V=5</span></td></tr>
<tr><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">X=10</span></td></tr>
<tr><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">L=50</span></td></tr>
<tr><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">C=100</span></td></tr>
<tr><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">D=500</span></td></tr>
<tr><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">M=1000</span></td></tr>
</tbody></table></div><div style="color: #330066; line-height: 22px;"><span class="Apple-style-span" style="font-family: inherit;">There are four basic principles for reading and writing Roman numerals:</span></div><ul id="A0908426" style="color: #330066; line-height: 22px;"><li style="padding-bottom: 1em;"><span class="Apple-style-span" style="font-family: inherit;">1. A letter repeats its value that many times (XXX = 30, CC = 200, etc.). A letter can only be repeated three times.</span></li>
<li style="padding-bottom: 1em;"><span class="Apple-style-span" style="font-family: inherit;">2. If one or more letters are placed after another letter of greater value, add that amount.</span><blockquote><span class="Apple-style-span" style="font-family: inherit;">VI = 6 (5 + 1 = 6)</span></blockquote><blockquote><span class="Apple-style-span" style="font-family: inherit;">LXX = 70 (50 + 10 + 10 = 70)</span></blockquote><blockquote><span class="Apple-style-span" style="font-family: inherit;">MCC = 1200 (1000 + 100 + 100 = 1200)</span></blockquote></li>
<span class="Apple-style-span" style="font-family: inherit;"> 3. If a letter is placed before another letter of greater value, subtract that amount.</span><blockquote><span class="Apple-style-span" style="font-family: inherit;">IV = 4 (5 – 1 = 4)</span></blockquote><blockquote><span class="Apple-style-span" style="font-family: inherit;">XC = 90 (100 – 10 = 90)</span></blockquote><blockquote><span class="Apple-style-span" style="font-family: inherit;">CM = 900 (1000 – 100 = 900)</span></blockquote><span class="Apple-style-span" style="font-family: inherit;">Several rules apply for subtracting amounts from Roman numerals:<a href="http://www.blogger.com/post-edit.g?blogID=8948305359118541243&postID=3551032769563965123" name="A0908427" style="color: #330066;"></a></span><ul class="nobull" style="list-style-type: none;"><li style="padding-bottom: 1em;"><span class="Apple-style-span" style="font-family: inherit;">a. Only subtract powers of ten (I, X, or C, but not V or L)</span><blockquote><span class="Apple-style-span" style="font-family: inherit;">For 95, do NOT write VC (100 – 5).</span><br />
<span class="Apple-style-span" style="font-family: inherit;">DO write XCV (XC + V or 90 + 5)</span></blockquote></li>
<li style="padding-bottom: 1em;"><span class="Apple-style-span" style="font-family: inherit;">b. Only subtract one number from another.</span><blockquote><span class="Apple-style-span" style="font-family: inherit;">For 13, do NOT write IIXV (15 – 1 - 1).</span><br />
<span class="Apple-style-span" style="font-family: inherit;">DO write XIII (X + I + I + I or 10 + 3)</span></blockquote></li>
<li style="padding-bottom: 1em;"><span class="Apple-style-span" style="font-family: inherit;">c. Do not subtract a number from one that is more than 10 times greater (that is, you can subtract 1 from 10 [IX] but not 1 from 20—there is no such number as IXX.)</span><blockquote><span class="Apple-style-span" style="font-family: inherit;">For 99, do NOT write IC (C – I or 100 - 1).</span><br />
<span class="Apple-style-span" style="font-family: inherit;">DO write XCIX (XC + IX or 90 + 9)</span></blockquote></li>
</ul><li style="padding-bottom: 1em;"><span class="Apple-style-span" style="font-family: inherit;">4. A bar placed on top of a letter or string of letters increases the numeral's value by 1,000 times.</span><blockquote><span class="Apple-style-span" style="font-family: inherit;">XV = 15, <img alt="(X-bar)" src="http://www.factmonster.com/images/chars/Xbar.gif" style="border-bottom-style: none; border-color: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;" /><img alt="(V-bar)" src="http://www.factmonster.com/images/chars/Vbar.gif" style="border-bottom-style: none; border-color: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;" />= 15,000</span></blockquote></li>
</ul><div class="center" style="color: #330066; line-height: 22px; text-align: center;"><table border="1" class="sgmltable" id="A0769548" style="border-bottom-color: rgb(51, 51, 153); border-collapse: collapse; border-left-color: rgb(51, 51, 153); border-right-color: rgb(51, 51, 153); border-top-color: rgb(51, 51, 153);"><tbody>
<tr><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">One</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">I</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Eleven</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">XI</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Thirty</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">XXX</span></td></tr>
<tr><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Two</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">II</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Twelve</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">XII</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Forty</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">XL</span></td></tr>
<tr><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Three</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">III</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Thirteen</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">XIII</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Fifty</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">L</span></td></tr>
<tr><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Four</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">IV</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Fourteen</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">XIV</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Sixty</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">LX</span></td></tr>
<tr><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Five</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">V</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Fifteen</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">XV</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Seventy</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">LXX</span></td></tr>
<tr><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Six</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">VI</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Sixteen</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">XVI</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Eighty</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">LXXX</span></td></tr>
<tr><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Seven</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">VII</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Seventeen</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">XVII</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Ninety</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">XC</span></td></tr>
<tr><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Eight</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">VIII</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Eighteen</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">XVIII</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">One hundred</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">C</span></td></tr>
<tr><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Nine</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">IX</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Nineteen</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">XIX</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Five hundred</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">D</span></td></tr>
<tr><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Ten</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">X</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">Twenty</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">XX</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">One thousand</span></td><td align="left" style="background-color: #ccccff; color: black; padding-bottom: 3px; padding-left: 4px; padding-right: 4px; padding-top: 3px; vertical-align: top !important;" valign="top"><span class="Apple-style-span" style="font-family: inherit;">M</span></td></tr>
</tbody></table></div><div class="center" style="line-height: 22px; text-align: right;"><div style="text-align: center;"><span class="Apple-style-span" style="color: #330066; font-family: inherit;"><br />
</span></div><div style="text-align: left;"><span class="Apple-style-span" style="color: #330066; font-family: inherit;">By Husnina</span></div></div>4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com0tag:blogger.com,1999:blog-8948305359118541243.post-91069553693098068572011-07-21T20:44:00.002+08:002011-07-23T18:27:28.382+08:00The Beauty of Mathematics<div style="text-align: center;"><b><u><span style="color: red; font-family: Arial; font-size: large;"><span style="color: red; font-family: Arial; font-size: 18pt; font-weight: bold;">The Beauty of Mathematics </span></span></u></b></div><div style="text-align: center;"><b><u><span style="color: red; font-family: Arial; font-size: large;"><span style="color: red; font-family: Arial; font-size: 18pt; font-weight: bold;"> </span></span></u></b></div><div style="text-align: center;"><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">1 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">8</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> + </span></span><span style="color: blue; font-family: Arial; font-size: small;"><span style="color: blue; font-family: Arial;">1</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> = 9 </span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">12 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">8</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> + </span></span><span style="color: blue; font-family: Arial; font-size: small;"><span style="color: blue; font-family: Arial;">2</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> = 98 </span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">123 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">8</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> + </span></span><span style="color: blue; font-family: Arial; font-size: small;"><span style="color: blue; font-family: Arial;">3</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> = 987 </span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">1234 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">8</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> + </span></span><span style="color: blue; font-family: Arial; font-size: small;"><span style="color: blue; font-family: Arial;">4</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> = 9876 </span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">12345 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">8</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> + </span></span><span style="color: blue; font-family: Arial; font-size: small;"><span style="color: blue; font-family: Arial;">5</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> = 98765 </span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">123456 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">8</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> + </span></span><span style="color: blue; font-family: Arial; font-size: small;"><span style="color: blue; font-family: Arial;">6</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> = 987654 </span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">1234567 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">8 </span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">+ </span></span><span style="color: blue; font-family: Arial; font-size: small;"><span style="color: blue; font-family: Arial;">7</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> = 9876543 </span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">12345678 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">8</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> + </span></span><span style="color: blue; font-family: Arial; font-size: small;"><span style="color: blue; font-family: Arial;">8</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> = 98765432 </span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">123456789 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">8</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> + </span></span><span style="color: blue; font-family: Arial; font-size: small;"><span style="color: blue; font-family: Arial;">9</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> = 987654321 </span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"></span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">1 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">9</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> + 2 = </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">11</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> </span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">12 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">9</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> + 3 = </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">111</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> </span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">123 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">9</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> + 4 = </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">1111 </span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"></span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">1234 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">9</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> + 5 = </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">11111</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> </span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">12345 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">9</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> + 6 = </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">111111 </span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"></span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">123456 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">9</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> + 7 = </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">1111111</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> </span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">1234567 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">9</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> + 8 = </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">11111111 </span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"></span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">12345678 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">9</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> + 9 = </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">111111111 </span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"></span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">123456789 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">9</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> +10= </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">1111111111 </span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"></span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"></span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">9 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">9</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> + 7 = </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">88</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> </span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">98 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">9</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> + 6 = </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">888 </span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"></span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">987 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">9</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> </span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">+ 5 = </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">8888 </span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"></span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">9876 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">9 </span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">+ 4 = </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">88888 </span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"></span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">98765 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">9</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> + 3 = </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">888888 </span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"></span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">987654 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">9</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> + 2 = </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">8888888 </span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"></span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">9876543 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">9 </span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">+ 1 = </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">88888888</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> </span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;">98765432 x </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">9</span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"> + 0 = </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">888888888</span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;"> </span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"></span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"></span></span><span style="color: #ff8100; font-family: Arial; font-size: small;"><span style="color: #ff8100; font-family: Arial;"></span></span><br />
<span style="color: #ff8100; font-family: Arial; font-size: small;"><span style="color: #ff8100; font-family: Arial;"></span></span><span style="font-size: small;"><u><span style="color: #ff8100; font-family: Arial;"><span style="color: #ff8100; font-family: Arial;">Brilliant, isn't it?</span></span></u></span><span style="color: #ff8100; font-family: Arial; font-size: small;"><span style="color: #ff8100; font-family: Arial;"> </span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"></span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"></span></span><span style="color: blue; font-family: Arial; font-size: small;"><span style="color: blue; font-family: Arial;"></span></span><br />
<span style="color: blue; font-family: Arial; font-size: small;"><span style="color: blue; font-family: Arial;"></span></span><span style="color: blue; font-family: Arial; font-size: small;"><span style="color: blue; font-family: Arial;">And look at this symmetry: </span></span><span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"></span></span><br />
<span style="color: black; font-family: Arial; font-size: small;"><span style="color: windowtext; font-family: Arial;"></span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;"></span></span><br />
<span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;"></span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">1 x 1 = </span></span><span style="color: blue; font-family: Arial; font-size: small;"><span style="color: blue; font-family: Arial;">1</span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;"> </span></span><br />
<span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">11 x 11 = </span></span><span style="color: blue; font-family: Arial; font-size: small;"><span style="color: blue; font-family: Arial;">121 </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;"></span></span><br />
<span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;"></span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">111 x 111 = </span></span><span style="color: blue; font-family: Arial; font-size: small;"><span style="color: blue; font-family: Arial;">12321</span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;"> </span></span><br />
<span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">1111 x 1111 = </span></span><span style="color: blue; font-family: Arial; font-size: small;"><span style="color: blue; font-family: Arial;">1234321 </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;"></span></span><br />
<span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;"></span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">11111 x 11111 = </span></span><span style="color: blue; font-family: Arial; font-size: small;"><span style="color: blue; font-family: Arial;">123454321 </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;"></span></span><br />
<span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;"></span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">111111 x 111111 = </span></span><span style="color: blue; font-family: Arial; font-size: small;"><span style="color: blue; font-family: Arial;">12345654321</span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;"> </span></span><br />
<span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">1111111 x 1111111 = </span></span><span style="color: blue; font-family: Arial; font-size: small;"><span style="color: blue; font-family: Arial;">1234567654321 </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;"></span></span><br />
<span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;"></span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">11111111 x 11111111 = </span></span><span style="color: blue; font-family: Arial; font-size: small;"><span style="color: blue; font-family: Arial;">123456787654321 </span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;"></span></span><br />
<span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;"></span></span><span style="color: red; font-family: Arial; font-size: small;"><span style="color: red; font-family: Arial;">111111111 x 111111111 = </span></span><span style="color: blue; font-family: Arial; font-size: small;"><span style="color: blue; font-family: Arial;">12345678987654321</span></span></div><div style="text-align: center;"></div><div style="text-align: center;"><div style="text-align: center;"><span style="color: #ff8100; font-family: Arial; font-size: large;"><span style="color: #ff8100; font-family: Arial;">Mind Boggling...</span></span><span style="color: black; font-family: Arial; font-size: large;"><span style="color: windowtext; font-family: Arial;"> </span></span> <span style="color: black; font-family: Arial; font-size: large;"><span style="font-family: Arial;"> </span></span><br />
<span style="color: black; font-family: Arial; font-size: large;"><span style="font-family: Arial;"></span></span><span style="color: blue; font-family: Arial; font-size: large;"><span style="color: blue; font-family: Arial;">Now, take a look at this...</span></span><span style="font-family: Arial; font-size: large;"><span style="font-family: Arial;"> </span></span><br />
<span style="font-family: Arial; font-size: large;"><span style="font-family: Arial;"></span></span><span style="color: red; font-family: Arial; font-size: large;"><span style="color: red; font-family: Arial;"></span></span><br />
<span style="color: red; font-family: Arial; font-size: large;"><span style="color: red; font-family: Arial;">100%</span></span><span style="font-family: Arial; font-size: large;"><span style="font-family: Arial;"> </span></span><br />
<span style="font-family: Arial; font-size: large;"><span style="font-family: Arial;"></span></span><span style="color: blue; font-family: Arial; font-size: large;"><span style="color: blue; font-family: Arial;"></span></span><br />
<span style="color: blue; font-family: Arial; font-size: large;"><span style="color: blue; font-family: Arial;">From a strictly mathematical viewpoint:</span></span><span style="font-family: Arial; font-size: large;"><span style="font-family: Arial;"> </span></span><br />
<span style="font-family: Arial; font-size: large;"><span style="font-family: Arial;"></span></span><span style="color: blue; font-family: Arial; font-size: large;"><span style="color: blue; font-family: Arial;"></span></span><br />
<span style="color: blue; font-family: Arial; font-size: large;"><span style="color: blue; font-family: Arial;">What Equals</span></span><span style="font-family: Arial; font-size: large;"><span style="font-family: Arial;"> </span></span><span style="color: red; font-family: Arial; font-size: large;"><span style="color: red; font-family: Arial;">100%</span></span><span style="color: blue; font-family: Arial; font-size: large;"><span style="color: blue; font-family: Arial;">? </span></span><span style="font-family: Arial; font-size: large;"><span style="font-family: Arial;"></span></span><br />
<span style="font-family: Arial; font-size: large;"><span style="font-family: Arial;"></span></span><span style="color: blue; font-family: Arial; font-size: large;"><span style="color: blue; font-family: Arial;"></span></span><br />
<span style="color: blue; font-family: Arial; font-size: large;"><span style="color: blue; font-family: Arial;">What does it mean to give MORE than</span></span><span style="font-family: Arial; font-size: large;"><span style="font-family: Arial;"> </span></span><span style="color: red; font-family: Arial; font-size: large;"><span style="color: red; font-family: Arial;">100%</span></span><span style="color: blue; font-family: Arial; font-size: large;"><span style="color: blue; font-family: Arial;">?</span></span><span style="font-family: Arial; font-size: x-large;"><span style="font-family: Arial; font-size: 24pt;"> </span></span></div><div style="text-align: center;"><br />
</div><span style="color: blue; font-family: Arial; font-size: large;"><span style="color: blue;">What equals</span></span><span style="font-family: Arial; font-size: large;"> </span><span style="color: red; font-family: Arial; font-size: large;"><span style="color: red;">100%</span></span><span style="font-family: Arial; font-size: large;"> </span><span style="color: blue; font-family: Arial; font-size: large;"><span style="color: blue;">in life?</span></span><span style="font-family: Arial; font-size: large;"> </span><span style="color: blue; font-family: Arial; font-size: large;"><span style="color: blue;"><br />
Here's a little mathematical formula that might help<br />
answer these questions:</span></span><span style="font-family: Arial; font-size: large;"> </span><span style="color: #ff8100; font-family: Arial; font-size: large;"><span style="color: #ff8100;"><br />
If:</span></span><span style="font-family: Arial; font-size: large;"><br />
<br />
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z </span><span style="color: #ff8100; font-family: Arial; font-size: large;"><span style="color: #ff8100;"><br />
Is represented as:</span></span><span style="font-family: Arial; font-size: large;"> </span><span style="color: red; font-family: Arial; font-size: large;"><span style="color: red;"><br />
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26. </span></span><span style="font-family: Arial; font-size: large;"><br />
</span><span style="color: #ff8100; font-family: Arial; font-size: large;"><span style="color: #ff8100;"><br />
Then: </span></span><span style="font-family: Georgia, 'Times New Roman', serif; font-size: large;"><b><span style="font-weight: bold;"><br />
<br />
H-A-R-D-W-O- R- K </span></b></span><span style="font-family: Arial; font-size: large;"><br />
</span><span style="color: red; font-family: Arial; font-size: large;"><span style="color: red;"><br />
8+1+18+4+23+ 15+18+11 = 98%</span></span><span style="font-family: Arial; font-size: large;"> </span><span style="color: #ff8100; font-family: Arial; font-size: large;"><span style="color: #ff8100;"><br />
And: </span></span><span style="font-family: Georgia, 'Times New Roman', serif; font-size: large;"><b><span style="font-weight: bold;"><br />
<br />
K-N-O-W-L-E- D-G-E </span></b></span><span style="font-family: Arial; font-size: large;"><br />
</span><span style="color: red; font-family: Arial; font-size: large;"><span style="color: red;"><br />
11+14+15+23+ 12+5+4+7+ 5 = 96%</span></span><span style="font-family: Arial; font-size: large;"> </span><span style="color: #ff8100; font-family: Arial; font-size: large;"><span style="color: #ff8100;"><br />
But:</span></span><span style="font-family: Arial; font-size: large;"> <b><span style="font-weight: bold;"><br />
<br />
A-T-T-I-T-U- D-E </span></b></span><span style="color: red; font-family: Arial; font-size: large;"><span style="color: red;"><br />
1+20+20+9+20+ 21+4+5 = 100% </span></span><span style="font-family: Arial; font-size: large;"></span></div><span class="Apple-style-span" style="font-family: inherit;"><br />
</span><br />
<span class="Apple-style-span" style="font-family: inherit;">T<span style="font-size: small;">herefore, one can conclude with mathematical certainty that:<br />
While</span><span style="font-size: small;"> <span style="color: black;">Hard Work and Knowledge</span> will get you close, and <span style="color: black;">Attitude</span> will get you there</span></span><br />
<span style="font-family: inherit; font-size: small;"><br />
</span><br />
<span style="font-size: small;"><span style="font-family: inherit;">By Sook Lynn</span></span>4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com1tag:blogger.com,1999:blog-8948305359118541243.post-66196374630861698882011-07-21T16:55:00.005+08:002011-07-23T18:25:21.185+08:00Math Anxiety<div style="color: black;"><div class="separator" style="clear: both; text-align: center;"></div><span class="Apple-style-span" style="font-size: large;">Math Anxiety</span> is a phenomenon that is often considered when examining students’ problems in mathematics.<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-LnnlxFZbKTA/TikydrA6UiI/AAAAAAAAAAk/JnUWYcRVxIY/s1600/images+%25282%2529.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="137" src="http://2.bp.blogspot.com/-LnnlxFZbKTA/TikydrA6UiI/AAAAAAAAAAk/JnUWYcRVxIY/s320/images+%25282%2529.jpg" width="320" /></a></div> Mark H. Ashcraft, Ph.D. defines math anxiety as “a feeling of tension, apprehension, or fear that interferes with math performance” (2002, p. 1). The first math anxiety measurement scale was developed by Richardson and Suinn in 1972. Since this development, several researchers have examined math anxiety in empirical studies. Hembree (1990) conducted a thorough meta-analysis of 151 studies concerning math anxiety. It determined that math anxiety is related to poor math performance on math achievement tests and that math anxiety is related to negative attitudes concerning math. Hembree also suggests that math anxiety is directly connected with math avoidance.</div><div style="color: black;"> <br />
Ashcraft<sup><span style="font-size: x-small;"> </span></sup>(2002) suggests that highly anxious math students will avoid situations in which they have to perform mathematical equations. Unfortunately, math avoidance results in less competency, exposure and math practice, leaving students more anxious and mathematically unprepared to achieve. In college and university, anxious math students take fewer math courses and tend to feel negatively towards math. In fact, Ashcraft found that the correlation between math anxiety and variables such as confidence and motivation are strongly negative.</div><div style="color: black;"> <br />
According to Ashcraft, because math anxiety can cause math avoidance, an empirical dilemma arises. For instance, when a highly math-anxious student performs disappointingly on a math question, it could be due to math anxiety, or the lack of competency in math because of math avoidance. Ashcraft determined that by administering a test that becomes increasingly more mathematically challenging, he noticed that even highly math-anxious individuals do well on the first portion of the test measuring performance. However, on the latter and more difficult portion of the test, there was a stronger negative relationship between accuracy and math anxiety.<span class="mw-headline" id="Performance_anxiety"> </span></div><div style="color: black;"><br />
</div><div><a href="http://3.bp.blogspot.com/-_n02y36gqhQ/TikxSF-OlKI/AAAAAAAAAAc/PTdCPlz0fD0/s1600/images.jpg" imageanchor="1" style="clear: right; color: black; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="200" src="http://3.bp.blogspot.com/-_n02y36gqhQ/TikxSF-OlKI/AAAAAAAAAAc/PTdCPlz0fD0/s200/images.jpg" width="140" /></a><u><b><span class="mw-headline" id="Performance_anxiety" style="color: #351c75; font-family: Times, 'Times New Roman', serif;">Performance Anxiety</span></b></u></div><div style="color: black;"><br />
</div><div style="color: black;">People's fear of math can be related to test taking and performance anxiety. Some scholars have suggested a strong relation between math anxiety and math performance. Current research in math anxiety concerns working memory.<span class="mw-headline" id="Anxiety_Rating_Scale"> </span></div><div style="color: black;"><br />
</div><div><u><b><span class="mw-headline" id="Anxiety_Rating_Scale" style="color: #351c75;">Anxiety Rating Scale</span></b></u></div><div style="color: black;"><br />
</div><div style="color: black;">A rating scale for mathematics anxiety was written about in 1972 by Richardson and Suinn. Richardson and Suinn defined mathematical anxiety as "feelings of apprehension and tension concerning manipulation of numbers and completion of mathematical problems in various contexts."<span class="mw-headline" id="Math_and_culture"> </span></div><div style="color: black;"><br />
</div><div><u><b><span class="mw-headline" id="Math_and_culture" style="color: #351c75;">Math And Culture</span></b></u></div><div style="color: black;"><br />
</div><div style="color: black;">While there are overarching similarities concerning the acquisition of math skills, researchers have shown that children’s mathematical abilities differ across countries. In Canada, students score substantially lower in math problem-solving and operations than students in Korea and Singapore. Researchers have conducted thorough comparisons between countries, and have determined that in countries such as Taiwan and Japan, parents place more emphasis on effort rather than one’s innate intellectual ability in school success. Moreover, parents in these countries tend to set higher expectations and standards for their children. In turn, students spend more time on homework and value homework more than American children. (Stevenson & Lee, 1990).<span class="mw-headline" id="Math_and_gender"> </span></div><div style="color: black;"><br />
</div><div><b><u><span class="mw-headline" id="Math_and_gender" style="color: #351c75;">Math And Gender</span></u></b></div><div style="color: black;"><br />
</div><div style="color: black;">Another difference in mathematic abilities often explored in research concerns gender disparities. There has been research examining gender difference in performance on standardized tests across various countries. Beller and Gafni’s have shown that children at approximately nine years of age do not show consistent gender difference in relation to math skills. However, in 17 out of the 20 countries examined in this study, 13 year old boys tended to score higher than girls. Moreover, mathematics is often labeled as a masculine ability; as a result, girls often have low confidence in their math capabilities. These gender stereotypes can reinforce low confidence in girls and can cause math anxiety as research has shown that performance on standardized math tests is affected by one’s confidence. As a result, educators have been trying and should continue to try to abolish this stereotype by fostering confidence in math in all students in order to avoid math anxiety.<span class="mw-headline" id="Mathematics_and_women"> </span></div><div style="color: black;"><br />
</div><div><u><b><span class="mw-headline" id="Mathematics_and_women" style="color: #351c75;">Mathematics And Women</span></b></u></div><div style="color: black;"><br />
</div><div style="color: black;">Related to this is gender and mathematics as younger female scholars are thought to develop anxiety towards mathematics and sciences when they become more interested in social relations in their teen years. It is thought that women experience more anxiety in mathematics as a group than men and this has also been suggested in regards computer programming. See for instance [Copper, Joel, & Weaver D, Kimberlee. <i>Gender and Computers: "Understanding the Digital Divide"</i> who explore computing and gender and especially have done experiments relating gender and anxiety.<span class="mw-headline" id="Math_pedagogy"> </span></div><div style="color: black;"><br />
</div><div><u><b><span class="mw-headline" id="Common_beliefs" style="color: #351c75;">Common Beliefs</span></b></u></div><div style="color: black;"><br />
</div><div style="color: black;"><div style="text-align: left;">In the United States, many people believe that only a few "gifted" individuals have "what it takes" to learn math, and that hard work cannot compensate for this. Studies have shown "When asked to explain why some children do better in math than others, Asian children, their teachers, and their parents point to hard work, their American counterparts to ability."</div></div><div style="color: black;"><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: left;">Women mathematicians in the United States have almost always been a minority according to Margaret Murray. Although the exact difference fluctuates with the times as she has explored in her book [<i>Women Becoming Mathematicians: Creating a Professional Identity in Post-World War II America</i>]. "Since 1980, women have earned over 17 percent of the mathematics doctorates.... [In The United States]". The trends in gender are by no means clear, but perhaps parity is still a way to go. Thus parity will take more work to overcome mathematical anxiety and this is one reason for women in mathematics being role model for younger women.<span class="mw-headline" id="Mathematical_anxiety_in_schools:_Causes_and_potential_solutions"> </span></div><div class="separator" style="clear: both; text-align: left;"><span class="mw-headline" id="Mathematical_anxiety_in_schools:_Causes_and_potential_solutions"><br />
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</span></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-xFR5tIyiJGI/TikxZ4eAATI/AAAAAAAAAAg/Ci-H7lSvlAk/s1600/images+%25283%2529.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="http://4.bp.blogspot.com/-xFR5tIyiJGI/TikxZ4eAATI/AAAAAAAAAAg/Ci-H7lSvlAk/s400/images+%25283%2529.jpg" width="400" /></a></div><div class="separator" style="clear: both; text-align: left;"><span class="mw-headline" id="Mathematical_anxiety_in_schools:_Causes_and_potential_solutions"><br />
</span></div><div class="separator" style="clear: both; text-align: left;">By Ching Khai Lin</div></div><sup class="reference" id="cite_ref-curtain-phillips_24-1"><a href="http://en.wikipedia.org/wiki/Mathematical_anxiety#cite_note-curtain-phillips-24"><span style="font-size: x-small;"><span style="color: #0645ad;"></span></span></a></sup>4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com0tag:blogger.com,1999:blog-8948305359118541243.post-23502019027507021492011-07-21T16:52:00.004+08:002011-07-23T18:25:53.852+08:00About MathsElementary mathematics was part of the <a href="http://en.wikipedia.org/wiki/Education" title="Education"><span style="color: #0645ad;">education system</span></a> in most ancient civilisations, including <a href="http://en.wikipedia.org/wiki/Ancient_Greece" title="Ancient Greece"><span style="color: #0645ad;">Ancient Greece</span></a>, the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Roman_empire" title="Roman empire"><span style="color: #0645ad;">Roman empire</span></a>, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Vedic_civilization" title="Vedic civilization"><span style="color: #0645ad;">Vedic society</span></a> and <a href="http://en.wikipedia.org/wiki/Ancient_Egypt" title="Ancient Egypt"><span style="color: #0645ad;">ancient Egypt</span></a>. In most cases, a formal education was only available to <a href="http://en.wikipedia.org/wiki/Male" title="Male"><span style="color: #0645ad;">male</span></a> children with a sufficiently high status, wealth or <a href="http://en.wikipedia.org/wiki/Caste" title="Caste"><span style="color: #0645ad;">caste</span></a>.<br />
In <a href="http://en.wikipedia.org/wiki/Plato" title="Plato"><span style="color: #0645ad;">Plato</span></a>'s division of the <a href="http://en.wikipedia.org/wiki/Liberal_arts" title="Liberal arts"><span style="color: #0645ad;">liberal arts</span></a> into the <a href="http://en.wikipedia.org/wiki/Trivium_(education)" title="Trivium (education)"><span style="color: #0645ad;">trivium</span></a> and the <a href="http://en.wikipedia.org/wiki/Quadrivium" title="Quadrivium"><span style="color: #0645ad;">quadrivium</span></a>, the quadrivium included the mathematical fields of <a href="http://en.wikipedia.org/wiki/Arithmetic" title="Arithmetic"><span style="color: #0645ad;">arithmetic</span></a> and <a href="http://en.wikipedia.org/wiki/Geometry" title="Geometry"><span style="color: #0645ad;">geometry</span></a>. This structure was continued in the structure of <a href="http://en.wikipedia.org/wiki/Medieval_university" title="Medieval university"><span style="color: #0645ad;">classical education</span></a> that was developed in medieval Europe. Teaching of geometry was almost universally based on <a href="http://en.wikipedia.org/wiki/Euclid" title="Euclid"><span style="color: #0645ad;">Euclid</span></a>'s <i>Elements</i>. Apprentices to trades such as masons, merchants and money-lenders could expect to learn such practical mathematics as was relevant to their profession.<br />
The first mathematics textbooks to be written in English and French were published by <a href="http://en.wikipedia.org/wiki/Robert_Recorde" title="Robert Recorde"><span style="color: #0645ad;">Robert Recorde</span></a>, beginning with <i>The Grounde of Artes</i> in 1540.<br />
In the <a href="http://en.wikipedia.org/wiki/Renaissance" title="Renaissance"><span style="color: #0645ad;">Renaissance</span></a> the academic status of mathematics declined, because it was strongly associated with trade and commerce. Although it continued to be taught in European universities, it was seen as subservient to the study of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Natural_Philosophy" title="Natural Philosophy"><span style="color: #0645ad;">Natural</span></a>, <a href="http://en.wikipedia.org/wiki/Metaphysics" title="Metaphysics"><span style="color: #0645ad;">Metaphysical</span></a> and <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Moral_philosophy" title="Moral philosophy"><span style="color: #0645ad;">Moral</span></a> Philosophy.<br />
This trend was somewhat reversed in the seventeenth century, with the <a href="http://en.wikipedia.org/wiki/University_of_Aberdeen" title="University of Aberdeen"><span style="color: #0645ad;">University of Aberdeen</span></a> creating a Mathematics Chair in 1613, followed by the Chair in Geometry being set up in <a href="http://en.wikipedia.org/wiki/University_of_Oxford" title="University of Oxford"><span style="color: #0645ad;">University of Oxford</span></a> in 1619 and the <a href="http://en.wikipedia.org/wiki/Lucasian_Professor_of_Mathematics" title="Lucasian Professor of Mathematics"><span style="color: #0645ad;">Lucasian Chair of Mathematics</span></a> being established by the <a href="http://en.wikipedia.org/wiki/University_of_Cambridge" title="University of Cambridge"><span style="color: #0645ad;">University of Cambridge</span></a> in 1662. However, it was uncommon for mathematics to be taught outside of the universities. <a href="http://en.wikipedia.org/wiki/Isaac_Newton" title="Isaac Newton"><span style="color: #0645ad;">Isaac Newton</span></a>, for example, received no formal mathematics teaching until he joined <a href="http://en.wikipedia.org/wiki/Trinity_College,_Cambridge" title="Trinity College, Cambridge"><span style="color: #0645ad;">Trinity College, Cambridge</span></a> in 1661.<br />
In the 18th and 19th centuries the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Industrial_revolution" title="Industrial revolution"><span style="color: #0645ad;">industrial revolution</span></a> led to an enormous increase in <a href="http://en.wikipedia.org/wiki/Urban_area" title="Urban area"><span style="color: #0645ad;">urban</span></a> populations. Basic numeracy skills, such as the ability to tell the time, count money and carry out simple <a href="http://en.wikipedia.org/wiki/Arithmetic" title="Arithmetic"><span style="color: #0645ad;">arithmetic</span></a>, became essential in this new urban lifestyle. Within the new <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Public_education" title="Public education"><span style="color: #0645ad;">public education</span></a> systems, mathematics became a central part of the curriculum from an early age.<br />
By the twentieth century mathematics was part of the core curriculum in all <a href="http://en.wikipedia.org/wiki/Developed_country" title="Developed country"><span style="color: #0645ad;">developed countries</span></a>.<br />
During the twentieth century mathematics education was established as an independent field of research. Here are some of the main events in this development:<br />
<ul><li>In 1893 a Chair in mathematics education was created at the University of Göttingen, under the administration of <a href="http://en.wikipedia.org/wiki/Felix_Klein" title="Felix Klein"><span style="color: #0645ad;">Felix Klein</span></a></li>
<li>The <a href="http://en.wikipedia.org/wiki/International_Commission_on_Mathematical_Instruction" title="International Commission on Mathematical Instruction"><span style="color: #0645ad;">International Commission on Mathematical Instruction</span></a> (ICMI) was founded in 1908, and Felix Klein became the first president of the organisation</li>
<li>A new interest in mathematics education emerged in the 1960s, and the commission was revitalised</li>
<li>In 1968, the <a class="new" href="http://en.wikipedia.org/w/index.php?title=Shell_Centre_for_Mathematical_Education&action=edit&redlink=1" title="Shell Centre for Mathematical Education (page does not exist)"><span style="color: #ba0000;">Shell Centre for Mathematical Education</span></a> was established in <a href="http://en.wikipedia.org/wiki/Nottingham" title="Nottingham"><span style="color: #0645ad;">Nottingham</span></a></li>
<li>The first <a href="http://en.wikipedia.org/wiki/International_Congress_on_Mathematical_Education" title="International Congress on Mathematical Education"><span style="color: #0645ad;">International Congress on Mathematical Education</span></a> (ICME) was held in <a href="http://en.wikipedia.org/wiki/Lyon" title="Lyon"><span style="color: #0645ad;">Lyon</span></a> in 1969. The second congress was in <a href="http://en.wikipedia.org/wiki/Exeter" title="Exeter"><span style="color: #0645ad;">Exeter</span></a> in 1972, and after that it has been held every four years</li>
</ul>In the 20th century, the cultural impact of the "<a class="new" href="http://en.wikipedia.org/w/index.php?title=Electric_age&action=edit&redlink=1" title="Electric age (page does not exist)"><span style="color: #ba0000;">electric age</span></a>" (McLuhan) was also taken up by <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Educational_theory" title="Educational theory"><span style="color: #0645ad;">educational theory</span></a> and the teaching of mathematics. While previous approach focused on "working with specialized 'problems' in <a href="http://en.wikipedia.org/wiki/Arithmetic" title="Arithmetic"><span style="color: #0645ad;">arithmetic</span></a>", the emerging structural approach to knowledge had "small children meditating about <a href="http://en.wikipedia.org/wiki/Number_theory" title="Number theory"><span style="color: #0645ad;">number theory</span></a> and '<a href="http://en.wikipedia.org/wiki/Set_theory" title="Set theory"><span style="color: #0645ad;">sets</span></a>'."<sup class="reference" id="cite_ref-0"><a href="http://en.wikipedia.org/wiki/Mathematics_education#cite_note-0"><span style="font-size: x-small;"><span style="color: #0645ad;">[1]</span></span></a></sup><br />
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematics_education&action=edit&section=2" title="Edit section: Objectives"><span style="color: #0645ad;">edit</span></a>]</span> <span class="mw-headline" id="Objectives">Objectives</span></h2>At different times and in different cultures and countries, mathematics education has attempted to achieve a variety of different objectives. These objectives have included:<br />
<ul><li>The teaching of basic <a href="http://en.wikipedia.org/wiki/Numeracy" title="Numeracy"><span style="color: #0645ad;">numeracy</span></a> skills to all pupils</li>
<li>The teaching of practical mathematics (<a href="http://en.wikipedia.org/wiki/Arithmetic" title="Arithmetic"><span style="color: #0645ad;">arithmetic</span></a>, <a href="http://en.wikipedia.org/wiki/Elementary_algebra" title="Elementary algebra"><span style="color: #0645ad;">elementary algebra</span></a>, plane and solid <a href="http://en.wikipedia.org/wiki/Geometry" title="Geometry"><span style="color: #0645ad;">geometry</span></a>, <a href="http://en.wikipedia.org/wiki/Trigonometry" title="Trigonometry"><span style="color: #0645ad;">trigonometry</span></a>) to most pupils, to equip them to follow a trade or craft</li>
<li>The teaching of abstract mathematical concepts (such as <a href="http://en.wikipedia.org/wiki/Set_(mathematics)" title="Set (mathematics)"><span style="color: #0645ad;">set</span></a> and <a href="http://en.wikipedia.org/wiki/Function_(mathematics)" title="Function (mathematics)"><span style="color: #0645ad;">function</span></a>) at an early age</li>
<li>The teaching of selected areas of mathematics (such as <a href="http://en.wikipedia.org/wiki/Euclidean_geometry" title="Euclidean geometry"><span style="color: #0645ad;">Euclidean geometry</span></a>) as an example of an <a href="http://en.wikipedia.org/wiki/Axiomatic_system" title="Axiomatic system"><span style="color: #0645ad;">axiomatic system</span></a> and a model of <a href="http://en.wikipedia.org/wiki/Deductive_reasoning" title="Deductive reasoning"><span style="color: #0645ad;">deductive reasoning</span></a></li>
<li>The teaching of selected areas of mathematics (such as <a href="http://en.wikipedia.org/wiki/Calculus" title="Calculus"><span style="color: #0645ad;">calculus</span></a>) as an example of the intellectual achievements of the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Modern_world" title="Modern world"><span style="color: #0645ad;">modern world</span></a></li>
<li>The teaching of advanced mathematics to those pupils who wish to follow a career in <a href="http://en.wikipedia.org/wiki/STEM_fields" title="STEM fields"><span style="color: #0645ad;">Science, Technology, Engineering, and Mathematics (STEM)</span></a> fields.</li>
<li>The teaching of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Heuristics" title="Heuristics"><span style="color: #0645ad;">heuristics</span></a> and other problem-solving strategies to solve non-routine problems.</li>
</ul>Methods of teaching mathematics have varied in line with changing objectives.<br />
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematics_education&action=edit&section=3" title="Edit section: Research"><span style="color: #0645ad;">edit</span></a>]</span> <span class="mw-headline" id="Research">Research</span></h2>An increasing amount of research has been done in the area of mathematics education in the last few decades. Here are a few of the major results.<br />
<dl><dt>Formative assessment<sup class="reference" id="cite_ref-ResearchClips_1-0"><a href="http://en.wikipedia.org/wiki/Mathematics_education#cite_note-ResearchClips-1"><span style="font-size: x-small;"><span style="color: #0645ad;">[2]</span></span></a></sup></dt>
<dd><a href="http://en.wikipedia.org/wiki/Formative_assessment" title="Formative assessment"><span style="color: #0645ad;">Formative assessment</span></a> is both the best and cheapest way to boost student achievement, student engagement and teacher professional satisfaction. Results surpass those of reducing class size or increasing teachers' content knowledge. Only short-term (within and between lessons) and medium-term (within and between units) assessment is effective. Effective assessment is based on clarifying what students should know, creating appropriate activities to obtain the evidence needed, giving good feedback, encouraging students to take control of their learning and letting students be resources for one another.</dd></dl><dl><dt>Effective instruction<sup class="reference" id="cite_ref-ResearchClips_1-1"><a href="http://en.wikipedia.org/wiki/Mathematics_education#cite_note-ResearchClips-1"><span style="font-size: x-small;"><span style="color: #0645ad;">[2]</span></span></a></sup></dt>
<dd>The two most important criteria for helping students gain conceptual understanding are making connections and intentionally struggling with important ideas. Research in the 70s and 80s concluded that skill efficiency is best attained by rapid pacing, <a href="http://en.wikipedia.org/wiki/Direct_instruction" title="Direct instruction"><span style="color: #0645ad;">direct traditional teaching</span></a> and a smooth transition from teacher modeling to error-free practice. More recent research shows that students who learn skills in conceptually-oriented instruction are better able to adapt their skills to new situations.</dd></dl><dl><dt>Students with difficulties<sup class="reference" id="cite_ref-ResearchClips_1-2"><a href="http://en.wikipedia.org/wiki/Mathematics_education#cite_note-ResearchClips-1"><span style="font-size: x-small;"><span style="color: #0645ad;">[2]</span></span></a></sup></dt>
<dd>Students with genuine difficulties (unrelated to motivation or past instruction) struggle with <a href="http://en.wikipedia.org/wiki/Recall_of_facts" title="Recall of facts"><span style="color: #0645ad;">basic facts</span></a>, answer impulsively, struggle with mental representations, have poor <a href="http://en.wikipedia.org/wiki/Number_sense" title="Number sense"><span style="color: #0645ad;">number sense</span></a> and have poor short-term memory. Techniques that have been found productive for helping such students include peer-assisted learning, explicit teaching with visual aids, instruction informed by <a href="http://en.wikipedia.org/wiki/Formative_assessment" title="Formative assessment"><span style="color: #0645ad;">formative assessment</span></a> and encouraging students to think aloud.</dd></dl><dl><dt>Homework<sup class="reference" id="cite_ref-ResearchClips_1-3"><a href="http://en.wikipedia.org/wiki/Mathematics_education#cite_note-ResearchClips-1"><span style="font-size: x-small;"><span style="color: #0645ad;">[2]</span></span></a></sup></dt>
<dd>Homework which leads students to practice past lessons or prepare future lessons are more effective than those going over today's lesson. Assignments should be a mix of easy and hard problems and ideally based on the student's learning style. Students must receive feedback. Students with learning disabilities or low motivation may profit from rewards. Shorter homework is better than long homework, and group homework is sometimes effective, though these findings depend on grade level. Homework helps simple skills, but not broader measures of achievement.</dd></dl><dl><dt>Bilingualism<sup class="reference" id="cite_ref-ResearchClips_1-4"><a href="http://en.wikipedia.org/wiki/Mathematics_education#cite_note-ResearchClips-1"><span style="font-size: x-small;"><span style="color: #0645ad;">[2]</span></span></a></sup></dt>
<dd>Most bilingual adults switch languages when calculating. Such <a href="http://en.wikipedia.org/wiki/Code-switching" title="Code-switching"><span style="color: #0645ad;">code-switching</span></a> has no impact on math ability and should not be discouraged.</dd></dl><dl><dt>Learning statistics<sup class="reference" id="cite_ref-ResearchClips_1-5"><a href="http://en.wikipedia.org/wiki/Mathematics_education#cite_note-ResearchClips-1"><span style="font-size: x-small;"><span style="color: #0645ad;">[2]</span></span></a></sup></dt>
<dd>When studying <a href="http://en.wikipedia.org/wiki/Statistics" title="Statistics"><span style="color: #0645ad;">statistics</span></a>, children need time to explore, study and share reasoning about centers, shape, spread and variability. The ability to calculate <a href="http://en.wikipedia.org/wiki/Average" title="Average"><span style="color: #0645ad;">averages</span></a> does not mean students understand the concept of averages, which students conceptualize in a variety of ways—from a simplistic "typical value" to a deeper idea of "representative value." Learning when to use <a href="http://en.wikipedia.org/wiki/Mean" title="Mean"><span style="color: #0645ad;">mean</span></a>, <a href="http://en.wikipedia.org/wiki/Median" title="Median"><span style="color: #0645ad;">median</span></a> and <a href="http://en.wikipedia.org/wiki/Mode_(statistics)" title="Mode (statistics)"><span style="color: #0645ad;">mode</span></a> is difficult.</dd></dl><dl><dt>Algebraic reasoning<sup class="reference" id="cite_ref-ResearchClips_1-6"><a href="http://en.wikipedia.org/wiki/Mathematics_education#cite_note-ResearchClips-1"><span style="font-size: x-small;"><span style="color: #0645ad;">[2]</span></span></a></sup></dt>
<dd>It is important for elementary school children to spend a long time learning to express algebraic properties without symbols before learning algebraic notation. When learning symbols, many students believe letters always represent unknowns and struggle with the concept of <a href="http://en.wikipedia.org/wiki/Variable_(mathematics)" title="Variable (mathematics)"><span style="color: #0645ad;">variable</span></a>. They prefer arithmetic reasoning to algebraic equations for solving word problems. It takes time to move from arithmetic to algebraic generalizations to describe patterns. Students often have trouble with the minus sign and understand the <a href="http://en.wikipedia.org/wiki/Equals_sign" title="Equals sign"><span style="color: #0645ad;">equals sign</span></a> to mean "the answer is...."</dd></dl><dl><dt>What can we learn from research?<sup class="reference" id="cite_ref-ResearchClips_1-7"><a href="http://en.wikipedia.org/wiki/Mathematics_education#cite_note-ResearchClips-1"><span style="font-size: x-small;"><span style="color: #0645ad;">[2]</span></span></a></sup></dt>
<dd>Instead of just looking at whether a particular program works, we must also look at <i>why</i> and <i>under what conditions</i> it works. Teachers can adapt tasks used in studies for their own classrooms. Individual studies are often inconclusive, so it is important to look at a consensus of many studies to draw conclusions. Theory can put practice in a new perspective. For example, research shows that when students invent their own algorithms first, and then learn the standard algorithm, they understand better and make fewer errors. Such findings can have an impact on classroom practice.</dd></dl><h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematics_education&action=edit&section=4" title="Edit section: Methodology controversy"><span style="color: #0645ad;">edit</span></a>]</span> <span class="mw-headline" id="Methodology_controversy">Methodology controversy</span></h3>In the scholarly literature in mathematics education, relatively few articles report random experiments in which teaching methods were randomly assigned to classes.<sup class="reference" id="cite_ref-Cook_2-0"><a href="http://en.wikipedia.org/wiki/Mathematics_education#cite_note-Cook-2"><span style="font-size: x-small;"><span style="color: #0645ad;">[3]</span></span></a></sup><sup class="reference" id="cite_ref-Scheaffer_3-0"><a href="http://en.wikipedia.org/wiki/Mathematics_education#cite_note-Scheaffer-3"><span style="font-size: x-small;"><span style="color: #0645ad;">[4]</span></span></a></sup> Randomized experiments have been done to evaluate non-pedagogical programs in schools, such as programs to reduce teenage-pregnancy and to reduce drug use.<sup class="reference" id="cite_ref-4"><a href="http://en.wikipedia.org/wiki/Mathematics_education#cite_note-4"><span style="font-size: x-small;"><span style="color: #0645ad;">[5]</span></span></a></sup> Randomized experiments were commonly done in educational psychology in the 1880s at American universities.<sup class="reference" id="cite_ref-stigler_5-0"><a href="http://en.wikipedia.org/wiki/Mathematics_education#cite_note-stigler-5"><span style="font-size: x-small;"><span style="color: #0645ad;">[6]</span></span></a></sup><sup class="reference" id="cite_ref-dehue_6-0"><a href="http://en.wikipedia.org/wiki/Mathematics_education#cite_note-dehue-6"><span style="font-size: x-small;"><span style="color: #0645ad;">[7]</span></span></a></sup> However, randomized experiments have been relatively rare in education in recent decades. In other disciplines concerned with human subjects, like biomedicine, psychology, and policy evaluation, controlled, randomized experiments remain the preferred method of evaluating treatments.<sup class="reference" id="cite_ref-7"><a href="http://en.wikipedia.org/wiki/Mathematics_education#cite_note-7"><span style="font-size: x-small;"><span style="color: #0645ad;">[8]</span></span></a></sup><sup class="reference" id="cite_ref-8"><a href="http://en.wikipedia.org/wiki/Mathematics_education#cite_note-8"><span style="font-size: x-small;"><span style="color: #0645ad;">[9]</span></span></a></sup> Educational statisticians and some mathematics educators have been working to increase the use of randomized experiments to evaluate teaching methods.<sup class="reference" id="cite_ref-Scheaffer_3-1"><a href="http://en.wikipedia.org/wiki/Mathematics_education#cite_note-Scheaffer-3"><span style="font-size: x-small;"><span style="color: #0645ad;">[4]</span></span></a></sup><br />
On the other hand, many scholars in educational schools have argued against increasing the number of randomized experiments, often because of philosophical objections.<sup class="reference" id="cite_ref-Cook_2-1"><a href="http://en.wikipedia.org/wiki/Mathematics_education#cite_note-Cook-2"><span style="font-size: x-small;"><span style="color: #0645ad;">[3]</span></span></a></sup> Unlike medicine, where volunteers understand the risks, it is often unethical to assign children to experimental methods without their consent. Because certain types of students or teachers may be more willing than others to give such consent, such studies would not be truly randomized.<br />
Whereas randomized studies remain the "gold standard" for testing whether certain educational interventions have an impact or not, observational studies are often needed to explore the sociological and psychologial reasons of how and why various educational factors have an impact. Non-randomized qualitative studies are also useful for discovering new principles or theories, which can eventually be tested by more rigorous methods. The scientific community recognizes that observational studies remain valuable in education—just as observational studies remain valuable in <a href="http://en.wikipedia.org/wiki/Epidemiology" title="Epidemiology"><span style="color: #0645ad;">epidemiology</span></a>, political science, economics, <a href="http://en.wikipedia.org/wiki/Sociology" title="Sociology"><span style="color: #0645ad;">sociology</span></a>, <a href="http://en.wikipedia.org/wiki/Anthropology" title="Anthropology"><span style="color: #0645ad;">anthropology</span></a>, etc.<sup class="reference" id="cite_ref-9"><a href="http://en.wikipedia.org/wiki/Mathematics_education#cite_note-9"><span style="font-size: x-small;"><span style="color: #0645ad;">[10]</span></span></a></sup><br />
In the United States, the <a class="new" href="http://en.wikipedia.org/w/index.php?title=National_Mathematics_Advisory_Panel&action=edit&redlink=1" title="National Mathematics Advisory Panel (page does not exist)"><span style="color: #ba0000;">National Mathematics Advisory Panel</span></a> (NMAP) published a report in 2008 based on only studies using randomized experiments; the NMAP report's exclusive reliance on randomized experiments received criticism from some scholars.<sup class="reference" id="cite_ref-10"><a href="http://en.wikipedia.org/wiki/Mathematics_education#cite_note-10"><span style="font-size: x-small;"><span style="color: #0645ad;">[11]</span></span></a></sup> In 2010, the <a class="new" href="http://en.wikipedia.org/w/index.php?title=What_Works_Clearinghouse&action=edit&redlink=1" title="What Works Clearinghouse (page does not exist)"><span style="color: #ba0000;">What Works Clearinghouse</span></a> (essentially the research arm for the <a href="http://en.wikipedia.org/wiki/United_States_Department_of_Education" title="United States Department of Education"><span style="color: #0645ad;">Department of Education</span></a>) responded to ongoing controversy by extending its research base to include non-experimental studies, including regression-discontinuity designs and certain single-case studies. <sup class="reference" id="cite_ref-11"><a href="http://en.wikipedia.org/wiki/Mathematics_education#cite_note-11"><span style="font-size: x-small;"><span style="color: #0645ad;">[12]</span></span></a></sup><br />
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematics_education&action=edit&section=5" title="Edit section: Standards"><span style="color: #0645ad;">edit</span></a>]</span> <span class="mw-headline" id="Standards">Standards</span></h2>Throughout most of history, standards for mathematics education were set locally, by individual schools or teachers, depending on the levels of achievement that were relevant to, realistic for, and considered socially appropriate for their pupils.<br />
In modern times there has been a move towards regional or national standards, usually under the umbrella of a wider standard school curriculum. In <a href="http://en.wikipedia.org/wiki/England" title="England"><span style="color: #0645ad;">England</span></a>, for example, standards for mathematics education are set as part of the <a class="external text" href="http://www.nc.uk.net/webdav/servlet/XRM?Page/@id=6004&Subject/@id=22" rel="nofollow"><span style="color: #3366bb;">National Curriculum for England</span></a>, while Scotland maintains its own educational system.<br />
Ma (2000) summarised the research of others who found, based on nationwide data, that students with higher scores on standardised math tests had taken more mathematics courses in high school. This led some states to require three years of math instead of two. But because this requirement was often met by taking another lower level math course, the additional courses had a “diluted” effect in raising achievement levels.<sup class="reference" id="cite_ref-12"><a href="http://en.wikipedia.org/wiki/Mathematics_education#cite_note-12"><span style="font-size: x-small;"><span style="color: #0645ad;">[13]</span></span></a></sup><br />
In North America, the <a class="external text" href="http://www.nctm.org/" rel="nofollow"><span style="color: #3366bb;">National Council of Teachers of Mathematics (NCTM)</span></a> has published the <a href="http://en.wikipedia.org/wiki/Principles_and_Standards_for_School_Mathematics" title="Principles and Standards for School Mathematics"><span style="color: #0645ad;">Principles and Standards for School Mathematics</span></a>. In 2006, they released the <a class="external text" href="http://www.nctm.org/focalpoints/" rel="nofollow"><span style="color: #3366bb;">Curriculum Focal Points</span></a>, which recommend the most important mathematical topics for each grade level through grade 8. However, these standards are not nationally enforced in US schools.<br />
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematics_education&action=edit&section=6" title="Edit section: Content and age levels"><span style="color: #0645ad;">edit</span></a>]</span> <span class="mw-headline" id="Content_and_age_levels">Content and age levels</span></h2>Different levels of mathematics are taught at different ages and in somewhat different sequences in different countries. Sometimes a class may be taught at an earlier age than typical as a special or "honors" class.<br />
Elementary mathematics in most countries is taught in a similar fashion, though there are differences. In the <a href="http://en.wikipedia.org/wiki/Mathematics_education_in_the_United_States" title="Mathematics education in the United States"><span style="color: #0645ad;">United States</span></a> fractions are typically taught starting from 1st grade, whereas in other countries they are usually taught later, since the metric system does not require young children to be familiar with them. Most countries tend to cover fewer topics in greater depth than in the United States.<sup class="reference" id="cite_ref-13"><a href="http://en.wikipedia.org/wiki/Mathematics_education#cite_note-13"><span style="font-size: x-small;"><span style="color: #0645ad;">[14]</span></span></a></sup><br />
In most of the US, algebra, geometry and analysis (pre-calculus and calculus) are taught as separate courses in different years of high school. Mathematics in most other countries (and in a few US states) is integrated, with topics from all branches of mathematics studied every year. Students in many countries choose an option or pre-defined course of study rather than choosing courses <i>à la carte</i> as in the United States. Students in science-oriented curricula typically study differential calculus and trigonometry at age 16-17 and integral calculus, complex numbers, analytic geometry, exponential and logarithmic functions, and infinite series in their final year of secondary school.<br />
<br />
By Haripriya Vanoo4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com0tag:blogger.com,1999:blog-8948305359118541243.post-65924929591333698622011-07-20T23:12:00.001+08:002011-07-23T18:23:24.137+08:00Quotes about Mathematics<div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-layout-grid-align: none;"><span class="Apple-style-span" style="font-family: CMSS10;">“Education should be started with mathematics. For it forms well</span></div><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-layout-grid-align: none;"><span style="font-family: CMSS10;"><span style="font-family: Calibri;">designed brains that are able to reason right. It is even admitted that</span></span></div><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-layout-grid-align: none;"><span style="font-family: CMSS10;"><span style="font-family: Calibri;">those who have studied mathematics during their childhood should be</span></span></div><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-layout-grid-align: none;"><span style="font-family: CMSS10;"><span style="font-family: Calibri;">trusted, for they have acquired solid bases for arguing which become</span></span></div><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-layout-grid-align: none;"><span style="font-family: CMSS10;"><span style="font-family: Calibri;">to them a sort of second nature”</span></span><span style="font-family: 'Times New Roman', serif;">.</span></div><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-layout-grid-align: none;"><i><span style="font-family: 'Times New Roman', serif;">Ibn Khaldun, al-Muqaddima (born in 1332, Tunis), historian, sociologist, philosopher</span></i></div><div class="MsoNormal" style="margin: 0in 0in 10pt;"><i><span style="font-family: 'Times New Roman', serif; line-height: 115%;">Strongest personalities of Arabo-Muslim culture in the period of its deline.</span></i></div><div class="MsoNormal" style="margin: 0in 0in 10pt;"><span class="Apple-style-span" style="font-family: 'Times New Roman', serif;">Mathematics in everyday life</span></div><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-layout-grid-align: none;"><span style="font-family: 'Times New Roman', serif;">1</span><span style="font-family: 'Times New Roman', serif;">. </span><span style="font-family: CMSS10;"><span style="font-family: Calibri;">On a basic level you need to be able to count, multiply, substract and divide. Mathematics is around us. It is present in different forms whenever we pick up the phone, manage the money, travel to some place, play soccer, meet new friends; unintentionally in all these things mathematics is involved.</span></span></div><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-layout-grid-align: none;"><br />
</div><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-layout-grid-align: none;"><span style="font-family: 'Times New Roman', serif;">2. </span><span style="font-family: CMSS10;"><span style="font-family: Calibri;">There are huge illustrations that testify the presence of mathematics in everything that we are doing. We hope you will enjoy the following slides that illustrate mathematics in everyday life. Enjoy!</span></span></div><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-layout-grid-align: none;"><br />
</div><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-layout-grid-align: none;"><span style="font-family: 'Times New Roman', serif;">Cooking: the idea of proportion</span></div><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-layout-grid-align: none;"><span style="font-family: CMSS10;"><span style="font-family: Calibri;">For a Chocolate cake: 5 eggs,3/4 cup of sugar, 1/2 cup of vegetable</span></span></div><div class="MsoNormal" style="margin: 0in 0in 10pt;"><span style="font-family: CMSS10; line-height: 115%;"><span style="font-family: Calibri;">oil,...</span></span></div><div class="MsoNormal" style="margin: 0in 0in 10pt;"><span style="font-family: 'Times New Roman', serif; line-height: 115%;">Percentage</span></div><div class="MsoNormal" style="margin: 0in 0in 10pt;"><span style="font-family: 'Times New Roman', serif; line-height: 115%;">Medicine/Pharmacy</span></div><div class="MsoNormal" style="margin: 0in 0in 10pt;"><span style="font-family: 'Times New Roman', serif;">Bank: savings and credit</span></div><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-layout-grid-align: none;"><span style="font-family: CMSS10;"><span style="font-family: Calibri;">With some good understanding of simple and compound interest,</span></span></div><div class="MsoNormal" style="margin: 0in 0in 10pt;"><span style="font-family: CMSS10; line-height: 115%;"><span style="font-family: Calibri;">you can manage the way your money grows.</span></span></div><div class="MsoNormal" style="margin: 0in 0in 10pt;"><span style="font-family: 'Times New Roman', serif;">Chance to win in lottery: Probalility</span></div><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-layout-grid-align: none;"><span style="font-family: CMSS10;"><span style="font-family: Calibri;">The mathematical concept that deals with the chance of winning</span></span></div><div class="MsoNormal" style="margin: 0in 0in 10pt;"><span style="font-family: Calibri;"><span style="font-family: CMSS10; line-height: 115%;">a lottery game is probability...</span></span><br />
<span class="Apple-style-span" style="font-family: 'Times New Roman', serif; line-height: 18px;">Area</span><br />
<span class="Apple-style-span" style="font-family: 'Times New Roman', serif; line-height: 18px;">Geometry in clothing</span></div><div class="MsoNormal" style="margin: 0in 0in 10pt;"><span style="font-family: 'Times New Roman', serif; line-height: 115%;">By </span><span class="Apple-style-span" style="font-family: 'Times New Roman', serif; line-height: 18px;">Neesahantani</span></div><div class="MsoNormal" style="margin: 0in 0in 10pt;"><b style="mso-bidi-font-weight: normal;"><shape id="Picture_x0020_7" o:spid="_x0000_i1025" style="height: 222.75pt; mso-wrap-style: square; visibility: visible; width: 203.25pt;" type="#_x0000_t75"><imagedata o:title="" src="file:///C:\DOCUME~1\USER\LOCALS~1\Temp\msohtmlclip1\01\clip_image013.emz"></imagedata></shape></b></div>4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com0tag:blogger.com,1999:blog-8948305359118541243.post-70416067334284769472011-07-20T22:45:00.001+08:002011-07-23T18:20:57.293+08:00Prehistoric Mathematics<h2><span class="Apple-style-span" style="font-size: small; font-weight: normal;">The origins of mathematical thought lie in the concepts of number, magnitude, and form.</span><span class="Apple-style-span" style="font-size: small; font-weight: normal;"><sup class="reference" id="cite_ref-Boyer_1991_loc.3DOrigins_p._3_10-0"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-Boyer_1991_loc.3DOrigins_p._3-10"></a></sup></span><span class="Apple-style-span" style="font-size: small; font-weight: normal;"> Modern studies of animal cognition have shown that these concepts are not unique to humans. Such concepts would have been part of everyday life in hunter-gatherer societies. That the concept of number evolved gradually over time is evident in that some languages today preserve the distinction between "one", "two", and "many", but not of numbers larger than two</span></h2><div style="color: black;">The oldest known possibly mathematical object is the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Lebombo_bone" title="Lebombo bone">Lebombo bone</a>, discovered in the Lebombo mountains of <a href="http://en.wikipedia.org/wiki/Swaziland" title="Swaziland">Swaziland</a> and dated to approximately 35,000 BC.<sup class="reference" id="cite_ref-11"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-11"></a></sup> It consists of 29 distinct notches cut into a baboon's fibula.<sup class="reference" id="cite_ref-Diaspora_12-0"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-Diaspora-12"></a></sup> Also <a href="http://en.wikipedia.org/wiki/Prehistory" title="Prehistory">prehistoric</a> <a href="http://en.wikipedia.org/wiki/Artifact_%28archaeology%29" title="Artifact (archaeology)">artifacts</a> discovered in Africa and <a href="http://en.wikipedia.org/wiki/France" title="France">France</a>, dated between <a class="mw-redirect" href="http://en.wikipedia.org/wiki/35000_BC" title="35000 BC">35,000</a> and <a href="http://en.wikipedia.org/wiki/Upper_Paleolithic" title="Upper Paleolithic">20,000</a> years old,<sup class="reference" id="cite_ref-13"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-13"></a></sup> suggest early attempts to <a href="http://en.wikipedia.org/wiki/Quantification" title="Quantification">quantify</a> time.<sup class="reference" id="cite_ref-14"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-14"></a></sup>In the book <i>How Mathematics Happened: The First 50,000 Years</i>, Peter Rudman argues that the development of the concept of prime numbers could only have come about after the concept of division, which he dates to after 10,000 BC, with prime numbers probably not being understood until about 500 BC. He also writes that "no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10."<sup class="reference" id="cite_ref-15"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-15"></a></sup></div><div style="color: black;">The <a href="http://en.wikipedia.org/wiki/Ishango_bone" title="Ishango bone">Ishango bone</a>, found near the headwaters of the <a href="http://en.wikipedia.org/wiki/Nile" title="Nile">Nile</a> river (northeastern <a href="http://en.wikipedia.org/wiki/Democratic_Republic_of_the_Congo" title="Democratic Republic of the Congo">Congo</a>), may be as much as <a href="http://en.wikipedia.org/wiki/Upper_Paleolithic" title="Upper Paleolithic">20,000</a> years old and consists of a series of tally marks carved in three columns running the length of the bone. Common interpretations are that the Ishango bone shows either the earliest known demonstration of <a href="http://en.wikipedia.org/wiki/Sequence" title="Sequence">sequences</a> of <a href="http://en.wikipedia.org/wiki/Prime_number" title="Prime number">prime numbers</a><sup class="reference" id="cite_ref-Diaspora_12-1"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-Diaspora-12"></a></sup> or a six month lunar calendar.<sup class="reference" id="cite_ref-Marshack_16-0"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-Marshack-16"></a></sup> <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Predynastic_Egypt" title="Predynastic Egypt">Predynastic Egyptians</a> of the 5th millennium BC pictorially represented <a href="http://en.wikipedia.org/wiki/Geometry" title="Geometry">geometric</a> designs. It has been claimed that <a href="http://en.wikipedia.org/wiki/Megalith" title="Megalith">megalithic</a> monuments in <a href="http://en.wikipedia.org/wiki/England" title="England">England</a> and <a href="http://en.wikipedia.org/wiki/Scotland" title="Scotland">Scotland</a>, dating from the 3rd millennium BC, incorporate geometric ideas such as <a href="http://en.wikipedia.org/wiki/Circle" title="Circle">circles</a>, <a href="http://en.wikipedia.org/wiki/Ellipse" title="Ellipse">ellipses</a>, and <sup class="reference" id="cite_ref-17"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-17"></a></sup></div><div class="thumb tright" style="color: black;"><div class="thumbinner" style="width: 402px;"><a class="image" href="http://en.wikipedia.org/wiki/File:Ishango_bone.jpg"><img alt="" class="thumbimage" height="178" src="http://upload.wikimedia.org/wikipedia/commons/thumb/0/08/Ishango_bone.jpg/400px-Ishango_bone.jpg" width="400" /></a> <br />
<div class="thumbcaption"><div class="magnify"><a class="internal" href="http://en.wikipedia.org/wiki/File:Ishango_bone.jpg" title="Enlarge"><img alt="" height="11" src="http://bits.wikimedia.org/skins-1.17/common/images/magnify-clip.png" width="15" /></a></div>The <a href="http://en.wikipedia.org/wiki/Ishango_bone" title="Ishango bone">Ishango bone</a>,dating to perhaps 18,000 to 20,000 <a href="http://en.wikipedia.org/wiki/Anno_Domini" title="Anno Domini">B.C.</a></div></div></div><h2 style="color: black; font-weight: normal;"><span class="mw-headline" id="Mesopotamian_mathematics">Mesopotamian mathematics</span></h2><div class="rellink boilerplate seealso" style="color: black;"><br />
<a href="http://en.wikipedia.org/wiki/Plimpton_322" title="Plimpton 322"> </a></div><div class="thumb tright" style="color: black;"><div class="thumbinner" style="width: 252px;"><a class="image" href="http://en.wikipedia.org/wiki/File:Plimpton_322.jpg"><img alt="" class="thumbimage" height="173" src="http://upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Plimpton_322.jpg/250px-Plimpton_322.jpg" width="250" /></a> <br />
<div class="thumbcaption"><div class="magnify"><a class="internal" href="http://en.wikipedia.org/wiki/File:Plimpton_322.jpg" title="Enlarge"><img alt="" height="11" src="http://bits.wikimedia.org/skins-1.17/common/images/magnify-clip.png" width="15" /></a></div>The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.</div></div></div><div style="color: black;"><a href="http://en.wikipedia.org/wiki/Babylonia" title="Babylonia">Babylonian</a> mathematics refers to any mathematics of the people of <a href="http://en.wikipedia.org/wiki/Mesopotamia" title="Mesopotamia">Mesopotamia</a> (modern <a href="http://en.wikipedia.org/wiki/Iraq" title="Iraq">Iraq</a>) from the days of the early <a href="http://en.wikipedia.org/wiki/Sumer" title="Sumer">Sumerians</a> through the <a href="http://en.wikipedia.org/wiki/Hellenistic_period" title="Hellenistic period">Hellenistic period</a> almost to the dawn of <a href="http://en.wikipedia.org/wiki/Christianity" title="Christianity">Christianity</a>.<sup class="reference" id="cite_ref-18"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-18"></a></sup> It is named Babylonian mathematics due to the central role of <a href="http://en.wikipedia.org/wiki/Babylon" title="Babylon">Babylon</a> as a place of study. Later under the <a href="http://en.wikipedia.org/wiki/Caliphate" title="Caliphate">Arab Empire</a>, Mesopotamia, especially <a href="http://en.wikipedia.org/wiki/Baghdad" title="Baghdad">Baghdad</a>, once again became an important center of study for <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Islamic_mathematics" title="Islamic mathematics">Islamic mathematics</a>.</div><div style="color: black;">In contrast to the sparsity of sources in <a href="http://en.wikipedia.org/wiki/Egyptian_mathematics" title="Egyptian mathematics">Egyptian mathematics</a>, our knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s.<sup class="reference" id="cite_ref-19"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-19"></a></sup> Written in <a href="http://en.wikipedia.org/wiki/Cuneiform_script" title="Cuneiform script">Cuneiform script</a>, tablets were inscribed whilst the clay was moist, and baked hard in an oven or by the heat of the sun. Some of these appear to be graded homework.</div><div style="color: black;">The earliest evidence of written mathematics dates back to the ancient <a href="http://en.wikipedia.org/wiki/Sumer" title="Sumer">Sumerians</a>, who built the earliest civilization in Mesopotamia. They developed a complex system of <a href="http://en.wikipedia.org/wiki/Metrology" title="Metrology">metrology</a> from 3000 BC. From around 2500 BC onwards, the Sumerians wrote <a href="http://en.wikipedia.org/wiki/Multiplication_table" title="Multiplication table">multiplication tables</a> on clay tablets and dealt with <a href="http://en.wikipedia.org/wiki/Geometry" title="Geometry">geometrical</a><a href="http://en.wikipedia.org/wiki/Division_%28mathematics%29" title="Division (mathematics)">division</a> problems. The earliest traces of the Babylonian numerals also date back to this period.<sup class="reference" id="cite_ref-20"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-20">[21]</a></sup> exercises and </div><div style="color: black;">The majority of recovered clay tablets date from 1800 to 1600 BC, and cover topics which include fractions, algebra, quadratic and cubic equations, and the calculation of <a href="http://en.wikipedia.org/wiki/Regular_number" title="Regular number">regular</a> <a href="http://en.wikipedia.org/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocal</a> <a href="http://en.wikipedia.org/wiki/Twin_prime" title="Twin prime">pairs</a>.<sup class="reference" id="cite_ref-21"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-21"></a></sup> The tablets also include multiplication tables and methods for solving <a href="http://en.wikipedia.org/wiki/Linear_equation" title="Linear equation">linear</a> and <a href="http://en.wikipedia.org/wiki/Quadratic_equation" title="Quadratic equation">quadratic equations</a>. The Babylonian tablet YBC 7289 gives an approximation to √2 accurate to five decimal places.</div><div style="color: black;">Babylonian mathematics were written using a <a href="http://en.wikipedia.org/wiki/Sexagesimal" title="Sexagesimal">sexagesimal</a> (base-60) <a href="http://en.wikipedia.org/wiki/Numeral_system" title="Numeral system">numeral system</a>. From this derives the modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle, as well as the use of seconds and minutes of arc to denote fractions of a degree. Babylonian advances in mathematics were facilitated by the fact that 60 has many divisors. Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a true place-value system, where digits written in the left column represented larger values, much as in the <a href="http://en.wikipedia.org/wiki/Decimal" title="Decimal">decimal</a> system. They lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context.</div><h2 style="color: black; font-weight: normal;"><span class="editsection"></span> <span class="mw-headline" id="Egyptian_mathematics">Egyptian mathematics</span></h2><div class="rellink relarticle mainarticle" style="color: black;"><br />
<a href="http://en.wikipedia.org/wiki/Egyptian_mathematics" title="Egyptian mathematics"> </a></div><div class="thumb tright" style="color: black;"><div class="thumbinner" style="width: 302px;"><a class="image" href="http://en.wikipedia.org/wiki/File:Moskou-papyrus.jpg"><img alt="" class="thumbimage" height="144" src="http://upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Moskou-papyrus.jpg/300px-Moskou-papyrus.jpg" width="300" /></a> <br />
<div class="thumbcaption"><div class="magnify"><a class="internal" href="http://en.wikipedia.org/wiki/File:Moskou-papyrus.jpg" title="Enlarge"><img alt="" height="11" src="http://bits.wikimedia.org/skins-1.17/common/images/magnify-clip.png" width="15" /></a></div>Image of Problem 14 from the <a href="http://en.wikipedia.org/wiki/Moscow_Mathematical_Papyrus" title="Moscow Mathematical Papyrus">Moscow Mathematical Papyrus</a>. The problem includes a diagram indicating the dimensions of the truncated pyramid.</div></div></div><div style="color: black;"><a href="http://en.wikipedia.org/wiki/Egypt" title="Egypt">Egyptian</a> mathematics refers to mathematics written in the <a href="http://en.wikipedia.org/wiki/Egyptian_language" title="Egyptian language">Egyptian language</a>. From the <a href="http://en.wikipedia.org/wiki/Hellenistic_period" title="Hellenistic period">Hellenistic period</a>, <a href="http://en.wikipedia.org/wiki/Greek_language" title="Greek language">Greek</a> replaced Egyptian as the written language of <a href="http://en.wikipedia.org/wiki/Egyptians" title="Egyptians">Egyptian</a> scholars. Mathematical study in <a href="http://en.wikipedia.org/wiki/Egypt" title="Egypt">Egypt</a> later continued under the <a href="http://en.wikipedia.org/wiki/Caliphate" title="Caliphate">Arab Empire</a> as part of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Islamic_mathematics" title="Islamic mathematics">Islamic mathematics</a>, when <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Arabic" title="Arabic">Arabic</a> became the written language of Egyptian scholars.</div><div style="color: black;">The most extensive Egyptian mathematical text is the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Rhind_papyrus" title="Rhind papyrus">Rhind papyrus</a> (sometimes also called the Ahmes Papyrus after its author), dated to c. 1650 BC but likely a copy of an older document from the <a href="http://en.wikipedia.org/wiki/Middle_Kingdom_of_Egypt" title="Middle Kingdom of Egypt">Middle Kingdom</a> of about 2000-1800 BC.<sup class="reference" id="cite_ref-Boyer_1991_loc.3DEgypt_p._11_22-0"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-Boyer_1991_loc.3DEgypt_p._11-22"></a></sup> It is an instruction manual for students in arithmetic and geometry. In addition to giving area formulas and methods for multiplication, division and working with unit fractions, it also contains evidence of other mathematical knowledge,<sup class="reference" id="cite_ref-23"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-23"></a></sup> including <a href="http://en.wikipedia.org/wiki/Composite_number" title="Composite number">composite</a> and <a href="http://en.wikipedia.org/wiki/Prime_number" title="Prime number">prime numbers</a>; <a href="http://en.wikipedia.org/wiki/Arithmetic_mean" title="Arithmetic mean">arithmetic</a>, <a href="http://en.wikipedia.org/wiki/Geometric_mean" title="Geometric mean">geometric</a> and <a href="http://en.wikipedia.org/wiki/Harmonic_mean" title="Harmonic mean">harmonic means</a>; and simplistic understandings of both the <a href="http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes" title="Sieve of Eratosthenes">Sieve of Eratosthenes</a> and <a href="http://en.wikipedia.org/wiki/Perfect_number" title="Perfect number">perfect number theory</a> (namely, that of the number 6).<sup class="reference" id="cite_ref-24"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-24"></a></sup> It also shows how to solve first order <a href="http://en.wikipedia.org/wiki/Linear_equation" title="Linear equation">linear equations</a><sup class="reference" id="cite_ref-25"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-25"> </a></sup><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Arithmetic_series" title="Arithmetic series">arithmetic</a> and <a href="http://en.wikipedia.org/wiki/Geometric_series" title="Geometric series">geometric series</a>.<sup class="reference" id="cite_ref-26"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-26"></a></sup> as well as </div><div style="color: black;">Another significant Egyptian mathematical text is the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Moscow_papyrus" title="Moscow papyrus">Moscow papyrus</a>, also from the <a href="http://en.wikipedia.org/wiki/Middle_Kingdom_of_Egypt" title="Middle Kingdom of Egypt">Middle Kingdom</a><sup class="reference" id="cite_ref-Boyer_1991_loc.3DEgypt_p._19_27-0"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-Boyer_1991_loc.3DEgypt_p._19-27"></a></sup> It consists of what are today called <i>word problems</i> or <i>story problems</i>, which were apparently intended as entertainment. One problem is considered to be of particular importance because it gives a method for finding the volume of a <a href="http://en.wikipedia.org/wiki/Frustum" title="Frustum">frustum</a>: "If you are told: A truncated pyramid of 6 for the vertical height by 4 on the base by 2 on the top. You are to square this 4, result 16. You are to double 4, result 8. You are to square 2, result 4. You are to add the 16, the 8, and the 4, result 28. You are to take one third of 6, result 2. You are to take 28 twice, result 56. See, it is 56. You will find it right." period, dated to c. 1890 BC.</div><div style="color: black;">Finally, the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Berlin_papyrus" title="Berlin papyrus">Berlin papyrus</a> (c. 1300 BC <sup class="reference" id="cite_ref-28"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-28"></a></sup>) shows that ancient Egyptians could solve a second-order <a href="http://en.wikipedia.org/wiki/Algebraic_equation" title="Algebraic equation">algebraic equation</a>.<sup class="reference" id="cite_ref-29"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-29"></a></sup></div><h2 style="color: black; font-weight: normal;"><span class="editsection"></span> <span class="mw-headline" id="African_mathematics">African mathematics</span></h2><div class="rellink boilerplate seealso" style="color: black;"><br />
<a href="http://en.wikipedia.org/wiki/Science_and_technology_in_Africa#Mathematics" title="Science and technology in Africa"></a></div><div style="color: black;">The <a href="http://en.wikipedia.org/wiki/Nubia" title="Nubia">ancient Nubians</a> used a trigonometric methodology similar to the Egyptians.<sup class="reference" id="cite_ref-30"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-30">[31]</a></sup> During the Meroitic Era the Nubians established a system of geometry including early versions of sun clocks.<sup class="reference" id="cite_ref-31"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-31"></a></sup><sup class="reference" id="cite_ref-32"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-32"></a></sup></div><div style="color: black;">All mathematical learning of the Islamic world during the medieval period were available to <a href="http://en.wikipedia.org/wiki/Timbuktu" title="Timbuktu">Timbuktu</a> scholars arithmetic, algebra, geometry, and trigonometry during the great Islamic reigns of the <a href="http://en.wikipedia.org/wiki/Mali_Empire" title="Mali Empire">Mali Empire</a> and the <a href="http://en.wikipedia.org/wiki/Songhai_Empire" title="Songhai Empire">Songhai Empire</a>.</div><h2 style="color: black; font-weight: normal;"><span class="editsection"></span> <span class="mw-headline" id="Greek_Mathematics">Greek Mathematics</span></h2><div class="thumb tleft"><div class="thumbinner" style="width: 202px;"><a class="image" href="http://en.wikipedia.org/wiki/File:Pythagorean.svg"><img alt="" class="thumbimage" height="159" src="http://upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Pythagorean.svg/200px-Pythagorean.svg.png" width="200" /></a> <br />
<div class="thumbcaption"><div class="magnify"><a class="internal" href="http://en.wikipedia.org/wiki/File:Pythagorean.svg" title="Enlarge"><img alt="" height="11" src="http://bits.wikimedia.org/skins-1.17/common/images/magnify-clip.png" width="15" /></a></div>The <a href="http://en.wikipedia.org/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>. The <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Pythagoreans" title="Pythagoreans">Pythagoreans</a> are generally credited with the first proof of the theorem.</div></div></div><div class="thumbinner" style="width: 202px;"><a class="image" href="http://en.wikipedia.org/wiki/File:Pythagorean.svg"></a> <br />
<div class="thumbcaption"><div class="magnify"><a class="internal" href="http://en.wikipedia.org/wiki/File:Pythagorean.svg" title="Enlarge"><img alt="" height="11" src="http://bits.wikimedia.org/skins-1.17/common/images/magnify-clip.png" width="15" /></a></div>The <a href="http://en.wikipedia.org/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>. The <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Pythagoreans" title="Pythagoreans">Pythagoreans</a> are generally credited with the first proof of the theorem.</div></div><div class="thumb tleft" style="color: black;"></div><div style="color: black;">Greek mathematics refers to the mathematics written in the <a href="http://en.wikipedia.org/wiki/Greek_language" title="Greek language">Greek language</a> from the time of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Thales_of_Miletus" title="Thales of Miletus">Thales of Miletus</a><a href="http://en.wikipedia.org/wiki/Platonic_Academy" title="Platonic Academy">Academy of Athens</a> in 529 AD.<sup class="reference" id="cite_ref-35"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-35"></a></sup> Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to North Africa, but were united by culture and language. Greek mathematics of the period following <a href="http://en.wikipedia.org/wiki/Alexander_the_Great" title="Alexander the Great">Alexander the Great</a> is sometimes called Hellenistic mathematics.<sup class="reference" id="cite_ref-36"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-36"></a></sup> (~600 BC) to the closure of the </div><div style="color: black;">Greek mathematics was much more sophisticated than the mathematics that had been developed by earlier cultures. All surviving records of pre-Greek mathematics show the use of inductive reasoning, that is, repeated observations used to establish rules of thumb. Greek mathematicians, by contrast, used deductive reasoning. The Greeks used logic to derive conclusions from definitions and axioms, and used <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Mathematical_rigor" title="Mathematical rigor">mathematical rigor</a> to <a href="http://en.wikipedia.org/wiki/Mathematical_proof" title="Mathematical proof">prove</a> them.<sup class="reference" id="cite_ref-37"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-37"></a></sup></div><div style="color: black;">Greek mathematics is thought to have begun with <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Thales_of_Miletus" title="Thales of Miletus">Thales of Miletus</a> (c. 624–c.546 BC) and <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Pythagoras_of_Samos" title="Pythagoras of Samos">Pythagoras of Samos</a> (c. 582–c. 507 BC). Although the extent of the influence is disputed, they were probably inspired by <a href="http://en.wikipedia.org/wiki/Egyptian_mathematics" title="Egyptian mathematics">Egyptian</a> and <a href="http://en.wikipedia.org/wiki/Babylonian_mathematics" title="Babylonian mathematics">Babylonian mathematics</a>. According to legend, Pythagoras traveled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests.</div><div class="thumb tright" style="color: black;"><div class="thumbinner" style="width: 222px;"><a class="image" href="http://en.wikipedia.org/wiki/File:Archimedes_pi.svg"><img alt="" class="thumbimage" height="73" src="http://upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Archimedes_pi.svg/220px-Archimedes_pi.svg.png" width="220" /></a> <br />
<div class="thumbcaption"><div class="magnify"><a class="internal" href="http://en.wikipedia.org/wiki/File:Archimedes_pi.svg" title="Enlarge"><img alt="" height="11" src="http://bits.wikimedia.org/skins-1.17/common/images/magnify-clip.png" width="15" /></a></div>Archimedes used the <a href="http://en.wikipedia.org/wiki/Method_of_exhaustion" title="Method of exhaustion">method of exhaustion</a> to approximate the value of <a href="http://en.wikipedia.org/wiki/Pi" title="Pi">pi</a>.</div></div></div><div style="color: black;">Thales used <a href="http://en.wikipedia.org/wiki/Geometry" title="Geometry">geometry</a> to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Thales%27_Theorem" title="Thales' Theorem">Thales' Theorem</a>. As a result, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed.<sup class="reference" id="cite_ref-38"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-38"></a></sup> Pythagoras established the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Pythagoreans" title="Pythagoreans">Pythagorean School</a>, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number".<sup class="reference" id="cite_ref-39"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-39"></a></sup> It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The Pythagoreans are credited with the first proof of the <a href="http://en.wikipedia.org/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>,<sup class="reference" id="cite_ref-40"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-40"></a></sup> though the statement of the theorem has a long history, and with the proof of the existence of <sup class="reference" id="cite_ref-42"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-42"></a></sup></div><div class="thumb tright" style="color: black;"><div class="thumbinner" style="width: 222px;"><a class="image" href="http://en.wikipedia.org/wiki/File:P._Oxy._I_29.jpg"><img alt="" class="thumbimage" height="134" src="http://upload.wikimedia.org/wikipedia/commons/thumb/8/8d/P._Oxy._I_29.jpg/220px-P._Oxy._I_29.jpg" width="220" /></a> <br />
<div class="thumbcaption"><div class="magnify"><a class="internal" href="http://en.wikipedia.org/wiki/File:Oxyrhynchus_papyrus_with_Euclid%27s_Elements.jpg" title="Enlarge"><img alt="" height="11" src="http://bits.wikimedia.org/skins-1.17/common/images/magnify-clip.png" width="15" /></a></div>One of the oldest surviving fragments of Euclid's <i>Elements</i>, found at <a href="http://en.wikipedia.org/wiki/Oxyrhynchus" title="Oxyrhynchus">Oxyrhynchus</a> and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.<sup class="reference" id="cite_ref-43"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-43"></a></sup></div></div></div><div style="color: black;"><a href="http://en.wikipedia.org/wiki/Plato" title="Plato">Plato</a> (428/427 BC – 348/347 BC) is important in the history of mathematics for inspiring and guiding others.<sup class="reference" id="cite_ref-44"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-44"></a></sup> His <a href="http://en.wikipedia.org/wiki/Platonic_Academy" title="Platonic Academy">Platonic Academy</a>, in <a href="http://en.wikipedia.org/wiki/Athens" title="Athens">Athens</a>, became the mathematical center of the world in the 4th century BC, and it was from this school that the leading mathematicians of the day, such as <a href="http://en.wikipedia.org/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus of Cnidus</a>, came from.<sup class="reference" id="cite_ref-45"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-45"></a></sup> Plato also discussed the foundations of mathematics, clarified some of the definitions (e.g. that of a line as "breadthless length"), and reorganized the assumptions.<sup class="reference" id="cite_ref-46"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-46"></a></sup> The <a href="http://en.wikipedia.org/wiki/Mathematical_analysis" title="Mathematical analysis">analytic method</a> is ascribed to Plato, while a formula for obtaining Pythagorean triples bears his name.<sup class="reference" id="cite_ref-47"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-47"></a></sup></div><div style="color: black;"><a href="http://en.wikipedia.org/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus</a> (408–c.355 BC) developed the <a href="http://en.wikipedia.org/wiki/Method_of_exhaustion" title="Method of exhaustion">method of exhaustion</a>, a precursor of modern <a href="http://en.wikipedia.org/wiki/Integral" title="Integral">integration</a><sup class="reference" id="cite_ref-48"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-48">[49]</a></sup> and a theory of ratios that avoided the problem of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Incommensurable_magnitudes" title="Incommensurable magnitudes">incommensurable magnitudes</a>.<sup class="reference" id="cite_ref-49"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-49"></a></sup> The former allowed the calculations of areas and volumes of curvilinear figures,<sup class="reference" id="cite_ref-50"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-50"></a></sup> while the latter enabled subsequent geometers to make significant advances in geometry. Though he made no specific technical mathematical discoveries, <a href="http://en.wikipedia.org/wiki/Aristotle" title="Aristotle">Aristotle</a> (384—c.322 BC) contributed significantly to the development of mathematics by laying the foundation of <a href="http://en.wikipedia.org/wiki/Logic" title="Logic">logic</a>.<sup class="reference" id="cite_ref-51"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-51"></a></sup></div><div style="color: black;">In the 3rd century BC, the premier center of mathematical education and research was the <a href="http://en.wikipedia.org/wiki/Musaeum" title="Musaeum">Musaeum</a> of <a href="http://en.wikipedia.org/wiki/Alexandria" title="Alexandria">Alexandria</a>.<sup class="reference" id="cite_ref-52"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-52"></a></sup> It was there that <a href="http://en.wikipedia.org/wiki/Euclid" title="Euclid">Euclid</a> (c. 300 BC) taught, and wrote the <a href="http://en.wikipedia.org/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a>, widely considered the most successful and influential textbook of all time<sup class="reference" id="cite_ref-53"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-53"></a></sup> The <i>Elements</i> introduced <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Mathematical_rigor" title="Mathematical rigor">mathematical rigor</a> through the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Axiomatic_method" title="Axiomatic method">axiomatic method</a> and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the <i>Elements</i> were already known, Euclid arranged them into a single, coherent logical framework.<sup class="reference" id="cite_ref-54"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-54"></a></sup> The <i>Elements</i> was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today.<sup class="reference" id="cite_ref-55"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-55"></a></sup> In addition to the familiar theorems of <a href="http://en.wikipedia.org/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, the <i>Elements</i> was meant as an introductory textbook to all mathematical subjects of the time, such as <a href="http://en.wikipedia.org/wiki/Number_theory" title="Number theory">number theory</a>, <a href="http://en.wikipedia.org/wiki/Algebra" title="Algebra">algebra</a> and <a href="http://en.wikipedia.org/wiki/Solid_geometry" title="Solid geometry">solid geometry</a>,<sup class="reference" id="cite_ref-56"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-56"></a></sup> including proofs that the square root of two is irrational and that there are infinitely many prime numbers. Euclid also <a href="http://en.wikipedia.org/wiki/Euclid#Other_works" title="Euclid">wrote extensively</a> on other subjects, such as <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Conic_sections" title="Conic sections">conic sections</a>, <a href="http://en.wikipedia.org/wiki/Optics" title="Optics">optics</a>, <a href="http://en.wikipedia.org/wiki/Spherical_geometry" title="Spherical geometry">spherical geometry</a>, and mechanics, but only half of his writings survive<sup class="reference" id="cite_ref-57"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-57"></a></sup></div><div style="color: black;"><a href="http://en.wikipedia.org/wiki/Archimedes" title="Archimedes">Archimedes</a> (c.287–212 BC) of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Syracuse,_Italy" title="Syracuse, Italy">Syracuse</a> used the <a href="http://en.wikipedia.org/wiki/Method_of_exhaustion" title="Method of exhaustion">method of exhaustion</a> to calculate the <a href="http://en.wikipedia.org/wiki/Area" title="Area">area</a> under the arc of a <a href="http://en.wikipedia.org/wiki/Parabola" title="Parabola">parabola</a> with the <a href="http://en.wikipedia.org/wiki/Series_%28mathematics%29" title="Series (mathematics)">summation of an infinite series</a>, and gave remarkably accurate approximations of <a href="http://en.wikipedia.org/wiki/Pi" title="Pi">Pi</a>.<sup class="reference" id="cite_ref-58"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-58"></a></sup><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Archimedes_spiral" title="Archimedes spiral">spiral</a> bearing his name, formulas for the <a href="http://en.wikipedia.org/wiki/Volume" title="Volume">volumes</a> of <a href="http://en.wikipedia.org/wiki/Surface_of_revolution" title="Surface of revolution">surfaces of revolution</a>, and an ingenious system for expressing very large numbers. He also studied t</div><div style="color: black;"><br />
</div><h2><span class="mw-headline" id="Chinese_mathematics">Chinese mathematics</span></h2><div class="rellink relarticle mainarticle" style="color: black;"><a href="http://en.wikipedia.org/wiki/Chinese_mathematics" title="Chinese mathematics"><br />
</a></div><div class="thumb tright" style="color: black;"><div class="thumbinner" style="width: 282px;"><a class="image" href="http://en.wikipedia.org/wiki/File:Chounumerals.jpg"><img alt="" class="thumbimage" height="100" src="http://upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Chounumerals.jpg/280px-Chounumerals.jpg" width="280" /></a> <br />
<div class="thumbcaption"><div class="magnify"><a class="internal" href="http://en.wikipedia.org/wiki/File:Chounumerals.jpg" title="Enlarge"><img alt="" height="11" src="http://bits.wikimedia.org/skins-1.17/common/images/magnify-clip.png" width="15" /></a></div>Counting rod numerals</div></div></div><div class="thumb tright" style="color: black;"><div class="thumbinner" style="width: 202px;"><a class="image" href="http://en.wikipedia.org/wiki/File:%E4%B9%9D%E7%AB%A0%E7%AE%97%E8%A1%93.gif"><img alt="" class="thumbimage" height="261" src="http://upload.wikimedia.org/wikipedia/commons/thumb/8/88/%E4%B9%9D%E7%AB%A0%E7%AE%97%E8%A1%93.gif/200px-%E4%B9%9D%E7%AB%A0%E7%AE%97%E8%A1%93.gif" width="200" /></a> <br />
<div class="thumbcaption"><div class="magnify"><a class="internal" href="http://en.wikipedia.org/wiki/File:%E4%B9%9D%E7%AB%A0%E7%AE%97%E8%A1%93.gif" title="Enlarge"><img alt="" height="11" src="http://bits.wikimedia.org/skins-1.17/common/images/magnify-clip.png" width="15" /></a></div><i><a href="http://en.wikipedia.org/wiki/The_Nine_Chapters_on_the_Mathematical_Art" title="The Nine Chapters on the Mathematical Art">The Nine Chapters on the Mathematical Art</a></i>, one of the earliest surviving mathematical texts from <a href="http://en.wikipedia.org/wiki/China" title="China">China</a> (2nd century AD).</div></div></div><div style="color: black;">Early Chinese mathematics is so different from that of other parts of the world that it is reasonable to assume independent development.<sup class="reference" id="cite_ref-59"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-59"></a></sup> The oldest extant mathematical text from China is the <i><a href="http://en.wikipedia.org/wiki/Zhou_Bi_Suan_Jing" title="Zhou Bi Suan Jing">Chou Pei Suan Ching</a></i>, variously dated to between 1200 BC and 100 BC, though a date of about 300 BC appears reasonable.<sup class="reference" id="cite_ref-Boyer_1991_loc.3DChina_and_India_p._196_60-0"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-Boyer_1991_loc.3DChina_and_India_p._196-60">[61]</a></sup></div><div style="color: black;">Of particular note is the use in Chinese mathematics of a decimal positional notation system, the so-called "rod numerals" in which distinct ciphers were used for numbers between 1 and 10, and additional ciphers for powers of ten.<sup class="reference" id="cite_ref-61"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-61"></a></sup> Thus, the number 123 would be written using the symbol for "1", followed by the symbol for "100", then the symbol for "2" followed by the symbol for "10", followed by the symbol for "3". This was the most advanced number system in the world at the time, apparently in use several centuries before the common era and well before the development of the Indian numeral system.<sup class="reference" id="cite_ref-62"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-62"></a></sup> Rod numerals allowed the representation of numbers as large as desired and allowed calculations to be carried out on the <i><a href="http://en.wikipedia.org/wiki/Suanpan" title="Suanpan">suan pan</a></i>, or (Chinese abacus). The date of the invention of the <i>suan pan</i> is not certain, but the earliest written mention dates from AD 190, in Xu Yue's <i>Supplementary Notes on the Art of Figures</i>.</div><div style="color: black;">The oldest existent work on <a href="http://en.wikipedia.org/wiki/Geometry" title="Geometry">geometry</a> in China comes from the philosophical <a href="http://en.wikipedia.org/wiki/Mohism" title="Mohism">Mohist</a> canon c. 330 BC, compiled by the followers of <a href="http://en.wikipedia.org/wiki/Mozi" title="Mozi">Mozi</a> (470–390 BC). The <i>Mo Jing</i> described various aspects of many fields associated with physical science, and provided a small number of geometrical theorems as well.<sup class="reference" id="cite_ref-63"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-63">[64]</a></sup></div><div style="color: black;">In 212 BC, the Emperor <a href="http://en.wikipedia.org/wiki/Qin_Shi_Huang" title="Qin Shi Huang">Qin Shi Huang</a> (Shi Huang-ti) commanded all books in the Qin Empire other than officially sanctioned ones be burned. This decree was not universally obeyed, but as a consequence of this order little is known about ancient Chinese mathematics before this date. After the <a href="http://en.wikipedia.org/wiki/Burning_of_books_and_burying_of_scholars" title="Burning of books and burying of scholars">book burning</a> of 212 BC, the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Han_dynasty" title="Han dynasty">Han dynasty</a> (202 BC–220 AD) produced works of mathematics which presumably expanded on works that are now lost. The most important of these is <i><a href="http://en.wikipedia.org/wiki/The_Nine_Chapters_on_the_Mathematical_Art" title="The Nine Chapters on the Mathematical Art">The Nine Chapters on the Mathematical Art</a></i>, the full title of which appeared by AD 179, but existed in part under other titles beforehand. It consists of 246 word problems involving agriculture, business, employment of geometry to figure height spans and dimension ratios for <a href="http://en.wikipedia.org/wiki/Chinese_pagoda" title="Chinese pagoda">Chinese pagoda</a> towers, engineering, <a href="http://en.wikipedia.org/wiki/Surveying" title="Surveying">surveying</a>, and includes material on <a href="http://en.wikipedia.org/wiki/Right_triangle" title="Right triangle">right triangles</a> and values of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/%CE%A0" title="Π">π</a>.<sup class="reference" id="cite_ref-Boyer_1991_loc.3DChina_and_India_p._196_60-1"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-Boyer_1991_loc.3DChina_and_India_p._196-60"></a></sup><a href="http://en.wikipedia.org/wiki/Cavalieri%27s_principle" title="Cavalieri's principle">Cavalieri's principle</a> on volume more than a thousand years before Cavalieri would propose it in the West.<sup class="Template-Fact" style="white-space: nowrap;" title="This claim needs references to reliable sources from April 2010">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a></i>]</sup> It created mathematical proof for the <a href="http://en.wikipedia.org/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>, and a mathematical formula for <a href="http://en.wikipedia.org/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a>. <a href="http://en.wikipedia.org/wiki/Liu_Hui" title="Liu Hui">Liu Hui</a> commented on the work by the 3rd century AD, and gave a value of π accurate to 5 decimal places. Though more of a matter of computational stamina than theoretical insight, in the 5th century AD <a href="http://en.wikipedia.org/wiki/Zu_Chongzhi" title="Zu Chongzhi">Zu Chongzhi</a> computed the value of π to seven decimal places, which remained the most accurate value of π for almost the next 1000 years.<sup class="reference" id="cite_ref-Boyer_1991_loc.3DChina_and_India_p._202_64-0"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-Boyer_1991_loc.3DChina_and_India_p._202-64"></a></sup> It also made use of </div><div style="color: black;">The high water mark of Chinese mathematics occurs in the 13th century, with the development of Chinese algebra. The most important text from that period is the <i>Precious Mirror of the Four Elements</i> by Chu Shih-chieh (fl. 1280-1303), dealing with the solution of simultaneous higher order algebraic equations using a method similar to <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Horner%27s_method" title="Horner's method">Horner's method</a>.<sup class="reference" id="cite_ref-Boyer_1991_loc.3DChina_and_India_p._202_64-1"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-Boyer_1991_loc.3DChina_and_India_p._202-64"></a></sup> The <i>Precious Mirror</i> also contains a diagram of <a href="http://en.wikipedia.org/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a> with coefficients of binomial expansions through the eighth power, though both appear in Chinese works as early as 1100.<sup class="reference" id="cite_ref-Boyer_1991_loc.3DChina_and_India_p._205_65-0"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-Boyer_1991_loc.3DChina_and_India_p._205-65"></a></sup> The Chinese also made use of the complex combinatorial diagram known as the <a href="http://en.wikipedia.org/wiki/Magic_square" title="Magic square">magic square</a><a href="http://en.wikipedia.org/wiki/Magic_circle_%28mathematics%29" title="Magic circle (mathematics)">magic circles</a>, </div><br />
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</div><h2><span class="mw-headline" id="Indian_mathematics">Indian mathematics</span></h2><div class="rellink relarticle mainarticle" style="color: black;">Main article: <a href="http://en.wikipedia.org/wiki/Indian_mathematics" title="Indian mathematics">Indian mathematics</a></div><div class="rellink boilerplate seealso" style="color: black;">See also: <a class="mw-redirect" href="http://en.wikipedia.org/wiki/History_of_the_Hindu-Arabic_numeral_system" title="History of the Hindu-Arabic numeral system">History of the Hindu-Arabic numeral system</a></div><div class="thumb tright" style="color: black;"><div class="thumbinner" style="width: 352px;"><a class="image" href="http://en.wikipedia.org/wiki/File:Bakhshali_numerals_2.jpg"><img alt="" class="thumbimage" height="53" src="http://upload.wikimedia.org/wikipedia/commons/thumb/5/58/Bakhshali_numerals_2.jpg/350px-Bakhshali_numerals_2.jpg" width="350" /></a> <br />
<div class="thumbcaption"><div class="magnify"><a class="internal" href="http://en.wikipedia.org/wiki/File:Bakhshali_numerals_2.jpg" title="Enlarge"><img alt="" height="11" src="http://bits.wikimedia.org/skins-1.17/common/images/magnify-clip.png" width="15" /></a></div>The numerals used in the <a href="http://en.wikipedia.org/wiki/Bakhshali_manuscript" title="Bakhshali manuscript">Bakhshali manuscript</a>, dated between the 2nd century BCE and the 2nd century CE.</div></div></div><div class="thumb tright" style="color: black;"><div class="thumbinner" style="width: 345px;"><a class="image" href="http://en.wikipedia.org/wiki/File:Indian_numerals_100AD.svg"><img alt="" class="thumbimage" height="72" src="http://upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Indian_numerals_100AD.svg/343px-Indian_numerals_100AD.svg.png" width="343" /></a> <br />
<div class="thumbcaption"><a href="http://en.wikipedia.org/wiki/Brahmi_numeral" title="Brahmi numeral">Brahmi numerals</a> (lower row) in <a href="http://en.wikipedia.org/wiki/India" title="India">India</a> in the 1st century CE</div></div></div><div style="color: black;">The earliest civilization on the Indian subcontinent is the <a href="http://en.wikipedia.org/wiki/Indus_Valley_Civilization" title="Indus Valley Civilization">Indus Valley Civilization</a> that flourished between 2600 and 1900 BC in the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Indus_river" title="Indus river">Indus river</a> basin. Their cities were laid out with geometric regularity, but no known mathematical documents survive from this civilization.<sup class="reference" id="cite_ref-66"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-66"></a></sup></div><div style="color: black;">The oldest extant mathematical records from India are the <i><a href="http://en.wikipedia.org/wiki/Shatapatha_Brahmana" title="Shatapatha Brahmana">Shatapatha Brahmana</a></i> (c. 9th century BC but estimates of the date vary widely). The <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Sulba_Sutras" title="Sulba Sutras">Sulba Sutras</a> (c. 800 BC–200 AD),<sup class="reference" id="cite_ref-Boyer_1991_loc.3DChina_and_India_p._207_67-0"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-Boyer_1991_loc.3DChina_and_India_p._207-67">[68]</a></sup> appendices to religious texts which give simple rules for constructing altars of various shapes, such as squares, rectangles, parallelograms, and others.<sup class="reference" id="cite_ref-68"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-68"></a></sup> The Sulba Sutras give methods for constructing a <a href="http://en.wikipedia.org/wiki/Squaring_the_circle" title="Squaring the circle">circle with approximately the same area as a given square</a>, which imply several different approximations of the value of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/%CE%A0" title="Π">π</a>,<sup class="reference" id="cite_ref-69"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-69"></a></sup><sup class="reference" id="cite_ref-70"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-70"></a></sup> In addition, they compute the <a href="http://en.wikipedia.org/wiki/Square_root" title="Square root">square root</a> of 2 to several decimal places, list Pythagorean triples, and give a statement of the <a href="http://en.wikipedia.org/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>.<sup class="reference" id="cite_ref-71"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-71"></a></sup> Mesopotamian influence at this stage is considered likely.<sup class="reference" id="cite_ref-Boyer_1991_loc.3DChina_and_India_p._207_67-1"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-Boyer_1991_loc.3DChina_and_India_p._207-67"></a></sup></div><div style="color: black;"><span class="Unicode"><a href="http://en.wikipedia.org/wiki/P%C4%81%E1%B9%87ini" title="Pāṇini">Pāṇini</a></span> (c. 5th century BC) formulated the rules for <a href="http://en.wikipedia.org/wiki/Sanskrit_grammar" title="Sanskrit grammar">Sanskrit grammar</a>.<sup class="reference" id="cite_ref-72"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-72"></a></sup> His notation was similar to modern mathematical notation, and used metarules, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Transformation_%28geometry%29" title="Transformation (geometry)">transformations</a>, and <a href="http://en.wikipedia.org/wiki/Recursion" title="Recursion">recursion</a>. <a href="http://en.wikipedia.org/wiki/Pingala" title="Pingala">Pingala</a> (roughly 3rd-1st centuries BC) in his treatise of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Prosody_%28poetry%29" title="Prosody (poetry)">prosody</a> uses a device corresponding to a <a href="http://en.wikipedia.org/wiki/Binary_numeral_system" title="Binary numeral system">binary numeral system</a>. His discussion of the <a href="http://en.wikipedia.org/wiki/Combinatorics" title="Combinatorics">combinatorics</a> of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Metre_%28music%29" title="Metre (music)">meters</a> corresponds to an elementary version of the <a href="http://en.wikipedia.org/wiki/Binomial_theorem" title="Binomial theorem">binomial theorem</a>. Pingala's work also contains the basic ideas of <a href="http://en.wikipedia.org/wiki/Fibonacci_number" title="Fibonacci number">Fibonacci numbers</a> (called <i>mātrāmeru</i>).<sup class="reference" id="cite_ref-73"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-73">[74]</a></sup></div><div style="color: black;">The <i><a href="http://en.wikipedia.org/wiki/Surya_Siddhanta" title="Surya Siddhanta">Surya Siddhanta</a></i> (c. 400) introduced the <a href="http://en.wikipedia.org/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a> of <a href="http://en.wikipedia.org/wiki/Sine" title="Sine">sine</a>, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Cosine" title="Cosine">cosine</a>, and inverse sine, and laid down rules to determine the true motions of the luminaries, which conforms to their actual positions in the sky.<sup class="reference" id="cite_ref-74"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-74"></a></sup> This work was translated into to Arabic and Latin during the Middle Ages.</div><div class="thumb tright" style="color: black;"><div class="thumbinner" style="width: 222px;"><a class="image" href="http://en.wikipedia.org/wiki/File:2064_aryabhata-crp.jpg"><img alt="" class="thumbimage" height="315" src="http://upload.wikimedia.org/wikipedia/commons/thumb/a/af/2064_aryabhata-crp.jpg/220px-2064_aryabhata-crp.jpg" width="220" /></a> <br />
<div class="thumbcaption"><div class="magnify"><a class="internal" href="http://en.wikipedia.org/wiki/File:2064_aryabhata-crp.jpg" title="Enlarge"><img alt="" height="11" src="http://bits.wikimedia.org/skins-1.17/common/images/magnify-clip.png" width="15" /></a></div>Statue of <a href="http://en.wikipedia.org/wiki/Aryabhata" title="Aryabhata">Āryabhaṭa</a>((आर्यभट)), author of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Aryabhatiya" title="Aryabhatiya">Aryabhatiya</a>, on the grounds of <a href="http://en.wikipedia.org/wiki/Inter-University_Centre_for_Astronomy_and_Astrophysics" title="Inter-University Centre for Astronomy and Astrophysics">IUCAA</a>, <a href="http://en.wikipedia.org/wiki/Pune" title="Pune">Pune</a></div></div></div><div style="color: black;">In the 5th century AD, <a href="http://en.wikipedia.org/wiki/Aryabhata" title="Aryabhata">Aryabhata</a> wrote the <i><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Aryabhatiya" title="Aryabhatiya">Aryabhatiya</a></i>, a slim volume, written in verse, intended to supplement the rules of calculation used in astronomy and mathematical mensuration, though with no feeling for logic or deductive methodology.<sup class="reference" id="cite_ref-75"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-75"></a></sup> Though about half of the entries are wrong, it is in the <i>Aryabhatiya</i><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Islamic_mathematics" title="Islamic mathematics">Muslim mathematician</a> <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Abu_Rayhan_Biruni" title="Abu Rayhan Biruni">Abu Rayhan Biruni</a> described the <i>Aryabhatiya</i> as a "mix of common pebbles and costly crystals".<sup class="reference" id="cite_ref-76"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-76"></a></sup> that the decimal place-value system first appears. Several centuries later, the </div><div style="color: black;">In the 7th century, <a href="http://en.wikipedia.org/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a> identified the <a href="http://en.wikipedia.org/wiki/Brahmagupta_theorem" title="Brahmagupta theorem">Brahmagupta theorem</a>, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Brahmagupta%27s_identity" title="Brahmagupta's identity">Brahmagupta's identity</a> and <a href="http://en.wikipedia.org/wiki/Brahmagupta%27s_formula" title="Brahmagupta's formula">Brahmagupta's formula</a>, and for the first time, in <i><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Brahmasphutasiddhanta" title="Brahmasphutasiddhanta">Brahma-sphuta-siddhanta</a></i>, he lucidly explained the use of <a href="http://en.wikipedia.org/wiki/0_%28number%29" title="0 (number)">zero</a> as both a placeholder and <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Decimal_digit" title="Decimal digit">decimal digit</a>, and explained the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Hindu-Arabic_numeral_system" title="Hindu-Arabic numeral system">Hindu-Arabic numeral system</a>.<sup class="reference" id="cite_ref-Boyer_Siddhanta_77-0"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-Boyer_Siddhanta-77"></a></sup> It was from a translation of this Indian text on mathematics (c. 770) that Islamic mathematicians were introduced to this numeral system, which they adapted as <a href="http://en.wikipedia.org/wiki/Arabic_numerals" title="Arabic numerals">Arabic numerals</a>. Islamic scholars carried knowledge of this number system to Europe by the 12th century, and it has now displaced all older number systems throughout the world. In the 10th century, <a href="http://en.wikipedia.org/wiki/Halayudha" title="Halayudha">Halayudha</a>'s commentary on <a href="http://en.wikipedia.org/wiki/Pingala" title="Pingala">Pingala</a>'s work contains a study of the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Fibonacci_sequence" title="Fibonacci sequence">Fibonacci sequence</a> and <a href="http://en.wikipedia.org/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a>, and describes the formation of a <a href="http://en.wikipedia.org/wiki/Matrix_%28mathematics%29" title="Matrix (mathematics)">matrix</a>.<sup class="Template-Fact" style="white-space: nowrap;" title="This claim needs references to reliable sources from April 2010"><i><a href="http://en.wikipedia.org/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><br />
</a></i></sup></div><div style="color: black;">In the 12th century, <a href="http://en.wikipedia.org/wiki/Bh%C4%81skara_II" title="Bhāskara II">Bhāskara II</a><sup class="reference" id="cite_ref-78"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-78"></a></sup> lived in southern India and wrote extensively on all then known branches of mathematic. His work contains mathematical objects equivalent or approximately equivalent to infinitesimals, derivatives, <a href="http://en.wikipedia.org/wiki/Mean_value_theorem" title="Mean value theorem">the mean value theorem</a> and the derivative of the sine function. To what extent he anticipated the invention of calculus is a controversial subject among historians of mathematics<sup class="reference" id="cite_ref-79"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-79"></a></sup></div><div style="color: black;">In the 14th century, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Madhava_of_Sangamagrama" title="Madhava of Sangamagrama">Madhava of Sangamagrama</a>, the founder of the so-called <a href="http://en.wikipedia.org/wiki/Kerala_school_of_astronomy_and_mathematics" title="Kerala school of astronomy and mathematics">Kerala School of Mathematics</a>, found the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Leibniz_formula_for_pi" title="Leibniz formula for pi">Madhava–Leibniz series</a>, and, using 21 terms, computed the value of π as 3.14159265359. Madhava also found <a href="http://en.wikipedia.org/wiki/Gregory%27s_series" title="Gregory's series">the Madhava-Gregory series</a> to determine the arctangent, the Madhava-Newton power series to determine sine and cosine and <a href="http://en.wikipedia.org/wiki/Taylor_series" title="Taylor series">the Taylor approximation</a> for sine and cosine functions .<sup class="reference" id="cite_ref-80"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-80"></a></sup> In the 16th century, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Jyesthadeva" title="Jyesthadeva">Jyesthadeva</a> consolidated many of the Kerala School's developments and theorems in the <i>Yukti-bhāṣā</i>.<sup class="reference" id="cite_ref-81"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-81">[82]</a></sup> However, the Kerala School did not formulate a systematic theory of <a href="http://en.wikipedia.org/wiki/Derivative" title="Derivative">differentiation</a> and <a href="http://en.wikipedia.org/wiki/Integral" title="Integral">integration</a>, nor is there any direct evidence of their results being transmitted outside Kerala.<sup class="reference" id="cite_ref-82"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-82"></a></sup><sup class="reference" id="cite_ref-83"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-83"></a></sup><sup class="reference" id="cite_ref-84"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-84"></a></sup><sup class="reference" id="cite_ref-85"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-85"></a></sup><a href="http://en.wikipedia.org/wiki/Muslim_conquest_in_the_Indian_subcontinent" title="Muslim conquest in the Indian subcontinent">Muslim rule in India</a>.<sup class="reference" id="cite_ref-86"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-86"></a></sup><sup class="reference" id="cite_ref-87"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-87"></a></sup> Progress in mathematics along with other fields of science stagnated in India with the establishment of </div><div style="color: black;">== Islamic mathematics == wada abbbbba</div><div class="rellink relarticle mainarticle" style="color: black;">Main article: <a href="http://en.wikipedia.org/wiki/Mathematics_in_medieval_Islam" title="Mathematics in medieval Islam">Mathematics in medieval Islam</a></div><div class="rellink boilerplate seealso" style="color: black;">See also: <a class="mw-redirect" href="http://en.wikipedia.org/wiki/History_of_the_Hindu-Arabic_numeral_system" title="History of the Hindu-Arabic numeral system">History of the Hindu-Arabic numeral system</a></div><div class="thumb tright" style="color: black;"><div class="thumbinner" style="width: 222px;"><a class="image" href="http://en.wikipedia.org/wiki/File:Image-Al-Kit%C4%81b_al-mu%E1%B8%ABta%E1%B9%A3ar_f%C4%AB_%E1%B8%A5is%C4%81b_al-%C4%9Fabr_wa-l-muq%C4%81bala.jpg"><img alt="" class="thumbimage" height="348" src="http://upload.wikimedia.org/wikipedia/commons/thumb/2/23/Image-Al-Kit%C4%81b_al-mu%E1%B8%ABta%E1%B9%A3ar_f%C4%AB_%E1%B8%A5is%C4%81b_al-%C4%9Fabr_wa-l-muq%C4%81bala.jpg/220px-Image-Al-Kit%C4%81b_al-mu%E1%B8%ABta%E1%B9%A3ar_f%C4%AB_%E1%B8%A5is%C4%81b_al-%C4%9Fabr_wa-l-muq%C4%81bala.jpg" width="220" /></a> <br />
<div class="thumbcaption"><div class="magnify"><a class="internal" href="http://en.wikipedia.org/wiki/File:Image-Al-Kit%C4%81b_al-mu%E1%B8%ABta%E1%B9%A3ar_f%C4%AB_%E1%B8%A5is%C4%81b_al-%C4%9Fabr_wa-l-muq%C4%81bala.jpg" title="Enlarge"><img alt="" height="11" src="http://bits.wikimedia.org/skins-1.17/common/images/magnify-clip.png" width="15" /></a></div>Page from <i><a href="http://en.wikipedia.org/wiki/The_Compendious_Book_on_Calculation_by_Completion_and_Balancing" title="The Compendious Book on Calculation by Completion and Balancing">The Compendious Book on Calculation by Completion and Balancing</a></i> by <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Muhammad_ibn_M%C5%ABs%C4%81_al-Khw%C4%81rizm%C4%AB" title="Muhammad ibn Mūsā al-Khwārizmī">Muhammad ibn Mūsā al-Khwārizmī</a> (c. AD 820)</div></div></div><div style="color: black;">The <a href="http://en.wikipedia.org/wiki/Caliphate" title="Caliphate">Islamic Empire</a> established across <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Persia" title="Persia">Persia</a>, the <a href="http://en.wikipedia.org/wiki/Middle_East" title="Middle East">Middle East</a>, <a href="http://en.wikipedia.org/wiki/Central_Asia" title="Central Asia">Central Asia</a>, <a href="http://en.wikipedia.org/wiki/North_Africa" title="North Africa">North Africa</a>, <a href="http://en.wikipedia.org/wiki/Iberian_Peninsula" title="Iberian Peninsula">Iberia</a>, and in parts of <a href="http://en.wikipedia.org/wiki/History_of_India" title="History of India">India</a> in the 8th century made significant contributions towards mathematics. Although most Islamic texts on mathematics were written in <a href="http://en.wikipedia.org/wiki/Arabic_language" title="Arabic language">Arabic</a>, most of them were not written by <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Arab" title="Arab">Arabs</a>, since much like the status of Greek in the Hellenistic world, Arabic was used as the written language of non-Arab scholars throughout the Islamic world at the time. <a href="http://en.wikipedia.org/wiki/Persian_people" title="Persian people">Persians</a> contributed to the world of Mathematics alongside Arabs.</div><div style="color: black;">In the 9th century, the <a href="http://en.wikipedia.org/wiki/Persian_people" title="Persian people">Persian</a> mathematician <span class="Unicode"><a href="http://en.wikipedia.org/wiki/Mu%E1%B8%A5ammad_ibn_M%C5%ABs%C4%81_al-Khw%C4%81rizm%C4%AB" title="Muḥammad ibn Mūsā al-Khwārizmī">Muḥammad ibn Mūsā al-Khwārizmī</a></span> wrote several important books on the Hindu-Arabic numerals and on methods for solving equations. His book <i>On the Calculation with Hindu Numerals</i>, written about 825, along with the work of <a href="http://en.wikipedia.org/wiki/Al-Kindi" title="Al-Kindi">Al-Kindi</a>, were instrumental in spreading <a href="http://en.wikipedia.org/wiki/Indian_mathematics" title="Indian mathematics">Indian mathematics</a> and <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Hindu-Arabic_numeral_system" title="Hindu-Arabic numeral system">Indian numerals</a> to the West. The word <i><a href="http://en.wikipedia.org/wiki/Algorithm" title="Algorithm">algorithm</a></i> is derived from the Latinization of his name, Algoritmi, and the word <i><a href="http://en.wikipedia.org/wiki/Algebra" title="Algebra">algebra</a></i> from the title of one of his works, <i><a href="http://en.wikipedia.org/wiki/The_Compendious_Book_on_Calculation_by_Completion_and_Balancing" title="The Compendious Book on Calculation by Completion and Balancing">Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala</a></i> (<i>The Compendious Book on Calculation by Completion and Balancing</i>). He gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,<sup class="reference" id="cite_ref-88"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-88"></a></sup> and he was the first to teach algebra in an <a href="http://en.wikipedia.org/wiki/Elementary_algebra" title="Elementary algebra">elementary form</a> and for its own sake.<sup class="reference" id="cite_ref-89"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-89"></a></sup> He also discussed the fundamental method of "<a href="http://en.wikipedia.org/wiki/Reduction_%28mathematics%29" title="Reduction (mathematics)">reduction</a>" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which al-Khwārizmī originally described as <i>al-jabr</i>.<sup class="reference" id="cite_ref-Boyer-229_90-0"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-Boyer-229-90"></a></sup> His algebra was also no longer concerned "with a series of <a href="http://en.wikipedia.org/wiki/Problem" title="Problem">problems</a> to be resolved, but an <a href="http://en.wikipedia.org/wiki/Expository_writing" title="Expository writing">exposition</a> which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."<sup class="reference" id="cite_ref-Rashed-Armstrong_91-0"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-Rashed-Armstrong-91"></a></sup></div><div style="color: black;">Further developments in algebra were made by <a href="http://en.wikipedia.org/wiki/Al-Karaji" title="Al-Karaji">Al-Karaji</a> in his treatise <i>al-Fakhri</i>, where he extends the methodology to incorporate integer powers and integer roots of unknown quantities. Something close to a <a href="http://en.wikipedia.org/wiki/Mathematical_proof" title="Mathematical proof">proof</a> by <a href="http://en.wikipedia.org/wiki/Mathematical_induction" title="Mathematical induction">mathematical induction</a> appears in a book written by Al-Karaji around 1000 AD, who used it to prove the <a href="http://en.wikipedia.org/wiki/Binomial_theorem" title="Binomial theorem">binomial theorem</a>, <a href="http://en.wikipedia.org/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a>, and the sum of <a href="http://en.wikipedia.org/wiki/Integral" title="Integral">integral</a> <a href="http://en.wikipedia.org/wiki/Cube_%28algebra%29" title="Cube (algebra)">cubes</a>.<sup class="reference" id="cite_ref-92"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-92"></a></sup> The <a href="http://en.wikipedia.org/wiki/Historian" title="Historian">historian</a> of mathematics, F. Woepcke,<sup class="reference" id="cite_ref-93"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-93"></a></sup> praised Al-Karaji for being "the first who introduced the <a href="http://en.wikipedia.org/wiki/Theory" title="Theory">theory</a> of <a href="http://en.wikipedia.org/wiki/Algebra" title="Algebra">algebraic</a> <a href="http://en.wikipedia.org/wiki/Calculus" title="Calculus">calculus</a>." Also in the 10th century, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Abul_Wafa" title="Abul Wafa">Abul Wafa</a> translated the works of <a href="http://en.wikipedia.org/wiki/Diophantus" title="Diophantus">Diophantus</a> into Arabic. <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Ibn_al-Haytham" title="Ibn al-Haytham">Ibn al-Haytham</a> was the first mathematician to derive the formula for the sum of the fourth powers, using a method that is readily generalizable for determining the general formula for the sum of any integral powers. He performed an integration in order to find the volume of a <a href="http://en.wikipedia.org/wiki/Paraboloid" title="Paraboloid">paraboloid</a>, and was able to generalize his result for the integrals of <a href="http://en.wikipedia.org/wiki/Polynomial" title="Polynomial">polynomials</a> up to the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Quartic_polynomial" title="Quartic polynomial">fourth degree</a>. He thus came close to finding a general formula for the <a href="http://en.wikipedia.org/wiki/Integral" title="Integral">integrals</a> of polynomials, but he was not concerned with any polynomials higher than the fourth degree.<sup class="reference" id="cite_ref-Katz_94-0"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-Katz-94"></a></sup></div><div style="color: black;">In the late 11th century, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Omar_Khayyam" title="Omar Khayyam">Omar Khayyam</a> wrote <i>Discussions of the Difficulties in Euclid</i>, a book about what he perceived as flaws in <a href="http://en.wikipedia.org/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's <i>Elements</i></a>, especially the <a href="http://en.wikipedia.org/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a>. He was also the first to find the general geometric solution to <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Cubic_equation" title="Cubic equation">cubic equations</a>. He was also very influential in <a href="http://en.wikipedia.org/wiki/Calendar_reform" title="Calendar reform">calendar reform</a>.<sup class="Template-Fact" style="white-space: nowrap;" title="This claim needs references to reliable sources from March 2009"><i><a href="http://en.wikipedia.org/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><br />
</a></i></sup></div><div style="color: black;">In the 13th century, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Nasir_al-Din_Tusi" title="Nasir al-Din Tusi">Nasir al-Din Tusi</a> (Nasireddin) made advances in <a href="http://en.wikipedia.org/wiki/Spherical_trigonometry" title="Spherical trigonometry">spherical trigonometry</a>. He also wrote influential work on <a href="http://en.wikipedia.org/wiki/Euclid" title="Euclid">Euclid</a>'s <a href="http://en.wikipedia.org/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a>. In the 15th century, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Ghiyath_al-Kashi" title="Ghiyath al-Kashi">Ghiyath al-Kashi</a> computed the value of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/%CE%A0" title="Π">π</a><i>n</i>th roots, which was a special case of the methods given many centuries later by <a href="http://en.wikipedia.org/wiki/Paolo_Ruffini" title="Paolo Ruffini">Ruffini</a> and <a href="http://en.wikipedia.org/wiki/William_George_Horner" title="William George Horner">Horner</a>. to the 16th decimal place. Kashi also had an algorithm for calculating </div><div style="color: black;">Other achievements of Muslim mathematicians during this period include the addition of the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Decimal_point" title="Decimal point">decimal point</a><a href="http://en.wikipedia.org/wiki/Arabic_numerals" title="Arabic numerals">Arabic numerals</a>, the discovery of all the modern <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Trigonometric_function" title="Trigonometric function">trigonometric functions</a> besides the sine, <a href="http://en.wikipedia.org/wiki/Al-Kindi" title="Al-Kindi">al-Kindi</a>'s introduction of <a href="http://en.wikipedia.org/wiki/Cryptanalysis" title="Cryptanalysis">cryptanalysis</a> and <a href="http://en.wikipedia.org/wiki/Frequency_analysis" title="Frequency analysis">frequency analysis</a>, the development of <a href="http://en.wikipedia.org/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a> by <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Ibn_al-Haytham" title="Ibn al-Haytham">Ibn al-Haytham</a>, the beginning of <a href="http://en.wikipedia.org/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> by <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Omar_Khayyam" title="Omar Khayyam">Omar Khayyam</a> and the development of an <a href="http://en.wikipedia.org/wiki/Mathematical_notation" title="Mathematical notation">algebraic notation</a> by <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Ab%C5%AB_al-Hasan_ibn_Al%C4%AB_al-Qalas%C4%81d%C4%AB" title="Abū al-Hasan ibn Alī al-Qalasādī">al-Qalasādī</a>.<sup class="reference" id="cite_ref-Qalasadi_95-0"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-Qalasadi-95"></a></sup> notation to the </div><div style="color: black;">During the time of the <a href="http://en.wikipedia.org/wiki/Ottoman_Empire" title="Ottoman Empire">Ottoman Empire</a> and <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Safavid_Empire" title="Safavid Empire">Safavid Empire</a> from the 15th century, the development of Islamic mathematics became stagnant</div><div style="color: black;"></div><div style="color: black;"></div><div style="color: black;"></div><div style="color: black;"></div><div style="color: black;"></div><div style="color: black;"></div><div style="color: black;"></div><div style="color: black;"></div><div style="color: black;"></div><div style="color: black;"></div><div style="color: black;"></div><div style="color: black;"></div><div style="color: black;"></div><div style="color: black;"><br />
</div><h2><span class="mw-headline" id="Medieval_European_mathematics">Medieval European mathematics</span></h2><div style="color: black;">Medieval European interest in mathematics was driven by concerns quite different from those of modern mathematicians. One driving element was the belief that mathematics provided the key to understanding the created order of nature, frequently justified by <a href="http://en.wikipedia.org/wiki/Plato" title="Plato">Plato</a>'s <i><a href="http://en.wikipedia.org/wiki/Timaeus_%28dialogue%29" title="Timaeus (dialogue)">Timaeus</a></i> and the biblical passage (in the <i><a href="http://en.wikipedia.org/wiki/Book_of_Wisdom" title="Book of Wisdom">Book of Wisdom</a></i>) that God had <i>ordered all things in measure, and number, and weight</i>.<sup class="reference" id="cite_ref-96"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-96"></a></sup></div><div style="color: black;"><a href="http://en.wikipedia.org/wiki/Boethius" title="Boethius">Boethius</a> provided a place for mathematics in the curriculum in the 6th century when he coined the term <i><a href="http://en.wikipedia.org/wiki/Quadrivium" title="Quadrivium">quadrivium</a></i> to describe the study of arithmetic, geometry, astronomy, and music. He wrote <i>De institutione arithmetica</i>, a free translation from the Greek of <a href="http://en.wikipedia.org/wiki/Nicomachus" title="Nicomachus">Nicomachus</a>'s <i>Introduction to Arithmetic</i>; <i>De institutione musica</i>, also derived from Greek sources; and a series of excerpts from <a href="http://en.wikipedia.org/wiki/Euclid" title="Euclid">Euclid</a>'s <a href="http://en.wikipedia.org/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a>. His works were theoretical, rather than practical, and were the basis of mathematical study until the recovery of Greek and Arabic mathematical works.<sup class="reference" id="cite_ref-97"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-97"></a></sup><sup class="reference" id="cite_ref-98"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-98"></a></sup></div><div style="color: black;">In the 12th century, European scholars traveled to Spain and Sicily <a href="http://en.wikipedia.org/wiki/Latin_translations_of_the_12th_century" title="Latin translations of the 12th century">seeking scientific Arabic texts</a>, including <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Al-Khw%C4%81rizm%C4%AB" title="Al-Khwārizmī">al-Khwārizmī</a>'s <i><a href="http://en.wikipedia.org/wiki/The_Compendious_Book_on_Calculation_by_Completion_and_Balancing" title="The Compendious Book on Calculation by Completion and Balancing">The Compendious Book on Calculation by Completion and Balancing</a></i>, translated into Latin by <a href="http://en.wikipedia.org/wiki/Robert_of_Chester" title="Robert of Chester">Robert of Chester</a>, and the complete text of <a href="http://en.wikipedia.org/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's <i>Elements</i></a>, translated in various versions by <a href="http://en.wikipedia.org/wiki/Adelard_of_Bath" title="Adelard of Bath">Adelard of Bath</a>, <a href="http://en.wikipedia.org/wiki/Herman_of_Carinthia" title="Herman of Carinthia">Herman of Carinthia</a>, and <a href="http://en.wikipedia.org/wiki/Gerard_of_Cremona" title="Gerard of Cremona">Gerard of Cremona</a>.<sup class="reference" id="cite_ref-99"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-99">[100]</a></sup><sup class="reference" id="cite_ref-100"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-100">[101]</a></sup></div><div class="rellink boilerplate seealso" style="color: black;">See also: <a href="http://en.wikipedia.org/wiki/Latin_translations_of_the_12th_century" title="Latin translations of the 12th century">Latin translations of the 12th century</a></div><div style="color: black;">These new sources sparked a renewal of mathematics. <a href="http://en.wikipedia.org/wiki/Fibonacci" title="Fibonacci">Fibonacci</a>, writing in the <i><a href="http://en.wikipedia.org/wiki/Liber_Abaci" title="Liber Abaci">Liber Abaci</a></i>, in 1202 and updated in 1254, produced the first significant mathematics in Europe since the time of <a href="http://en.wikipedia.org/wiki/Eratosthenes" title="Eratosthenes">Eratosthenes</a>, a gap of more than a thousand years. The work introduced <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Hindu-Arabic_numerals" title="Hindu-Arabic numerals">Hindu-Arabic numerals</a> to Europe, and discussed many other mathematical problems.</div><div style="color: black;">The 14th century saw the development of new mathematical concepts to investigate a wide range of problems.<sup class="reference" id="cite_ref-101"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-101"></a></sup> One important contribution was development of mathematics of local motion.</div><div style="color: black;"><a href="http://en.wikipedia.org/wiki/Thomas_Bradwardine" title="Thomas Bradwardine">Thomas Bradwardine</a> proposed that speed (V) increases in arithmetic proportion as the ratio of force (F) to resistance (R) increases in geometric proportion. Bradwardine expressed this by a series of specific examples, but although the logarithm had not yet been conceived, we can express his conclusion anachronistically by writing: V = log (F/R).<sup class="reference" id="cite_ref-102"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-102"></a></sup> Bradwardine's analysis is an example of transferring a mathematical technique used by <a href="http://en.wikipedia.org/wiki/Al-Kindi" title="Al-Kindi">al-Kindi</a> and <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Arnald_of_Villanova" title="Arnald of Villanova">Arnald of Villanova</a> to quantify the nature of compound medicines to a different physical problem.<sup class="reference" id="cite_ref-103"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-103"></a></sup></div><div style="color: black;">One of the 14th-century <a href="http://en.wikipedia.org/wiki/Oxford_Calculators" title="Oxford Calculators">Oxford Calculators</a>, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/William_Heytesbury" title="William Heytesbury">William Heytesbury</a>, lacking <a href="http://en.wikipedia.org/wiki/Differential_calculus" title="Differential calculus">differential calculus</a> and the concept of <a href="http://en.wikipedia.org/wiki/Limit_of_a_function" title="Limit of a function">limits</a>, proposed to measure instantaneous speed "by the path that would be described by [a body] if... it were moved uniformly at the same degree of speed with which it is moved in that given instant".<sup class="reference" id="cite_ref-104"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-104">[105]</a></sup></div><div style="color: black;">Heytesbury and others mathematically determined the distance covered by a body undergoing uniformly accelerated motion (today solved by <a href="http://en.wikipedia.org/wiki/Integral" title="Integral">integration</a>), stating that "a moving body uniformly acquiring or losing that increment [of speed] will traverse in some given time a [distance] completely equal to that which it would traverse if it were moving continuously through the same time with the mean degree [of speed]".<sup class="reference" id="cite_ref-105"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-105">[106]</a></sup></div><div style="color: black;"><a href="http://en.wikipedia.org/wiki/Nicole_Oresme" title="Nicole Oresme">Nicole Oresme</a> at the <a href="http://en.wikipedia.org/wiki/University_of_Paris" title="University of Paris">University of Paris</a> and the Italian <a href="http://en.wikipedia.org/wiki/Giovanni_di_Casali" title="Giovanni di Casali">Giovanni di Casali</a> independently provided graphical demonstrations of this relationship, asserting that the area under the line depicting the constant acceleration, represented the total distance traveled.<sup class="reference" id="cite_ref-106"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-106">[107]</a></sup> In a later mathematical commentary on Euclid's <i>Elements</i>, Oresme made a more detailed general analysis in which he demonstrated that a body will acquire in each successive increment of time an increment of any quality that increases as the odd numbers. Since Euclid had demonstrated the sum of the odd numbers are the square numbers, the total quality acquired by the body increases as the square of the time.<sup class="reference" id="cite_ref-107"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-107"></a></sup></div><h2 style="color: black; font-weight: normal;"><span class="editsection"></span> <span class="mw-headline" id="Renaissance_mathematics">Renaissance mathematics</span></h2><div class="thumb tright" style="color: black;"><div class="thumbinner" style="width: 252px;"><a class="image" href="http://en.wikipedia.org/wiki/File:Pacioli.jpg"><img alt="" class="thumbimage" height="209" src="http://upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Pacioli.jpg/250px-Pacioli.jpg" width="250" /></a> <br />
<div class="thumbcaption"><div class="magnify"><a class="internal" href="http://en.wikipedia.org/wiki/File:Pacioli.jpg" title="Enlarge"><img alt="" height="11" src="http://bits.wikimedia.org/skins-1.17/common/images/magnify-clip.png" width="15" /></a></div><i><a href="http://en.wikipedia.org/wiki/Portrait_of_Luca_Pacioli" title="Portrait of Luca Pacioli">Portrait of Luca Pacioli</a></i>, a painting traditionally attributed to <a href="http://en.wikipedia.org/wiki/Jacopo_de%27_Barbari" title="Jacopo de' Barbari">Jacopo de' Barbari</a>, 1495, (<a class="mw-redirect" href="http://en.wikipedia.org/wiki/Museo_di_Capodimonte" title="Museo di Capodimonte">Museo di Capodimonte</a>).</div></div></div><div style="color: black;">During the <a href="http://en.wikipedia.org/wiki/Renaissance" title="Renaissance">Renaissance</a>, the development of mathematics and of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Accounting" title="Accounting">accounting</a> were intertwined.<sup class="reference" id="cite_ref-108"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-108"></a></sup> While there is no direct relationship between algebra and accounting, the teaching of the subjects and the books published often intended for the children of merchants who were sent to reckoning schools (in <a href="http://en.wikipedia.org/wiki/Flanders" title="Flanders">Flanders</a> and <a href="http://en.wikipedia.org/wiki/Germany" title="Germany">Germany</a>) or <a href="http://en.wikipedia.org/wiki/Abacus_school" title="Abacus school">abacus schools</a> (known as <i>abbaco</i> in Italy), where they learned the skills useful for trade and commerce. There is probably no need for algebra in performing <a href="http://en.wikipedia.org/wiki/Bookkeeping" title="Bookkeeping">bookkeeping</a> operations, but for complex bartering operations or the calculation of <a href="http://en.wikipedia.org/wiki/Compound_interest" title="Compound interest">compound interest</a>, a basic knowledge of arithmetic was mandatory and knowledge of algebra was very useful.</div><div style="color: black;"><a href="http://en.wikipedia.org/wiki/Luca_Pacioli" title="Luca Pacioli">Luca Pacioli</a>'s <i>"Summa de Arithmetica, Geometria, Proportioni et Proportionalità"</i> (Italian: "Review of <a href="http://en.wikipedia.org/wiki/Arithmetic" title="Arithmetic">Arithmetic</a>, <a href="http://en.wikipedia.org/wiki/Geometry" title="Geometry">Geometry</a>, <a href="http://en.wikipedia.org/wiki/Ratio" title="Ratio">Ratio</a> and <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Proportion" title="Proportion">Proportion</a>") was first printed and published in <a href="http://en.wikipedia.org/wiki/Venice" title="Venice">Venice</a> in 1494. It included a 27-page <a href="http://en.wikipedia.org/wiki/Treatise" title="Treatise">treatise</a> on <a href="http://en.wikipedia.org/wiki/Bookkeeping" title="Bookkeeping">bookkeeping</a>, <i>"Particularis de Computis et Scripturis"</i> (Italian: "Details of Calculation and Recording"). It was written primarily for, and sold mainly to, merchants who used the book as a reference text, as a source of pleasure from the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Mathematical_puzzles" title="Mathematical puzzles">mathematical puzzles</a> it contained, and to aid the education of their sons.<sup class="reference" id="cite_ref-109"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-109"></a></sup> In <i>Summa Arithmetica</i>, Pacioli introduced symbols for <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Plus_and_minus" title="Plus and minus">plus and minus</a> for the first time in a printed book, symbols that became standard notation in Italian Renaissance mathematics. <i>Summa Arithmetica</i> was also the first known book printed in Italy to contain <a href="http://en.wikipedia.org/wiki/Algebra" title="Algebra">algebra</a>. It is important to note that Pacioli himself had borrowed much of the work of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Piero_Della_Francesca" title="Piero Della Francesca">Piero Della Francesca</a> whom he plagiarized.</div><div style="color: black;">In Italy, during the first half of the 16th century, <a href="http://en.wikipedia.org/wiki/Scipione_del_Ferro" title="Scipione del Ferro">Scipione del Ferro</a> and <a href="http://en.wikipedia.org/wiki/Niccol%C3%B2_Fontana_Tartaglia" title="Niccolò Fontana Tartaglia">Niccolò Fontana Tartaglia</a> discovered solutions for <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Cubic_equations" title="Cubic equations">cubic equations</a>. <a href="http://en.wikipedia.org/wiki/Gerolamo_Cardano" title="Gerolamo Cardano">Gerolamo Cardano</a> published them in his 1545 book <i><a href="http://en.wikipedia.org/wiki/Ars_Magna_%28Gerolamo_Cardano%29" title="Ars Magna (Gerolamo Cardano)">Ars Magna</a></i>, together with a solution for the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Quartic_equations" title="Quartic equations">quartic equations</a>, discovered by his student <a href="http://en.wikipedia.org/wiki/Lodovico_Ferrari" title="Lodovico Ferrari">Lodovico Ferrari</a>. In 1572 <a href="http://en.wikipedia.org/wiki/Rafael_Bombelli" title="Rafael Bombelli">Rafael Bombelli</a><i>L'Algebra</i> in which he showed how to deal with the <a href="http://en.wikipedia.org/wiki/Imaginary_number" title="Imaginary number">imaginary quantities</a> that could appear in Cardano's formula for solving cubic equations. published his </div><div style="color: black;"><a href="http://en.wikipedia.org/wiki/Simon_Stevin" title="Simon Stevin">Simon Stevin</a>'s book <i>De Thiende</i> ('the art of tenths'), first published in Dutch in 1585, contained the first systematic treatment of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Decimal_notation" title="Decimal notation">decimal notation</a>, which influenced all later work on the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Real_number_system" title="Real number system">real number system</a>.</div><div style="color: black;">Driven by the demands of navigation and the growing need for accurate maps of large areas, <a href="http://en.wikipedia.org/wiki/Trigonometry" title="Trigonometry">trigonometry</a><a href="http://en.wikipedia.org/wiki/Bartholomaeus_Pitiscus" title="Bartholomaeus Pitiscus">Bartholomaeus Pitiscus</a> was the first to use the word, publishing his <i>Trigonometria</i> in 1595. Regiomontanus's table of sines and cosines was published in 1533.<sup class="reference" id="cite_ref-110"><a href="http://en.wikipedia.org/wiki/History_of_mathematics#cite_note-110"></a></sup> grew to be a major branch of mathematics. </div><div style="color: black;"><br />
By : Tharshini Devi Krishna Moorthy</div>4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com0tag:blogger.com,1999:blog-8948305359118541243.post-22803196706122544282011-07-20T21:57:00.000+08:002011-07-23T18:08:13.360+08:00The Greatest Mathematician<table cellspacing="0" style="margin-top: 5px;"><span style="background-color: white; color: red;"> </span><tbody><span style="background-color: white; color: red;"> </span>
<tr><span style="background-color: white; color: red;"> </span><td class="sqtdq" colspan="2"><span class="sqq"><span style="background-color: white; color: red;"></span></span></td></tr>
<tr> <td colspan="2"> <div style="padding-top: 3px;"><span class="sqb"><span style="color: black;"> </span></span></div><span class="sqb"><span style="color: black;"><table><tbody>
<tr> <td align="center"><img border="0" src="http://fabpedigree.com/james/ce1a.gif" /> <br />
<a href="http://fabpedigree.com/james/mathmen.htm#Newton">Isaac Newton</a> </td><td align="center"><img border="0" src="http://fabpedigree.com/james/ce2a.gif" /> <br />
<a href="http://fabpedigree.com/james/mathmen.htm#Gauss">Carl Gauss</a> </td><td align="center"><img border="0" src="http://fabpedigree.com/james/ce4a.gif" /> <br />
<a href="http://fabpedigree.com/james/mathmen.htm#Euler">Leonhard Euler</a> </td><td align="center"><img border="0" src="http://fabpedigree.com/james/ce3a.gif" /> <br />
<a href="http://fabpedigree.com/james/mathmen.htm#Archimedes">Archimedes</a> </td><td align="center"><img border="0" src="http://fabpedigree.com/james/ce6a.gif" /> <br />
<a href="http://fabpedigree.com/james/mathmen.htm#Riemann">Bernhard Riemann</a> </td><td align="center"><img border="0" src="http://fabpedigree.com/james/ce5a.gif" /> <br />
<a href="http://fabpedigree.com/james/mathmen.htm#Euclid">Euclid</a> </td></tr>
<tr> <td colspan="6"> </td></tr>
<tr> <td align="center"><img border="0" src="http://fabpedigree.com/james/ce7a.gif" /> <br />
<a href="http://fabpedigree.com/james/mathmen.htm#Poincare">Henri Poincaré</a> </td><td align="center"><img border="0" src="http://fabpedigree.com/james/ce9a.gif" /> <br />
<a href="http://fabpedigree.com/james/mathmen.htm#Lagrange">J.-L. Lagrange</a> </td><td align="center"><img border="0" src="http://fabpedigree.com/james/ce8a.gif" /> <br />
<a href="http://fabpedigree.com/james/mathmen.htm#Hilbert">David Hilbert</a> </td><td align="center"><img border="0" src="http://fabpedigree.com/james/celeib.gif" /> <br />
<a href="http://fabpedigree.com/james/mathmen.htm#Leibniz">G.W. Leibniz</a> </td><td align="center"><img border="0" src="http://fabpedigree.com/james/cegro.gif" /> <br />
<a href="http://fabpedigree.com/james/mathmen.htm#Grothendieck">Alex. Grothendieck</a> </td><td align="center"><img border="0" src="http://fabpedigree.com/james/ce10a.gif" /> <br />
<a href="http://fabpedigree.com/james/mathmen.htm#Fermat">Pierre de Fermat</a> </td></tr>
</tbody></table><div style="padding-top: 3px;"><br />
</div><hr /><div style="padding-top: 3px;"> </div><b><span style="color: #300080;"><a href="" name="top"></a>The Greatest Mathematicians of All Time</span></b> <br />
ranked in approximate order of "greatness." <br />
To qualify, the mathematician <span>must be born before 1930</span> and his work must have <br />
<span style="color: #300080;"><b>breadth</b></span>, <span style="color: #300080;"><b>depth</b></span>, and <span style="color: #300080;"><b>historical importance</b></span>. <br />
<br />
<br />
<table><tbody>
<tr> <td><span style="font-size: large;"> <ol><li><a href="http://fabpedigree.com/james/mathmen.htm#Newton">Isaac Newton</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Gauss">Carl F. Gauss</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Euler">Leonhard Euler</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Archimedes">Archimedes</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Riemann">Bernhard Riemann</a> </li>
</ol></span> </td><td> </td><td><span style="font-size: large;"> <ol start="6"><li><a href="http://fabpedigree.com/james/mathmen.htm#Euclid">Euclid</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Poincare">Henri Poincaré</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Lagrange">Joseph-Louis Lagrange</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Hilbert">David Hilbert</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Leibniz">Gottfried W. Leibniz</a> </li>
</ol></span> </td><td> </td><td><span style="font-size: large;"> <ol start="11"><li><a href="http://fabpedigree.com/james/mathmen.htm#Grothendieck">Alexandre Grothendieck</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Fermat">Pierre de Fermat</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Abel">Niels Abel</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Galois">Évariste Galois</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Neumann">John von Neumann</a> </li>
</ol></span> </td></tr>
<tr> <td colspan="5"> <hr width="80%" /> </td></tr>
<tr> <td><span style="font-size: large;"> <ol start="16"><li><a href="http://fabpedigree.com/james/mathmen.htm#Weier">Karl W. T. Weierstrass</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Descartes">René Déscartes</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Cauchy">Augustin Cauchy</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Jacobi">Carl G. J. Jacobi</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Ramanujan">Srinivasa Ramanujan</a> </li>
</ol></span> </td><td> </td><td><span style="font-size: large;"> <ol start="21"><li><a href="http://fabpedigree.com/james/mathmen.htm#Brahmagupta">Brahmagupta</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Dirichlet">Peter G. L. Dirichlet</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Weyl">Hermann K. H. Weyl</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Eudoxus">Eudoxus</a> of Cnidus </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Cantor">Georg Cantor</a> </li>
</ol></span> </td><td> </td><td><span style="font-size: large;"> <ol start="26"><li><a href="http://fabpedigree.com/james/mathmen.htm#Khowarizmi">Muhammed al-Khowârizmi</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Cayley">Arthur Cayley</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Noether">Emma Noether</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Pythagoras">Pythagoras</a> of Samos </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Fibonacci">Leonardo `Fibonacci'</a> </li>
</ol></span> </td></tr>
<tr> <td colspan="5"> <hr width="80%" /> <span style="font-size: small;"> </span> </td></tr>
<tr> <td><span style="font-size: large;"> <ol start="31"><li><a href="http://fabpedigree.com/james/mathmen.htm#Godel">Kurt Gödel</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Apollo">Apollonius</a> of Perga </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Aryabhatta">Aryabhatta</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Hermite">Charles Hermite</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Pascal">Blaise Pascal</a> </li>
</ol></span> </td><td> </td><td><span style="font-size: large;"> <ol start="36"><li><a href="http://fabpedigree.com/james/mathmen.htm#Laplace">Pierre-Simon Laplace</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Dedek">Richard Dedekind</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Diophantus">Diophantus</a> of Alexandria </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Klein">Felix Christian Klein</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Hamilton">William R. Hamilton</a> </li>
</ol></span> </td><td> </td><td><span style="font-size: large;"> <ol start="41"><li><a href="http://fabpedigree.com/james/mathmen.htm#Bhaskara">Bháscara</a> Áchárya </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Boole">George Boole</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Vieta">François Viète</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Banach">Stefan Banach</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Eisen">Ferdinand Eisenstein</a> </li>
</ol></span> </td></tr>
<tr> <td colspan="5"> <hr width="80%" /> </td></tr>
<tr> <td><span style="font-size: large;"> <ol start="46"><li><a href="http://fabpedigree.com/james/mathmen.htm#Alembert">Jean le Rond d'Alembert</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Hadam">Jacques Hadamard</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Monge">Gaspard Monge</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Kepler">Johannes Kepler</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Bernou">Jacob Bernoulli</a> </li>
</ol></span> </td><td> </td><td><span style="font-size: large;"> <ol start="51"><li><a href="http://fabpedigree.com/james/mathmen.htm#Cartan">Élie Cartan</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Fourier">Joseph Fourier</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Liu">Liu Hui</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Poncelet">Jean-Victor Poncelet</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Archytas">Archytas</a> of Tarentum </li>
</ol></span> </td><td> </td><td><span style="font-size: large;"> <ol start="56"><li><a href="http://fabpedigree.com/james/mathmen.htm#Weil">André Weil</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Hippar">Hipparchus</a> of Nicaea </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Hardy">Godfrey H. Hardy</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Poisson">Siméon-Denis Poisson</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Plucker">Julius Plücker</a> </li>
</ol></span> </td></tr>
<tr> <td colspan="5"> <hr width="80%" /> </td></tr>
<tr> <td><span style="font-size: large;"> <ol start="61"><li><a href="http://fabpedigree.com/james/mathmen.htm#Borel">F.E.J. Émile Borel</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Kolmogorov">Andrey N. Kolmogorov</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Huygens">Christiaan Huygens</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Alhazen">Alhazen ibn al-Haytham</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Wallis">John Wallis</a> </li>
</ol></span> </td><td> </td><td><span style="font-size: large;"> <ol start="66"><li><a href="http://fabpedigree.com/james/mathmen.htm#Brouwer">L.E.J. Brouwer</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Atiyah">Michael F. Atiyah</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Pappus">Pappus</a> of Alexandria </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Jordan">M. E. Camille Jordan</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Lebesgue">Henri Léon Lebesgue</a> </li>
</ol></span> </td><td> </td><td><span style="font-size: large;"> <ol start="71"><li><a href="http://fabpedigree.com/james/mathmen.htm#Littlew">John E. Littlewood</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Siegel">Carl Ludwig Siegel</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Steiner">Jakob Steiner</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Liouville">Joseph Liouville</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Cardano">Girolamo Cardano</a> </li>
</ol></span> </td></tr>
<tr> <td colspan="5"> <hr width="80%" /> </td></tr>
<tr> <td><span style="font-size: large;"> <ol start="76"><li><a href="http://fabpedigree.com/james/mathmen.htm#Grassm">Hermann G. Grassmann</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Serre">Jean-Pierre Serre</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Legendre">Adrien M. Legendre</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Peano">Giuseppe Peano</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Sylvester">James J. Sylvester</a> </li>
</ol></span> </td><td> </td><td><span style="font-size: large;"> <ol start="81"><li><a href="http://fabpedigree.com/james/mathmen.htm#Jbernou">Johann Bernoulli</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Selberg">Atle Selberg</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Cheby">Pafnuti Chebyshev</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Chern">Shiing-Shen Chern</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Caval">Francesco B. Cavalieri</a> </li>
</ol></span> </td><td> </td><td><span style="font-size: large;"> <ol start="86"><li><a href="http://fabpedigree.com/james/mathmen.htm#Khayyam">Omar al-Khayyám</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Minkow">Hermann Minkowski</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Hippoc">Hippocrates</a> of Chios </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Lambert">Johann H. Lambert</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Thabit">Thabit ibn Qurra</a> </li>
</ol></span> </td></tr>
<tr> <td colspan="5"> <hr width="80%" /> </td></tr>
<tr> <td><span style="font-size: large;"> <ol start="91"><li><a href="http://fabpedigree.com/james/mathmen.htm#Lie">Marius Sophus Lie</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Panini">Panini</a> (of Shalatula) </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Turing">Alan M. Turing</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Thales">Thales</a> of Miletus </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Erdos">Paul Erdös</a> </li>
</ol></span> </td><td> </td><td><span style="font-size: large;"> <ol start="96"><li><a href="http://fabpedigree.com/james/mathmen.htm#Kummer">Ernst E. Kummer</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Artin">Emil Artin</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Lobac">Nicolai Lobachevsky</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Clairaut">Alexis C. Clairaut</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Darboux">Jean Gaston Darboux</a> </li>
</ol></span> </td><td> </td><td><span style="font-size: large;"> <ul><li><a href="http://fabpedigree.com/james/mathmen.htm#Einstein">Albert Einstein</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Maxwell">James C. Maxwell</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Galileo">Galileo Galilei</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Aristotle">Aristotle</a> </li>
<li><a href="http://fabpedigree.com/james/mathmen.htm#Biruni">Abu Rayhan Biruni</a> </li>
</ul></span></td></tr>
</tbody></table></span></span><table cellspacing="0" style="margin-top: 5px;"><tbody>
<tr> <td class="sqtdq" colspan="2"><br />
<br />
<a href="" name="Early"></a><u><span style="color: #a00000;"><b>Earliest mathematicians</b></span></u> <br />
<blockquote>Little is known of the earliest mathematics, but the famous <i>Ishango Bone</i> from Early Stone-Age Africa has tally marks suggesting arithmetic. The markings include six prime numbers (5, 7, 11, 13, 17, 19) in order, though this is probably coincidence. The advanced artifacts of Egypt's Old Kingdom and the Indus-Harrapa civilization imply strong mathematical skill, but the first written evidence of advanced arithmetic dates from Sumeria, where 4500-year old clay tablets show multiplication and division problems; the first abacus may be about this old. By 3600 years ago, Mesopotamian tablets show tables of squares, cubes, reciprocals, and even logarithms, using a primitive place-value system (in base 60, not 10). Babylonians were familiar with the Pythagorean theorem, quadratic equations, even cubic equations (though they didn't have a general solution for these), and eventually even developed methods to estimate terms for compound interest. <br />
Also at least 3600 years ago, the Egyptian scribe Ahmes produced a famous manuscript (now called the <i>Rhind Papyrus</i>), itself a copy of a late Middle Kingdom text. It showed simple algebra methods and included a table giving optimal expressions using Egyptian fractions. (Today, Egyptian fractions lead to challenging number theory problems with no practical applications, but they may have had practical value for the Egyptians. To divide 17 grain bushels among 21 workers, the equation 17/21 = 1/2 + 1/6 + 1/7 has practical value, especially when compared with the "greedy" decomposition 17/21 = 1/2 + 1/4 + 1/17 + 1/1428.) <br />
While Egyptians may have had more advanced geometry, Babylon was much more advanced at arithmetic and algebra. This was probably due, at least in part, to their place-value system. But although their base-60 system survives (e.g. in the division of hours and degrees into minutes and seconds) the Babylonian notation, which used the equivalent of IIIIII XXXXXIIIIIII XXXXIII to denote 417+43/60, was unwieldy compared to the "ten digits of the Hindus." <br />
The Egyptians used the approximation <b>π ≈ (4/3)<sup>4</sup></b> (derived from the idea that a circle of diameter 9 has about the same area as a square of side 8). Although the ancient Hindu mathematician Apastambha had achieved a good approximation for <b>√2</b>, and the ancient Babylonians an ever better <b>√2</b>, neither of these ancient cultures achieved a <i>π</i> approximation as good as Egypt's, or better than <b>π ≈ 25/8</b>, until the Alexandrian era. </blockquote><br />
<a href="" name="Lagadha"></a><u><span style="color: #a00000;"><b>Early Vedic mathematicians</b></span></u> <br />
<blockquote>The greatest mathematics before the Golden Age of Greece was in India's early Vedic (Hindu) civilization. The Vedics understood relationships between geometry and arithmetic, developed astronomy, astrology, calendars, and used mathematical forms in some religious rituals. The earliest mathematician to whom definite teachings can be ascribed was Lagadha, who apparently lived about 1300 BC and used geometry and elementary trigonometry for his astronomy. Baudhayana lived about 800 BC and also wrote on algebra and geometry; Yajnavalkya lived about the same time and is credited with the then-best approximation to <b>π</b>. Apastambha did work summarized below; other early Vedic mathematicians solved quadratic and simultaneous equations. <br />
Other early cultures also developed some mathematics. The ancient Mayans apparently had a place-value system with zero before the Hindus did; Aztec architecture implies practical geometry skills. Ancient China certainly developed mathematics, though little written evidence survives prior to Chang Tshang's famous book. <br />
</blockquote><a href="" name="Pythagoras"></a><span><u><span style="color: #a00000;"><b>Pythagoras</b></span> of Samos</u></span> (ca 578-505 BC) Greek domain <br />
<blockquote>Pythagoras, who is sometimes called the "First Philosopher," studied under Anaximander, Egyptians, Babylonians, and the mystic Pherekydes (from whom Pythagoras acquired a belief in reincarnation); he became the most influential of early Greek mathematicians. He is credited with being first to use axioms and deductive proofs, so his influence on Plato and Euclid may be enormous. He and his students (the "Pythagoreans") were ascetic mystics for whom mathematics was partly a spiritual tool. (Some occultists treat Pythagoras as a wizard and founding mystic philosopher.) Pythagoras was very interested in astronomy and recognized that the Earth was a globe similar to the other planets. He believed thinking was located in the brain rather than heart. The words "philosophy" and "mathematics" are said to have been coined by Pythagoras. Despite Pythagoras' historical importance I may have ranked him too high: many results of the Pythagoreans were due to his students; none of their writings survive; and what is known is reported second-hand, and possibly exaggerated, by Plato and others. His students included Hippasus of Metapontum, perhaps the famous physician Alcmaeon, Milo of Croton, and Croton's daughter Theano (who may have been Pythagoras's wife). The term "Pythagorean" was also adopted by many disciples who lived later; these disciples include Philolaus of Croton, the natural philosopher Empedocles, and several other famous Greeks. Pythagoras' successor was apparently Theano herself: the Pythagoreans were one of the few ancient schools to practice gender equality. <br />
Pythagoras discovered that harmonious intervals in music are based on simple rational numbers. This led to a fascination with integers and mystic numerology; he is sometimes called the "Father of Numbers" and once said "Number rules the universe." (About the mathematical basis of music, Leibniz later wrote, "Music is the pleasure the human soul experiences from counting without being aware that it is counting." Other mathematicians who investigated the arithmetic of music included Huygens, Euler and Simon Stevin.) <br />
The Pythagorean Theorem was known long before Pythagoras, but he is often credited with the first <i>proof</i>. (Apastambha proved it in India at about the same time, and some theorize that Pythagoras journeyed to India and learned of the proof there.) He also discovered the simple parametric form of Pythagorean triplets <b>(xx-yy, 2xy, xx+yy)</b>. Other discoveries of the Pythagorean school include the concepts of perfect and amicable numbers, polygonal numbers, golden ratio (attributed to Theano), the five regular solids (attributed to Pythagoras himself), and irrational numbers (attributed to Hippasus). It is said that the discovery of irrational numbers upset the Pythagoreans so much they tossed Hippasus into the ocean! (Another version has Hippasus banished for revealing the secret for constructing the sphere which circumscribes a dodecahedron.) <br />
The famous successors of Thales and Pythagoras included Parmenides of Elea (ca 515-440 BC), Zeno of Elea (see below), Hippocrates of Chios (see below), Plato of Athens (ca 428-348 BC), Theaetetus (ca 414-369 BC), and Archytas (see below). These early Greeks ushered in a Golden Age of Mathematics and Philosophy unequaled in Europe until the Renaissance. The emphasis was on pure, rather than practical, mathematics. Plato (who ranks #40 on Michael Hart's famous list of the Most Influential Persons in History) decreed that his scholars should do geometric construction solely with compass and straight-edge rather than with "carpenter's tools" like rulers and protractors. </blockquote><br />
<center> </center> <br />
<a href="" name="Panini"></a><u><span style="color: #a00000;"><b>Panini</b></span> (of Shalatula)</u> (ca 520-460 BC) Gandhara (India) <br />
<blockquote>Panini's great accomplishment was his study of the Sanskrit language, especially in his text <i>Ashtadhyayi</i>. Although this work might be considered the very first study of linguistics or grammar, it used a non-obvious elegance that would not be equalled in the West until the 20th century. Linguistics may seem an unlikely qualification for a "great mathematician," but language theory is a field of mathematics. The works of eminent 20th-century linguists and computer scientists like Chomsky, Backus, Post and Church are seen to resemble Panini's work 24 centuries earlier. Panini's systematic study of Sanskrit may have inspired the development of Indian science and algebra. Panini has been called "the Indian Euclid" since the rigor of his grammar is comparable to Euclid's geometry. Although his great texts have been preserved, little else is known about Panini. Some scholars would place his dates a century later than shown here; he may or may not have been the same person as the famous poet Panini. In any case, he was the very last Vedic Sanskrit scholar by definition: his text formed the transition to the Classic Sanskrit period. Panini has been called "one of the most innovative people in the whole development of knowledge." </blockquote><a href="" name="Zeno"></a><u><span style="color: #a00000;"><b>Zeno</b></span> of Elea</u> (ca 495-435 BC) Greek domain <br />
<blockquote>Zeno, a student of Parmenides, had great fame in ancient Greece. This fame, which continues to the present-day, is largely due to his paradoxes of infinities and infinitesimals, e.g. his argument that Achilles can never catch the tortoise (whenever Achilles arrives at the tortoise's last position, the tortoise has moved on). Although some regard these paradoxes as simple fallacies, they have been contemplated for many centuries. It is due to these paradoxes that the use of infinitesimals, which provides the basis for mathematical analysis, has been regarded as a non-rigorous heuristic and is finally viewed as sound only after the work of the great 19th-century rigorists, Dedekind and Weierstrass. </blockquote><br />
<center> </center> <br />
<a href="" name="Hippoc"></a><u><span style="color: #a00000;"><b>Hippocrates</b></span> of Chios</u> (ca 470-410 BC) Greek domain <br />
<blockquote>Hippocrates (no relation to the famous physician) wrote his own <i>Elements</i> more than a century before Euclid. Only fragments survive but it apparently used axiomatic-based proofs similar to Euclid's and contains many of the same theorems. Hippocrates is said to have invented the <i>reductio ad absurdem</i> proof method. Hippocrates is most famous for his work on the three ancient geometric quandaries: his work on cube-doubling (the <i>Delian Problem</i>) laid the groundwork for successful efforts by Archytas and others; his circle quadrature was of course ultimately unsuccessful but he did prove ingenious theorems about "lunes" (certain circle fragments); and some claim Hippocrates was first to trisect the general angle. (Doubling the cube and angle trisection are often called "impossible," but they are impossible only when restricted to collapsing compass and unmarkable straightedge. There are ingenious solutions available with other tools.) Hippocrates also did work in algebra and rudimentary analysis. </blockquote><br />
<center> </center> <br />
<a href="" name="Archytas"></a><span><u><span style="color: #a00000;"><b>Archytas</b></span> of Tarentum</u></span> (ca 420-350 BC) Greek domain <br />
<blockquote>Archytas was an important statesman as well as philosopher. He studied under Philolaus of Croton, was a friend of Plato, and tutored Eudoxus and Menaechmus. In addition to discoveries always attributed to him, he may be the source of several of Euclid's theorems, and some works attributed to Eudoxus and perhaps Pythagoras. Recently it has been shown that the magnificent <i>Mechanical Problems</i> attributed to (pseudo-)Aristotle were probably actually written by Archytas, making him one of the greatest mathematicians of antiquity. Archytas introduced "motion" to geometry, rotating curves to produce solids. If his writings had survived he'd surely be considered one of the most brilliant and innovative geometers of antiquity. Archytas' most famous mathematical achievement was "doubling the cube" (constructing a line segment larger than another by the factor cube-root of two). Although others solved the problem with other techniques, Archytas' solution for cube doubling was astounding because it wasn't achieved in the plane, but involved the intersection of three-dimensional bodies. This construction (which introduced the "Archytas Curve") has been called "a tour de force of the spatial imagination." He invented the term "harmonic mean" and worked with geometric means as well (proving that consecutive integers never have rational geometric mean). He was a true polymath: he advanced the theory of music far beyond Pythagoras; studied sound, optics and cosmology; invented the pulley (and a rattle to occupy infants!); wrote about the lever; developed the curriculum called quadrivium; and is supposed to have built a steam-powered wooden bird which flew for 200 meters. Archytas is sometimes called the Father of Mathematical Mechanics. <br />
</blockquote><a href="" name="Conway"></a><span><u>John Horton <span style="color: #a00000;"><b>Conway</b></span></u></span> (1937-) Britain <br />
<blockquote>Conway has done pioneering work in a very broad range of mathematics including knot theory, number theory, group theory, lattice theory, combinatorial game theory, geometry, quaternions, tilings, and cellular automaton theory. He started his career by proving a case of Waring's conjecture, but achieved fame when he discovered the largest then-known sporadic group (the symmetry group of the Leech lattice); this sporadic group is now known to be second in size only to the "Monster Group," with which Conway also worked. Conway's fertile creativity has produced a cornucopia of fascinating inventions: markable straight-edge construction of the regular heptagon (a feat also achieved by Archimedes), a nowhere-continuous function that has the Intermediate Value property, the Conway box function, the aperiodic pinwheel tiling, a representation of symmetric polyhedra, his chained-arrow notation for large numbers, and many results and conjectures in recreational mathematics. He found the simplest proof for Morley's Trisector Theorem (sometimes called the best result in simple plane geometry since ancient Greece). He proved an unusual theorem about quantum physics: "If experimenters have free will, then so do elementary particles." His most famous construction is the computationally complete automaton known as the <i>Game of Life</i>. His most important theoretical invention, however, may be his <i>surreal numbers</i> incorporating infinitesimals; he invented them to solve combinatorial games like Go, but they have pure mathematical significance as the largest possible ordered field. Although Conway hasn't won the highest mathematics prizes, nor perhaps proved deep theoretical theorems, his great creativity and breadth qualifies him as one of the greatest living mathematicians. </blockquote><a href="" name="Ramanujan"></a><span><u>Srinivasa <span style="color: #a00000;"><b>Ramanujan</b></span> Iyengar</u></span> (1887-1920) India <br />
<blockquote>Like Abel, Ramanujan was a self-taught prodigy who lived in a country distant from his mathematical peers, and suffered from poverty: childhood dysentery and vitamin deficiencies probably led to his early death. Yet he produced 4000 theorems or conjectures in number theory, algebra, and combinatorics. He might be almost unknown today, except that his letter caught the eye of Godfrey Hardy, who saw remarkable, almost inexplicable formulae which "must be true, because if they were not true, no one would have had the imagination to invent them." Ramanujan's specialties included infinite series, elliptic functions, continued fractions, partition enumeration, definite integrals, modular equations, gamma functions, "mock theta" functions, hypergeometric series, and "highly composite" numbers. Much of his best work was done in collaboration with Hardy, for example a proof that almost all numbers <i>n</i> have about <i>log log n</i> prime factors (a result which inspired probabilistic number theory). Much of his methodology, including unusual ideas about divergent series, was his own invention. (As a young man he made the absurd claim that <b>1+2+3+4+... = -1/12</b>. Later it was noticed that this claim translates to a true statement about the Riemann zeta function, with which Ramanujan was unfamiliar.) Ramanujan's innate ability for algebraic manipulations equaled or surpassed that of Euler and Jacobi. Ramanujan's most famous work was with the partition enumeration function <i>p()</i>, Hardy guessing that some of these discoveries would have been delayed at least a century without Ramanujan. Together, Hardy and Ramanujan developed an analytic approximation to <i>p()</i>. (Rademacher and Selberg later discovered an exact expression to replace the Hardy-Ramanujan formula; when Ramanujan's notebooks were studied it was found he had anticipated their technique, but had deferred to his friend and mentor.) <br />
Many of Ramanujan's other results would also probably never have been discovered without him, and are so inspirational that there is a periodical dedicated to them. The theories of strings and crystals have benefited from Ramanujan's work. (Today some professors achieve fame just by finding a new proof for one of Ramanujan's many results.) Unlike Abel, who insisted on rigorous proofs, Ramanujan often omitted proofs. (Ramanujan may have had unrecorded proofs, poverty leading him to use chalk and erasable slate rather than paper.) Unlike Abel, much of whose work depended on the complex numbers, most of Ramanujan's work focused on real numbers. Despite these limitations, Ramanujan is considered one of the greatest geniuses ever. <br />
Because of its fast convergence, an odd-looking formula of Ramanujan is often used to calculate <b>π</b>: <br />
<b>99<sup>2</sup> / π = √8 ∑<sub>k=0,∞</sub> (4k! (1103+26390 k) / (k!<sup>4</sup> 396<sup>4k</sup>))</b><br />
</blockquote><a href="" name="Brahmagupta"></a><span><u><span style="color: #a00000;"><b>Brahmagupta</b></span> `Bhillamalacarya'</u></span> (589-668) Rajasthan (India) <br />
<blockquote> No one person gets unique credit for the invention of the decimal system but Brahmagupta's textbook <i>Brahmasphutasiddhanta</i> was very influential, and is sometimes considered the first textbook "to treat zero as a number in its own right." It also treated negative numbers. (Others claim these were first seen 800 years earlier in Chang Tshang's Chinese text and were implicit in what survives of earlier Hindu works, but Brahmagupta's text discussed them lucidly.) <br />
Brahmagupta Bhillamalacarya (`The Teacher from Bhillamala') made great advances in arithmetic, algebra, numeric analysis, and geometry. Several theorems bear his name, including the formula for the area of a cyclic quadrilateral: <br />
<b>16 A<sup>2</sup> = (a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d)</b> <br />
Another famous Brahmagupta theorem dealing with such quadrilaterals can be phrased "In a circle, if the chords AB and CD are perpendicular and intersect at E, then the line from E which bisects AC will be perpendicular to BD." Proving Brahmagupta's theorems are good challenges even today. <br />
In addition to his famous writings on practical mathematics and his ingenious theorems of geometry, Brahmagupta solved the general quadratic equation, and worked on Diophantine and Pell's equations. He proved the Brahmagupta-Fibonacci Identity (the set of sums of two squares is closed under multiplication). He applied mathematics to astronomy, predicting eclipses, etc. </blockquote><a href="" name="Khowarizmi"></a><span><u>Muhammed `Abu Jafar' ibn Musâ <span style="color: #a00000;"><b>al-Khowârizmi</b></span></u></span> (ca 780-850) Persia, Iraq <br />
<blockquote>Al-Khowârizmi (aka Mahomet ibn Moses) was a Persian who worked as a mathematician, astronomer and geographer early in the Golden Age of Islamic science. He introduced the Hindu decimal system to the Islamic world and Europe; invented the horary quadrant; improved the sundial; developed trigonometry tables; and improved on Ptolemy's astronomy and geography. He wrote the book <i>Al-Jabr</i>, which demonstrated simple algebra and geometry, and several other influential books. Unlike Diophantus' work, which dealt in specific examples, Al-Khowârizmi presented general methods. The word <i>algorithm</i> is borrowed from Al-Khowârizmi's name. There were several Muslim mathematicians who contributed to the development of Islamic science, and indirectly to Europe's later Renaissance, but Al-Khowârizmi was one of the earliest and most influential. </blockquote><br />
<br />
<a href="" name="Alkindus"></a><u>Abu Yusuf Ya'qub ibn Ishaq <span style="color: #a00000;"><b>al-Kindi</b></span></u> (803-873) Iraq <br />
<blockquote>Al-Kindi (called Alkindus in the West) wrote on diverse philosophical subjects, physics, optics, astronomy, music, psychology, medicine, chemistry, and more. He invented pharmaceutical methods, perfumes, and distilling of alcohol. In mathematics, he popularized the use of the decimal system, developed spherical geometry, wrote on many other topics and was a pioneer of cryptography (code-breaking). (Al-Kindi, called <i>The Arab Philosopher</i>, can not be considered among the greatest of mathematicians, but was one of the most influential general scientists between Aristotle and da Vinci.) </blockquote><br />
<br />
<a href="" name="Thabit"></a><u>Al-Sabi Thabit <span style="color: #a00000;"><b>ibn Qurra</b></span></u> al-Harrani (836-901) Harran, Iraq <br />
<blockquote>Thabit produced important books in philosophy (including perhaps the famous mystic work <i>De Imaginibus</i>), mechanics, astronomy, and especially several mathematical fields: analysis, non-Euclidean geometry, trigonometry, arithmetic, number theory. He developed an important new cosmology superior to Ptolemy's (and which, though it was not heliocentric, may have inspired Copernicus). He was perhaps the first great mathematician to take the important step of emphasizing real numbers rather than either rational numbers or geometric sizes. He worked in plane and spherical trigonometry, in cubic equations. He was an earlier practitioner of calculus and seems to have been first to take the integral of <b>√x</b>. He also may have been first to calculate the area of an ellipse, and first to calculate the volume of a paraboloid. He also worked in number theory where he is especially famous for his theorem about amicable numbers. While many of his discoveries in geometry, plane and spherical trigonometry, and analysis (parabola quadrature, trigonometric law, principle of lever) duplicated work by Archimedes and Pappus, Thabit's list of novel achievements is still impressive. Among the several great and famous Baghdad geometers, Thabit may have had the greatest genius. </blockquote> By: Tharshini Jayaseelan<br />
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</div><div class="separator" style="clear: both; text-align: center;">By: Elsa Athira Asri</div><div class="separator" style="clear: both; text-align: center;"><br />
</div>4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com0tag:blogger.com,1999:blog-8948305359118541243.post-71559190027123782352011-07-20T18:09:00.001+08:002011-07-23T18:17:10.632+08:00Math JokesAn infinite crowd of mathematicians enters a bar. <br />
The first one orders a pint, the second one a half pint, the third one a quarter pint... <br />
"I understand", says the bartender - and pours two pints. <br />
<hr />Q: When did Bourbaki stop writing books? <br />
A: When they realized that Serge Lang was a single person... <br />
<hr />Teacher: What is 2<i>k</i> + <i>k</i>? <br />
Student: 3000! <br />
<hr />Q: What do you get if you divide the cirucmference of a jack-o-lantern by its diameter? <br />
A: Pumpkin Pi! <br />
<hr />Q: Why do you rarely find mathematicians spending time at the beach? <br />
A: Because they have sine and cosine to get a tan and don't need the sun! <br />
<hr />Q: Why do mathematicians, after a dinner at a Chinese restaurant, always insist on taking the leftovers home? <br />
A: Because they know the Chinese remainder theorem! <br />
<hr />Teacher: "Who can tell me what 7 times 6 is?" <br />
Student: "It's 42!" <br />
Teacher: "Very good! - And who can tell me what 6 times 7 is?" <br />
Same student: "It's 24!" <br />
<hr />A math student is pestered by a classmate who wants to copy his homework assignment. The student hesitates, not only because he thinks it's wrong, but also because he doesn't want to be sanctioned for aiding and abetting. <br />
His classmate calms him down: "Nobody will be able to trace my homework to you: I'll be changing the names of all the constants and variables: <i>a</i> to <i>b</i>, <i>x</i> to <i>y</i>, and so on." <br />
Not quite convinced, but eager to be left alone, the student hands his completed assignment to the classmate for copying. <br />
After the deadline, the student asks: "Did you <i>really</i> change the names of <i>all</i> the variables?" <br />
"Sure!" the classmate replies. "When you called a function <i>f</i>, I called it <i>g</i>; when you called a variable <i>x</i>, I renamed it to <i>y</i>; and when you were writing about the log of <i>x</i>+1, I called it the timber of <i>x</i>+1..." <br />
<hr />The math teacher asks his students: "What is 9 times 7?" <br />
He gets several answers - all are either 62 or 65. <br />
"Come on - the correct answer can either be 62 or 65!" <br />
<hr />"That math prof's marriage is falling apart!" <br />
"No wonder! He's into scientific computing - and she's incalculable!" <br />
<hr />Q: How does one insult a mathematician? <br />
A: You say: "Your brain is smaller than any <img src="http://www.math.ualberta.ca/%7Erunde/images/epsilon.gif" />>0!" <br />
<hr />Q: What does a mathematician present to his fiancée when he wants to propose? <br />
A: A polynomial ring! <br />
<hr />The chef instructs his apprentice: "You take two thirds of water, one third of cream, one third of broth..." <br />
The apprentice: "But that makes four thirds already!" <br />
"Well - just take a larger pot!" <br />
<hr />A woman in a bar tries to pick up a mathematician. <br />
"How old, do you think, am I?" she asks coyly. <br />
"Well - 18 by that fire in your eyes, 19 by that glow on your cheeks, 20 by that radiance of your face, and adding that up is something you can probably do for yourself..." <br />
<hr />Q: What is the most erotic number? <br />
A: 2110593! <br />
Q: Why? <br />
A: When 2 are 1 and don't pay at10tion, they'll know within 5 weeks whether or not, after 9 months, they'll be 3... <br />
<hr /><b>Theorem.</b> A cat has nine tails. <br />
<i>Proof.</i> No cat has eight tails. Since one cat has one more tail than no cat, it must have nine tails. <br />
<hr />Trigonometry for farmers: swine and coswine... <br />
<hr />Q: What is the difference between a mathematician and a philosopher? <br />
A: The mathematician only needs paper, pencil, and a trash bin for his work - the philosopher can do without the trash bin... <br />
<hr />Q: What is a mathematician's pick when faced with the choice between poutine and eternal bliss in the afterlife? <br />
A: Poutine! Because nothing is better than eternal bliss in the afterlife, and poutine is better than nothing. <br />
<hr />Two math students, a boy and his girlfriend, are going to a fair. They are in line to ride the ferris wheel when it shuts down. <br />
The boy says: "It's a sin for those people to keep us waiting like this!" <br />
The girl replies: "No - it's a cosin, silly!!!" <br />
<hr />Life is complex: it has both real and imaginary components. <br />
<hr />The math professor just accepted a new position at a university in another city and has to move. He and his wife pack all their belongings into cardboard boxes and have them shipped off to their new home. To sort out some family matters, the wife stays behind for a few more days while her husband has already left for their new residence. <br />
The boxes arrive when the wife still hasn't rejoined her husband. When they talk on the phone in the evening, she asks him to count the boxes, just to make sure the movers didn't loose any of them. <br />
"Thirty nine boxes altogether", says the prof on the phone. <br />
"That can't be", the wife exclaims. "The movers picked up forty boxes at our old place." <br />
The prof counts once again, but again his count only reaches 39. <br />
The next morning, the wife calls the moving company and complains. The company promises to check; a few hours later, someone calls back and reports that all forty boxes did arrive. <br />
In the evening, when the prof and his wife are on the phone again, she asks: "I don't understand it. When you count, you get 39, and when they do, they get 40. That's more than strange..." <br />
"Well", the prof says. "This is a cordless phone, so you can stay on the line and count with me: zero, one, two, three,..." <br />
<hr />There are 10 kinds of mathematicians. Those who can think binarily and those who can't... <br />
<hr />Q: What is the difference between a Ph.D. in mathematics and a large pizza? <br />
A: A large pizza can feed a family of four... <br />
<hr /><i>New York (CNN)</i>. At John F. Kennedy International Airport today, a Caucasian male (later discovered to be a high school mathematics teacher) was arrested trying to board a flight while in possession of a compass, a protractor and a graphical calculator. <br />
According to law enforcement officials, he is believed to have ties to the Al-Gebra network. He will be charged with carrying weapons of math instruction. <br />
<hr />After her husband's death, the elderly lady decided to go back to school and get a degree in mathematics. <br />
A few weeks into the term, she storms into the dean's office, exclaiming: "I've been silent until now - but I'm not going to take these obscenities anymore!" <br />
"What obscenities are you talking about?" <br />
She reaches into her purse and pulls out a notebook. "I noted of all of them. In my presence, professors had the complete lack of decency to speak of" - she leafs through her notebook - "Bruhat-Tits spaces, a pumping lemma, and even degenerate colonels!" <br />
<hr />Two mathematicians are studying a convergent series. <br />
The first one says: "Do you realize that the series converges even when all the terms are made positive?" <br />
The second one asks: "Are you sure?" <br />
"Absolutely!" <br />
<hr />Q: How can you tell that Harvard was planned by a mathematician? <br />
A: The div school is right next to the grad school... <br />
<hr />A mathematician is asked by a friend who is a devout Christian: "Do you believe in one God?" <br />
He answers: "Yes - up to isomorphism." <br />
<br />
<hr />Q: How does a mathematician induce good behavior in her children? <br />
A: `I've told you <i>n</i> times, I've told you <i>n</i>+1 times...' <br />
<hr />Q: Do you know any catchy anagram of Banach-Tarski? <br />
A: Banach-Tarski Banach-Tarski... <br />
<hr />"Students nowadays are so clueless", the math professor complains to a colleague. "Yesterday, a student came to my office hours and wanted to know if General Calculus was a Roman war hero..." <br />
<hr />Mother to her daughter: "Why does the tablecloth you just put on the table have the word `truth' written on it?" <br />
Daughter: "Because I want to turn the table into a truth table!" <br />
<hr />A mathematician and his best friend, an engineer, attend a public lecture on geometry in thirteen-dimensional space. <br />
"How did you like it?" the mathematician wants to know after the talk. <br />
"My head's spinning", the engineer confesses. "How can you develop any intuition for thirteen-dimensional space?" <br />
"Well, it's not even difficult. All I do is visualize the situation in arbitrary <i>N</i>-dimensional space and then set <i>N</i> = 13." <br />
<hr />A newlywed husband is discouraged by his wife's obsession with mathematics. Afraid of being second fiddle to her profession, he finally confronts her: "Do you love math more than me?" <br />
"Of course not, dear - I love you much more!" <br />
Happy, although sceptical, he challenges her: "Well, then prove it!" <br />
Pondering a bit, she responds: "Ok... Let epsilon be greater than zero..." <br />
<hr />"So how's your boyfriend doing, the math student?" <br />
"Don't mention that crazy pervert to me anymore! We broke up." <br />
"How can you say such a nasty thing about him? He seemed to be such a nice boy." <br />
"Imagine! He was restless during the days and couldn't sleep at night - always trying to solve his math problems. When he had finally done it, he wasn't happy: he would call himself a complete idiot and throw all his notes into the garbage. One day, I couldn't take it anymore, and I told him to drop math. You know what he told me?" <br />
"No." <br />
"He said, he enjoyed it!!!" <br />
<hr />It is only two weeks into the term that, in a calculus class, a student raises his hand and asks: "Will we ever need this stuff in real life?" <br />
The professor gently smiles at him and says: "Of course not - if your real life will consist of flipping hamburgers at MacDonald's!" <br />
<hr />An American mathematician returns home from a conference in Moscow on real and complex analysis. <br />
The immigration officer at the airport glances at his landing card and says: "So, your trip to Russia was business related. What's the nature of your business?" <br />
"I am a professor of mathematics." <br />
"What kind of mathematics are you doing?" <br />
The professor ponders for a split second, trying to come up with something that would sound specific enough without making the immigration officer suspicious, and replies: "I am an analyst." <br />
The immigration officer nods with approval: "I think it's great that guys like you go to Russia to help those poor ex-commies to get their stock market on its feet..."<br />
<br />
By Thashini Kathir4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com0tag:blogger.com,1999:blog-8948305359118541243.post-49285046168539918312011-07-19T21:49:00.001+08:002011-07-23T18:15:27.888+08:00Surface Area of A Sphere - Fun Fact <u><b>Problem</b></u><br />
<br />
The area of a disk enclosed by a circle of radius R is Pi*R<sup>2</sup>. <br />
The formula for the circumference of a circle of radius R is 2*Pi*R. <br />
A simple calculus check reveals that the latter is the derivative of the former with respect to R. <br />
Similarly, the volume of a ball enclosed by a sphere of radius R is (4/3)*Pi*R<sup>3</sup>. <br />
And the formula for the surface area of a sphere of radius R is 4*Pi*R<sup>2</sup>. <br />
And, you can check that the latter is the derivative of the former with respect to R. <br />
Coincidence, or is there a reason? <br />
<br />
<b><u>The Math Behind the Fact</u></b><br />
<br />
Well, no, it is not a coincidence. For the ball, a small change in radius produces a change in volume of the ball which is equal to the volume of a spherical shell of radius R and thickness (delta R). The spherical shell's volume is thus approximately (surface area of the sphere)*(delta R). But the derivative is approximately the change in ball volume divided by (delta R), which is thus just (surface area of the sphere). <br />
So, if I tell you the 4-dimensional "volume" of the 4-dimensional ball is (1/2)*Pi<sup>2</sup>*R<sup>4</sup>, what is 3-dimensional volume of its boundary?<br />
<br />
Sourced from: <a href="http://www.math.hmc.edu/funfacts/ffiles/20004.2-3.shtml">Mudd Math Fun Facts - Surface Area of A Sphere</a> <br />
<br />
- Thivya4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com0tag:blogger.com,1999:blog-8948305359118541243.post-76824870838141958122011-07-18T23:13:00.002+08:002011-07-23T18:13:32.036+08:00Circles<div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/lWDha0wqbcI?feature=player_embedded' frameborder='0'></iframe> </div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: left;">- Nur Zalikha </div>4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com0tag:blogger.com,1999:blog-8948305359118541243.post-71239336803887097872011-07-18T19:33:00.004+08:002011-07-23T18:26:17.726+08:00About Maths<b>Mathematics</b> (from <a href="http://en.wikipedia.org/wiki/Greek_language" title="Greek language"><span style="color: #0645ad;">Greek</span></a> μάθημα (<i>máthēma</i>) — knowledge, study, learning) is the study of <a href="http://en.wikipedia.org/wiki/Quantity" title="Quantity"><span style="color: #0645ad;">quantity</span></a>, <a href="http://en.wikipedia.org/wiki/Structure" title="Structure"><span style="color: #0645ad;">structure</span></a>, <a href="http://en.wikipedia.org/wiki/Space" title="Space"><span style="color: #0645ad;">space</span></a>, and <a href="http://en.wikipedia.org/wiki/Calculus" title="Calculus"><span style="color: #0645ad;">change</span></a>. <a href="http://en.wikipedia.org/wiki/Mathematician" title="Mathematician"><span style="color: #0645ad;">Mathematicians</span></a> seek out <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Patterns" title="Patterns"><span style="color: #0645ad;">patterns</span></a>,<sup class="reference" id="cite_ref-1"><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-1"><span style="font-size: x-small;"><span style="color: #0645ad;">[2]</span></span></a></sup><sup class="reference" id="cite_ref-2"><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-2"><span style="font-size: x-small;"><span style="color: #0645ad;">[3]</span></span></a></sup> and formulate new <a href="http://en.wikipedia.org/wiki/Conjecture" title="Conjecture"><span style="color: #0645ad;">conjectures</span></a>. Mathematicians resolve the truth or falsity of conjectures by <a href="http://en.wikipedia.org/wiki/Mathematical_proof" title="Mathematical proof"><span style="color: #0645ad;">mathematical proofs</span></a>, which are arguments sufficient to convince other mathematicians of their validity. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. However, mathematical proofs are less formal and painstaking than proofs in <a href="http://en.wikipedia.org/wiki/Mathematical_logic" title="Mathematical logic"><span style="color: #0645ad;">mathematical logic</span></a>. Since the pioneering work of <a href="http://en.wikipedia.org/wiki/Giuseppe_Peano" title="Giuseppe Peano"><span style="color: #0645ad;">Giuseppe Peano</span></a>, <a href="http://en.wikipedia.org/wiki/David_Hilbert" title="David Hilbert"><span style="color: #0645ad;">David Hilbert</span></a>, and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing <a href="http://en.wikipedia.org/wiki/Truth" title="Truth"><span style="color: #0645ad;">truth</span></a> by <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Mathematical_rigour" title="Mathematical rigour"><span style="color: #0645ad;">rigorous</span></a> <a href="http://en.wikipedia.org/wiki/Deductive_reasoning" title="Deductive reasoning"><span style="color: #0645ad;">deduction</span></a> from appropriately chosen <a href="http://en.wikipedia.org/wiki/Axiom" title="Axiom"><span style="color: #0645ad;">axioms</span></a> and <a href="http://en.wikipedia.org/wiki/Definition" title="Definition"><span style="color: #0645ad;">definitions</span></a>. When those mathematical structures are good models of real phenomena, then mathematical reasoning often provides insight or predictions.<br />
Through the use of <a href="http://en.wikipedia.org/wiki/Abstraction_(mathematics)" title="Abstraction (mathematics)"><span style="color: #0645ad;">abstraction</span></a> and <a href="http://en.wikipedia.org/wiki/Logic" title="Logic"><span style="color: #0645ad;">logical</span></a> <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Reasoning" title="Reasoning"><span style="color: #0645ad;">reasoning</span></a>, mathematics evolved from <a href="http://en.wikipedia.org/wiki/Counting" title="Counting"><span style="color: #0645ad;">counting</span></a>, <a href="http://en.wikipedia.org/wiki/Calculation" title="Calculation"><span style="color: #0645ad;">calculation</span></a>, <a href="http://en.wikipedia.org/wiki/Measurement" title="Measurement"><span style="color: #0645ad;">measurement</span></a>, and the systematic study of the <a href="http://en.wikipedia.org/wiki/Shape" title="Shape"><span style="color: #0645ad;">shapes</span></a> and <a href="http://en.wikipedia.org/wiki/Motion_(physics)" title="Motion (physics)"><span style="color: #0645ad;">motions</span></a> of physical objects. Practical mathematics has been a human activity for as far back as <a class="mw-redirect" href="http://en.wikipedia.org/wiki/History_of_Mathematics" title="History of Mathematics"><span style="color: #0645ad;">written records</span></a> exist. <a href="http://en.wikipedia.org/wiki/Logic" title="Logic"><span style="color: #0645ad;">Rigorous arguments</span></a> first appeared in <a href="http://en.wikipedia.org/wiki/Greek_mathematics" title="Greek mathematics"><span style="color: #0645ad;">Greek mathematics</span></a>, most notably in <i><a href="http://en.wikipedia.org/wiki/Euclid%27s_Elements" title="Euclid's Elements"><span style="color: #0645ad;">Euclid's Elements</span></a></i>. Mathematics continued to develop, for example in China in 300 BC, in India in AD 100<sup class="Template-Fact" style="white-space: nowrap;" title="This claim needs references to reliable sources from June 2011"><span style="font-size: x-small;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span style="color: #0645ad;">citation needed</span></a></i>]</span></sup>, and in the <a href="http://en.wikipedia.org/wiki/Muslim_world" title="Muslim world"><span style="color: #0645ad;">Muslim world</span></a> in AD 800, until the <a href="http://en.wikipedia.org/wiki/Renaissance" title="Renaissance"><span style="color: #0645ad;">Renaissance</span></a>, when mathematical innovations interacting with new <a href="http://en.wikipedia.org/wiki/Timeline_of_scientific_discoveries" title="Timeline of scientific discoveries"><span style="color: #0645ad;">scientific discoveries</span></a> led to a rapid increase in the rate of mathematical discovery that continues to the present day.<sup class="reference" id="cite_ref-3"><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-3"><span style="font-size: x-small;"><span style="color: #0645ad;">[4]</span></span></a></sup><br />
The mathematician <a href="http://en.wikipedia.org/wiki/Benjamin_Peirce" title="Benjamin Peirce"><span style="color: #0645ad;">Benjamin Peirce</span></a> called mathematics "the science that draws necessary conclusions".<sup class="reference" id="cite_ref-4"><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-4"><span style="font-size: x-small;"><span style="color: #0645ad;">[5]</span></span></a></sup> David Hilbert defined mathematics as follows: <i>We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise.</i><sup class="reference" id="cite_ref-5"><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-5"><span style="font-size: x-small;"><span style="color: #0645ad;">[6]</span></span></a></sup> <a href="http://en.wikipedia.org/wiki/Albert_Einstein" title="Albert Einstein"><span style="color: #0645ad;">Albert Einstein</span></a> stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."<sup class="reference" id="cite_ref-certain_6-0"><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-certain-6"><span style="font-size: x-small;"><span style="color: #0645ad;">[7]</span></span></a></sup><br />
Mathematics is used throughout the world as an essential tool in many fields, including <a href="http://en.wikipedia.org/wiki/Natural_science" title="Natural science"><span style="color: #0645ad;">natural science</span></a>, <a href="http://en.wikipedia.org/wiki/Engineering" title="Engineering"><span style="color: #0645ad;">engineering</span></a>, <a href="http://en.wikipedia.org/wiki/Medicine" title="Medicine"><span style="color: #0645ad;">medicine</span></a>, and the <a href="http://en.wikipedia.org/wiki/Social_sciences" title="Social sciences"><span style="color: #0645ad;">social sciences</span></a>. <a href="http://en.wikipedia.org/wiki/Applied_mathematics" title="Applied mathematics"><span style="color: #0645ad;">Applied mathematics</span></a>, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as <a href="http://en.wikipedia.org/wiki/Statistics" title="Statistics"><span style="color: #0645ad;">statistics</span></a> and <a href="http://en.wikipedia.org/wiki/Game_theory" title="Game theory"><span style="color: #0645ad;">game theory</span></a>. Mathematicians also engage in <a href="http://en.wikipedia.org/wiki/Pure_mathematics" title="Pure mathematics"><span style="color: #0645ad;">pure mathematics</span></a>, or mathematics for its own sake, without having any application in mind, although practical applications for what began as pure mathematics are often discovered.<sup class="reference" id="cite_ref-7"><span style="font-size: x-small;"><span style="color: #0645ad;"><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-7">[8]</a></span></span></sup><br />
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<sup class="reference" id="cite_ref-7"><span class="Apple-style-span" style="font-size: small;"> By Nurhanim Jamaluddin</span></sup>4SC '11http://www.blogger.com/profile/14061996081022333631noreply@blogger.com0