The origins of mathematical thought lie in the concepts of number, magnitude, and form. 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
The oldest known possibly mathematical object is the
Lebombo bone, discovered in the Lebombo mountains of
Swaziland and dated to approximately 35,000 BC.
It consists of 29 distinct notches cut into a baboon's fibula.
Also
prehistoric artifacts discovered in Africa and
France, dated between
35,000 and
20,000 years old,
suggest early attempts to
quantify time.
In the book
How Mathematics Happened: The First 50,000 Years, 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."
The
Ishango bone, found near the headwaters of the
Nile river (northeastern
Congo), may be as much as
20,000 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
sequences of
prime numbers or a six month lunar calendar.
Predynastic Egyptians of the 5th millennium BC pictorially represented
geometric designs. It has been claimed that
megalithic monuments in
England and
Scotland, dating from the 3rd millennium BC, incorporate geometric ideas such as
circles,
ellipses, and
Mesopotamian mathematics
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
In contrast to the sparsity of sources in
Egyptian mathematics, our knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s.
Written in
Cuneiform script, 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.
The earliest evidence of written mathematics dates back to the ancient
Sumerians, who built the earliest civilization in Mesopotamia. They developed a complex system of
metrology from 3000 BC. From around 2500 BC onwards, the Sumerians wrote
multiplication tables on clay tablets and dealt with
geometricaldivision problems. The earliest traces of the Babylonian numerals also date back to this period.
[21] exercises and
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
regular reciprocal pairs.
The tablets also include multiplication tables and methods for solving
linear and
quadratic equations. The Babylonian tablet YBC 7289 gives an approximation to √2 accurate to five decimal places.
Babylonian mathematics were written using a
sexagesimal (base-60)
numeral system. 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
decimal 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.
Egyptian mathematics
Image of Problem 14 from the
Moscow Mathematical Papyrus. The problem includes a diagram indicating the dimensions of the truncated pyramid.
Another significant Egyptian mathematical text is the
Moscow papyrus, also from the
Middle Kingdom It consists of what are today called
word problems or
story problems, 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
frustum: "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.
African mathematics
The
ancient Nubians used a trigonometric methodology similar to the Egyptians.
[31] During the Meroitic Era the Nubians established a system of geometry including early versions of sun clocks.
All mathematical learning of the Islamic world during the medieval period were available to
Timbuktu scholars arithmetic, algebra, geometry, and trigonometry during the great Islamic reigns of the
Mali Empire and the
Songhai Empire.
Greek Mathematics
Greek mathematics refers to the mathematics written in the
Greek language from the time of
Thales of MiletusAcademy of Athens in 529 AD.
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
Alexander the Great is sometimes called Hellenistic mathematics.
(~600 BC) to the closure of the
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
mathematical rigor to
prove them.
Greek mathematics is thought to have begun with
Thales of Miletus (c. 624–c.546 BC) and
Pythagoras of Samos (c. 582–c. 507 BC). Although the extent of the influence is disputed, they were probably inspired by
Egyptian and
Babylonian mathematics. According to legend, Pythagoras traveled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests.
Thales used
geometry 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
Thales' Theorem. 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.
Pythagoras established the
Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number".
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
Pythagorean theorem,
though the statement of the theorem has a long history, and with the proof of the existence of
One of the oldest surviving fragments of Euclid's
Elements, found at
Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.
Plato (428/427 BC – 348/347 BC) is important in the history of mathematics for inspiring and guiding others.
His
Platonic Academy, in
Athens, 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
Eudoxus of Cnidus, came from.
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.
The
analytic method is ascribed to Plato, while a formula for obtaining Pythagorean triples bears his name.
Eudoxus (408–c.355 BC) developed the
method of exhaustion, a precursor of modern
integration[49] and a theory of ratios that avoided the problem of
incommensurable magnitudes.
The former allowed the calculations of areas and volumes of curvilinear figures,
while the latter enabled subsequent geometers to make significant advances in geometry. Though he made no specific technical mathematical discoveries,
Aristotle (384—c.322 BC) contributed significantly to the development of mathematics by laying the foundation of
logic.
In the 3rd century BC, the premier center of mathematical education and research was the
Musaeum of
Alexandria.
It was there that
Euclid (c. 300 BC) taught, and wrote the
Elements, widely considered the most successful and influential textbook of all time
The
Elements introduced
mathematical rigor through the
axiomatic method 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
Elements were already known, Euclid arranged them into a single, coherent logical framework.
The
Elements 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.
In addition to the familiar theorems of
Euclidean geometry, the
Elements was meant as an introductory textbook to all mathematical subjects of the time, such as
number theory,
algebra and
solid geometry,
including proofs that the square root of two is irrational and that there are infinitely many prime numbers. Euclid also
wrote extensively on other subjects, such as
conic sections,
optics,
spherical geometry, and mechanics, but only half of his writings survive
Chinese mathematics
Early Chinese mathematics is so different from that of other parts of the world that it is reasonable to assume independent development.
The oldest extant mathematical text from China is the
Chou Pei Suan Ching, variously dated to between 1200 BC and 100 BC, though a date of about 300 BC appears reasonable.
[61]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.
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.
Rod numerals allowed the representation of numbers as large as desired and allowed calculations to be carried out on the
suan pan, or (Chinese abacus). The date of the invention of the
suan pan is not certain, but the earliest written mention dates from AD 190, in Xu Yue's
Supplementary Notes on the Art of Figures.
The oldest existent work on
geometry in China comes from the philosophical
Mohist canon c. 330 BC, compiled by the followers of
Mozi (470–390 BC). The
Mo Jing described various aspects of many fields associated with physical science, and provided a small number of geometrical theorems as well.
[64]In 212 BC, the Emperor
Qin Shi Huang (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
book burning of 212 BC, the
Han dynasty (202 BC–220 AD) produced works of mathematics which presumably expanded on works that are now lost. The most important of these is
The Nine Chapters on the Mathematical Art, 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
Chinese pagoda towers, engineering,
surveying, and includes material on
right triangles and values of
π.
Cavalieri's principle on volume more than a thousand years before Cavalieri would propose it in the West.
[citation needed] It created mathematical proof for the
Pythagorean theorem, and a mathematical formula for
Gaussian elimination.
Liu Hui 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
Zu Chongzhi computed the value of π to seven decimal places, which remained the most accurate value of π for almost the next 1000 years.
It also made use of
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
Precious Mirror of the Four Elements by Chu Shih-chieh (fl. 1280-1303), dealing with the solution of simultaneous higher order algebraic equations using a method similar to
Horner's method.
The
Precious Mirror also contains a diagram of
Pascal's triangle with coefficients of binomial expansions through the eighth power, though both appear in Chinese works as early as 1100.
The Chinese also made use of the complex combinatorial diagram known as the
magic squaremagic circles,
Indian mathematics
The numerals used in the
Bakhshali manuscript, dated between the 2nd century BCE and the 2nd century CE.
The earliest civilization on the Indian subcontinent is the
Indus Valley Civilization that flourished between 2600 and 1900 BC in the
Indus river basin. Their cities were laid out with geometric regularity, but no known mathematical documents survive from this civilization.
The oldest extant mathematical records from India are the
Shatapatha Brahmana (c. 9th century BC but estimates of the date vary widely). The
Sulba Sutras (c. 800 BC–200 AD),
[68] appendices to religious texts which give simple rules for constructing altars of various shapes, such as squares, rectangles, parallelograms, and others.
The Sulba Sutras give methods for constructing a
circle with approximately the same area as a given square, which imply several different approximations of the value of
π,
In addition, they compute the
square root of 2 to several decimal places, list Pythagorean triples, and give a statement of the
Pythagorean theorem.
Mesopotamian influence at this stage is considered likely.
The
Surya Siddhanta (c. 400) introduced the
trigonometric functions of
sine,
cosine, and inverse sine, and laid down rules to determine the true motions of the luminaries, which conforms to their actual positions in the sky.
This work was translated into to Arabic and Latin during the Middle Ages.
In the 5th century AD,
Aryabhata wrote the
Aryabhatiya, 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.
Though about half of the entries are wrong, it is in the
AryabhatiyaMuslim mathematician Abu Rayhan Biruni described the
Aryabhatiya as a "mix of common pebbles and costly crystals".
that the decimal place-value system first appears. Several centuries later, the
In the 7th century,
Brahmagupta identified the
Brahmagupta theorem,
Brahmagupta's identity and
Brahmagupta's formula, and for the first time, in
Brahma-sphuta-siddhanta, he lucidly explained the use of
zero as both a placeholder and
decimal digit, and explained the
Hindu-Arabic numeral system.
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
Arabic numerals. 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,
Halayudha's commentary on
Pingala's work contains a study of the
Fibonacci sequence and
Pascal's triangle, and describes the formation of a
matrix.
In the 12th century,
Bhāskara II 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,
the mean value theorem and the derivative of the sine function. To what extent he anticipated the invention of calculus is a controversial subject among historians of mathematics
In the 14th century,
Madhava of Sangamagrama, the founder of the so-called
Kerala School of Mathematics, found the
Madhava–Leibniz series, and, using 21 terms, computed the value of π as 3.14159265359. Madhava also found
the Madhava-Gregory series to determine the arctangent, the Madhava-Newton power series to determine sine and cosine and
the Taylor approximation for sine and cosine functions .
In the 16th century,
Jyesthadeva consolidated many of the Kerala School's developments and theorems in the
Yukti-bhāṣā.
[82] However, the Kerala School did not formulate a systematic theory of
differentiation and
integration, nor is there any direct evidence of their results being transmitted outside Kerala.
Muslim rule in India.
Progress in mathematics along with other fields of science stagnated in India with the establishment of
== Islamic mathematics == wada abbbbba
The
Islamic Empire established across
Persia, the
Middle East,
Central Asia,
North Africa,
Iberia, and in parts of
India in the 8th century made significant contributions towards mathematics. Although most Islamic texts on mathematics were written in
Arabic, most of them were not written by
Arabs, 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.
Persians contributed to the world of Mathematics alongside Arabs.
In the 9th century, the
Persian mathematician
Muḥammad ibn Mūsā al-Khwārizmī wrote several important books on the Hindu-Arabic numerals and on methods for solving equations. His book
On the Calculation with Hindu Numerals, written about 825, along with the work of
Al-Kindi, were instrumental in spreading
Indian mathematics and
Indian numerals to the West. The word
algorithm is derived from the Latinization of his name, Algoritmi, and the word
algebra from the title of one of his works,
Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala (
The Compendious Book on Calculation by Completion and Balancing). He gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,
and he was the first to teach algebra in an
elementary form and for its own sake.
He also discussed the fundamental method of "
reduction" 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
al-jabr.
His algebra was also no longer concerned "with a series of
problems to be resolved, but an
exposition 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."
Further developments in algebra were made by
Al-Karaji in his treatise
al-Fakhri, where he extends the methodology to incorporate integer powers and integer roots of unknown quantities. Something close to a
proof by
mathematical induction appears in a book written by Al-Karaji around 1000 AD, who used it to prove the
binomial theorem,
Pascal's triangle, and the sum of
integral cubes.
The
historian of mathematics, F. Woepcke,
praised Al-Karaji for being "the first who introduced the
theory of
algebraic calculus." Also in the 10th century,
Abul Wafa translated the works of
Diophantus into Arabic.
Ibn al-Haytham 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
paraboloid, and was able to generalize his result for the integrals of
polynomials up to the
fourth degree. He thus came close to finding a general formula for the
integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree.
Medieval European mathematics
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
Plato's
Timaeus and the biblical passage (in the
Book of Wisdom) that God had
ordered all things in measure, and number, and weight.
Boethius provided a place for mathematics in the curriculum in the 6th century when he coined the term
quadrivium to describe the study of arithmetic, geometry, astronomy, and music. He wrote
De institutione arithmetica, a free translation from the Greek of
Nicomachus's
Introduction to Arithmetic;
De institutione musica, also derived from Greek sources; and a series of excerpts from
Euclid's
Elements. His works were theoretical, rather than practical, and were the basis of mathematical study until the recovery of Greek and Arabic mathematical works.
These new sources sparked a renewal of mathematics.
Fibonacci, writing in the
Liber Abaci, in 1202 and updated in 1254, produced the first significant mathematics in Europe since the time of
Eratosthenes, a gap of more than a thousand years. The work introduced
Hindu-Arabic numerals to Europe, and discussed many other mathematical problems.
The 14th century saw the development of new mathematical concepts to investigate a wide range of problems.
One important contribution was development of mathematics of local motion.
Thomas Bradwardine 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).
Bradwardine's analysis is an example of transferring a mathematical technique used by
al-Kindi and
Arnald of Villanova to quantify the nature of compound medicines to a different physical problem.
Heytesbury and others mathematically determined the distance covered by a body undergoing uniformly accelerated motion (today solved by
integration), 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]".
[106]Nicole Oresme at the
University of Paris and the Italian
Giovanni di Casali independently provided graphical demonstrations of this relationship, asserting that the area under the line depicting the constant acceleration, represented the total distance traveled.
[107] In a later mathematical commentary on Euclid's
Elements, 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.
Renaissance mathematics
During the
Renaissance, the development of mathematics and of
accounting were intertwined.
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
Flanders and
Germany) or
abacus schools (known as
abbaco in Italy), where they learned the skills useful for trade and commerce. There is probably no need for algebra in performing
bookkeeping operations, but for complex bartering operations or the calculation of
compound interest, a basic knowledge of arithmetic was mandatory and knowledge of algebra was very useful.
Luca Pacioli's
"Summa de Arithmetica, Geometria, Proportioni et Proportionalità" (Italian: "Review of
Arithmetic,
Geometry,
Ratio and
Proportion") was first printed and published in
Venice in 1494. It included a 27-page
treatise on
bookkeeping,
"Particularis de Computis et Scripturis" (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
mathematical puzzles it contained, and to aid the education of their sons.
In
Summa Arithmetica, Pacioli introduced symbols for
plus and minus for the first time in a printed book, symbols that became standard notation in Italian Renaissance mathematics.
Summa Arithmetica was also the first known book printed in Italy to contain
algebra. It is important to note that Pacioli himself had borrowed much of the work of
Piero Della Francesca whom he plagiarized.
Driven by the demands of navigation and the growing need for accurate maps of large areas,
trigonometryBartholomaeus Pitiscus was the first to use the word, publishing his
Trigonometria in 1595. Regiomontanus's table of sines and cosines was published in 1533.
grew to be a major branch of mathematics.
By : Tharshini Devi Krishna Moorthy
