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{{histOfScience}}
'''Indian mathematics'''&mdash;which here is the mathematics that emerged in [[South Asia]] (known as the [[Indian subcontinent]])<ref name=plofker/> from ancient times until the end of the 18th century&mdash;had its beginnings in the [[Bronze Age]]  [[Indus Valley civilization]] (2600-1900 [[BCE]]) and the [[Iron Age]] [[Vedic culture]] (1500-500 BCE).   In the classical period of Indian mathematics ([[400 CE]] to [[1200 CE]]), important contributions were made by scholars like [[Aryabhata]], [[Brahmagupta]], and [[Bhaskara II]].  Indian mathematicians made early contributions to the study of the [[Decimal Number System|decimal number system]],<ref name=irfah346>{{Harv|Ifrah|2000|p=346}}: "The measure of the genius of Indian civilisation, to which we owe our modern (number) system, is all the greater in that it was the only one in all history to have achieved this triumph.  Some cultures succeeded, earlier than the Indian, in discovering one or at best two of the characteristics of this intellectual feat.  But none of them managed to bring together into a complete and coherent system the necessary and sufficient conditions for a number-system with the same potential as our own." </ref> [[0 (number)|zero]],<ref name=bourbaki46> {{Harv|Bourbaki|1998|p=46}}: "...our decimal system, which (by the agency of the Arabs) is derived from Hindu mathematics, where its use is attested already from the first centuries of our era.  It must be noted moreover that the conception of zero as a number and not as a simple symbol of separation) and its introduction into calculations, also count amongst the original contribution of the Hindus."</ref> [[negative numbers]],<ref name=bourbaki49> {{Harv|Bourbaki|1998|p=49}}: "On this point, the Hindus are already conscious of the interpretation that negative numbers must have in certain cases (a debt in a commercial problem, for instance).  In the following centuries, as there is a diffusion into the West (by intermediary of the Arabs) of the methods and results of Greek and Hindu mathematics, one becomes more used to the handling of these numbers, and one begins to have other "representation" for them which are geometric or dynamic."</ref> [[arithmetic]], and [[algebra]].<ref name=concise-britannica/>  In addition, [[trigonometry]], having evolved in the [[Hellenistic civilization|Hellenistic world]] and having been introduced into [[ancient India]] through the translation of [[Greek language|Greek]] works,<ref>{{Harv|Pingree|2003|p=45}} Quote: "Geometry, and its branch trigonometry, was the mathematics Indian astronomers used most frequently. In fact, the Indian astronomers in the third or fourth century, using a pre-Ptolemaic Greek table of chords, produced tables of sines and versines, from which it was trivial to derive cosines. This new system of trigonometry, produced in India, was transmitted to the Arabs in the late eighth century and by them, in an expanded form, to the Latin West and the Byzantine East in the twelfth century."</ref>
was further advanced in India, and, in particular, the modern definitions of [[sine]] and [[cosine]] were developed there.<ref>{{Harv|Bourbaki|1998|p=126}}: "As for trigonometry, it is disdained by geometers and abandoned to surveyors and astronomers; it is these latter ([[Aristarchus of Samos|Aristarchus]], [[Hipparchus]], [[Ptolemy]]) who establish the fundamental relations between the sides and angles of a right angled triangle (plane or spherical) and draw up the first tables (they consist of tables giving the ''chord'' of the arc cut out by an angle <math>\theta < \pi</math> on a circle of radius ''r'', in other words the number <math> 2r\sin\left(\theta/2\right)</math>; the introduction of the sine, more easily handled, is due to Hindu mathematicians of the Middle Ages)."</ref>  These mathematical concepts were transmitted to the [[Middle East]], [[China]], and [[Europe]]<ref name=concise-britannica> "algebra"  2007.  [http://www.britannica.com/ebc/article-231064 ''Britannica Concise Encyclopedia''].  Encyclopædia Britannica Online. 16 May 2007. Quote: "A full-fledged decimal, positional system certainly existed in India by the 9th century (AD), yet many of its central ideas had been transmitted well before that time to China and the Islamic world. Indian arithmetic, moreover, developed consistent and correct rules for operating with positive and negative numbers and for treating zero like any other number, even in problematic contexts such as division. Several hundred years passed before European mathematicians fully integrated such ideas into the developing discipline of algebra."</ref>  and led to further developments that now form the foundations of many areas of mathematics.  

Ancient and medieval Indian mathematical works, all composed in [[Sanskrit]], usually consisted of a section of ''[[sutra]]s'' in which a set of rules or problems were stated with great economy in verse in order to aid memorization by a student.  This was followed by a second section consisting of a prose commentary (sometimes multiple commentaries by different scholars) that explained the problem in more detail and provided justification for the solution.  In the prose section, the form (and therefore its memorization) was not considered as important as the ideas involved.<ref name=filliozat-p140to143>{{Harv|Filliozat|2004|pp=140-143}}</ref><ref name=plofker>{{Harv|Encyclopaedia Britannica (Kim Plofker)|2007|p=1}}</ref> All mathematical works were orally transmitted until approximately 500 BCE; thereafter, they were transmitted both orally and in manuscript form.  The oldest extant mathematical ''document'' produced on the [[Indian subcontinent]] is the birch bark [[Bakhshali Manuscript]], discovered in 1881 in the village of Bakhshali, near [[Peshawar]] (modern day [[Pakistan]]); the manuscript is likely from the seventh century CE.<ref name=hayashi95>{{Harv|Hayashi|1995}}</ref><ref name=plofker-brit6>{{Harv|Encyclopaedia Britannica (Kim Plofker)|2007|p=6}}</ref> 

A later landmark in Indian mathematics was the development of the [[Series (mathematics)|series]] expansions for [[trigonometric function]]s (sine, cosine, and [[arc tangent]]) by mathematicians of the [[Kerala School]] in the fifteenth century CE.  Their remarkable work, completed two centuries before the invention of [[calculus]] in Europe, provided what is now considered the first example of a [[power series]] (apart from geometric series).<ref>{{Harv|Stillwell|2004|p=173}}</ref> However, they did not formulate a systematic theory of [[derivative|differentiation]] and [[integral|integration]], nor is there any direct evidence of their results being transmitted outside [[Kerala]].<ref> {{Harv|Bressoud|2002|p=12}}</ref> (However, see [[#Charges of Eurocentrism|Charges of Eurocentrism]] below for recent research in this area.)

==Fields of Indian mathematics==

Some of the areas of mathematics studied in ancient and medieval India include the following:
*[[Arithmetic]]: [[Decimal system]], [[Negative number]]s  (see [[Brahmagupta]]), [[0 (number)|Zero]] (see [[Hindu-Arabic numeral system]]), the modern [[positional notation]] [[numeral system]], [[Floating point]] numbers (see [[Kerala School]]), [[Number theory]], [[Infinity]] (see [[Yajur Veda]]), [[Transfinite number]]s, [[Irrational number]]s (see [[Sulba Sutras]])
*[[Geometry]]: [[Square root]]s (see [[Methods of computing square roots#Bakhshali approximation|Bakhshali approximation]]), [[Cube root]]s (see [[Mahavira (mathematician)|Mahavira]]), [[Pythagorean triples]] (see [[Sulba Sutras]]; [[Baudhayana]] and [[Apastamba]] state the [[Pythagorean theorem]] without proof), [[Transformation (mathematics)|Transformation]] (see [[Pāṇini|Panini]]), [[Pascal's triangle]] (see [[Pingala]])
*[[Algebra]]: [[Quadratic equation]]s (see [[Sulba Sutras]], [[Aryabhata]], and [[Brahmagupta]]), [[Cubic equation]]s (see [[Mahavira (mathematician)|Mahavira]] and [[Bhaskara]]), [[Quartic equation]]s (biquadratic equations; see [[Mahavira (mathematician)|Mahavira]] and [[Bhaskara]])
*[[Mathematical logic]]: [[Formal grammar]]s, [[Formal language|formal language theory]], the [[Panini-Backus form]] (see [[Pāṇini|Panini]]), [[Recursion]] (see [[Pāṇini|Panini]])
* General mathematics: [[Fibonacci number]]s (see [[Pingala]]), Earliest forms of [[Morse code]] (see [[Pingala]]), [[Logarithm]]s, [[Index (mathematics)|indices]] (see [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Jaina_mathematics.html Jaina mathematics]), [[Algorithm]]s, [[Algorism]] (see [[Aryabhata]] and [[Brahmagupta]])
*[[Trigonometry]]: [[Trigonometric function]]s (see [[Surya Siddhanta]] and [[Aryabhata]]), [[Trigonometric series]] (see [[Madhava of Sangamagrama|Madhava]] and [[Kerala School]])

==§Harappan Mathematics (2600 BCE - 1700 BCE)§==
{{See also|Indus Valley Civilization}}
The earliest evidence of the use of mathematics in [[South Asia]] is in the artifacts of the [[Indus Valley Civilization]] (IVC), also called the ''Harappan civilization''.  Excavations at [[Harappa]], [[Mohenjo-daro]] and other locations in the [[Indus River|Indus river]] valley have uncovered evidence of the use of practical mathematics.  The people of the IVC manufactured bricks whose dimensions were in the proportion 4:2:1, considered favorable for the stability of a brick structure.  They used a standardized system of weights based on the ratios: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, with the unit weight equaling approximately 28 grams (and approximately equal to the English ounce or Greek uncia).  They mass produced weights in regular [[geometrical]] shapes, which included [[hexahedron|hexahedra]], [[barrel]]s, [[cone (geometry)|cone]]s, and [[cylinder (geometry)|cylinder]]s, thereby demonstrating knowledge of basic [[geometry]].  

The inhabitants of Indus civilization also tried to standardize measurement of length to a high degree of accuracy.  They designed a ruler&mdash;the ''Mohenjo-daro ruler''&mdash;whose unit of length (approximately 1.32 inches) was divided into ten equal parts.  Bricks manufactured in ancient Mohenjo-daro often had dimensions that were integral multiples of this unit of length.

==The Oral Mathematical Tradition==
Mathematicians of ancient and early medieval India were almost all [[Sanskrit]] [[pandit]]s (''paṇḍita'' "learned man"),<ref name=filliozat-p137>{{Harv|Filliozat|2004|p=137}}</ref> who were trained in Sanskrit language and literature, and possessed "a common stock of knowledge in grammar ([[vyakarana|''vyākaraṇa'']]), [[exegesis]] ([[mimamsa|''mīmāṃsā'']]) and logic ([[nyaya|''nyāya'']])."<ref name=filliozat-p137/>  Memorization of "what is heard" (''[[śruti]]'' in Sanskrit) through recitation played a major role in the transmission of sacred texts in ancient India.  Memorization and recitation was also used to transmit philosophical and literary works, as well as treatises on ritual and grammar.  Modern scholars of ancient India have noted the "truly remarkable achievements of the Indian pandits who have preserved enormously bulky texts orally for millennia."<ref name=pingree1988a>{{Harv|Pingree|1988|p=637}}</ref>
 
;<big>Styles of Memorization</big>

Prodigous energy was expended by ancient Indian culture in ensuring that these texts were transmitted from generation to generation with inordinate fidelity.<ref>{{Harv|Staal|1986}}</ref> For example, memorization of the sacred ''[[Veda]]s'' included up to eleven forms of recitation of the same text.  The texts were subsequently "proof-read" by comparing the different recited versions.  Forms of recitation included the ''jaṭā-pāṭha'' (literally "mesh recitation") in which every two adjacent words in the text were first recited in their original order, then repeated in the reverse order, and finally repeated again in the original order.<ref name=filliozat-p139>{{Harv|Filliozat|2004|p=139}}</ref>  The recitation thus proceeded as:
<center> '''word1word2, word2word1, word1word2; word2word3, word3word2, word2word3; ...'''</center>
In another form of recitation, ''dvaja-pāṭha''<ref name=filliozat-p139/> (literally "flag recitation") a sequence of ''N'' words were recited (and memorized) by pairing the first two and last two words and then proceeding as:
<center> '''word1word2, word(N-1)wordN; word2word3, word(N-3)word(N-2); ...; word(N-1)wordN, word1word2;'''</center>
The most complex form of recitation, ''ghana-pāṭha'' (literally "dense recitation"), according to {{Harv|Filliozat|2004|p=139}}, took the form:
<center>'''word1word2, word2word1, word1word2word3, word3word2word1, word1word2word3; word2word3, word3word2, word2word3word4, word4word3word2, word2word3word4; ... '''</center>
That these methods have been effective, is testified to by the preservation of the most ancient Indian religious text, the ''[[Rigveda|Ṛgveda]]'' ([[circa|ca.]] 1500 BCE), as a single text, without any variant readings.<ref name=filliozat-p139/>  Similar methods were used for memorizing mathematical texts, whose transmission remained exclusively oral until the end of the [[Vedic period]] (ca. 500 BCE).

;<big>The ''Sūtra'' Genre</big>

Mathematical activity in ancient India began as a part of a "methodological reflexion" on the sacred [[Veda]]s, which took the form of works called [[Vedanga|''Vedāṇgas'']], or, "Ancillaries of the Veda" (7th-4th century BCE).<ref name=filliozat2004-p140-141>{{Harv|Filliozat|2004|pp=140-141}}</ref>  The need to conserve the sound of sacred text by use of [[shiksha|''śikṣā'']] ([[phonetics]]) and ''[[chandas]]'' ([[Metre (poetry)|metric]]s); to conserve its meaning by use of [[vyakarana|''vyākaraṇa'']] ([[grammar]]) and ''[[nirukta]]'' ([[etymology]]); and to correctly perform the rites at the correct time by the use of ''[[kalpa]]'' ([[ritual]]) and [[jyotisha|''jyotiṣa'']] ([[astronomy]]), gave rise to the six disciplines of the ''Vedāṇgas''.<ref name=filliozat2004-p140-141/>  Mathematics arose as a part of the last two disciplines, ritual and astronomy (which also included astrology).
Since the ''Vedāṇgas'' immediately preceded the use of writing in ancient India, they formed the last of the exclusively oral literature.  They were expressed in a highly compressed mnemonic form, the [[sutra|''sūtra'']] (literally, "thread"):

<blockquote>The knowers of the ''sūtra'' know it as having few phonemes, being devoid of ambiguity, containing the essence, facing everything, being without pause and unobjectionable.<ref name=filliozat2004-p140-141/></blockquote>
Extreme brevity was achieved through multiple means, which included using [[ellipsis]] "beyond the tolerance of natural language,"<ref name=filliozat2004-p140-141/> using technical names instead of longer descriptive names, abridging lists by only mentioning the first and last entries, and using markers and variables.<ref name=filliozat2004-p140-141/> The ''sūtras'' create the impression that communication through the text was "only a part of the whole instruction.  The rest of the instruction must have been transmitted by the so-called [[Guru-shishya tradition|''Guru-shishya parampara'']], 'uninterrupted succession from  teacher (''guru'') to the student (''śisya''),' and it was not open to the general public" and perhaps even kept secret.<ref>{{Harv|Yano|2006|p=146}}</ref>  The brevity achieved in a ''sūtra'' is demonstrated in the following example from the Baudhāyana ''Śulba Sūtra'' (700 BCE). 
[[Image:Domestic fire altar.jpg|thumb|right|300px|The design of the domestic fire altar in the ''Śulba Sūtra'']]
The domestic fire-altar in the [[Vedic period]] was required by ritual to have a square base and be constituted of five layers of bricks with 21 bricks in each layer.  One method of constructing the altar was to divide one side of the square into three equal parts using a cord or rope, to next divide the transverse (or perpendicular) side into seven equal parts, and thereby sub-divide the square into 21 congruent rectangles.  The bricks were then designed to be of the shape of the constituent rectangle and the layer was created.  To form the next layer, the same formula was used, but the bricks were arranged transversely.<ref name=filliozat2004-p143-144>{{Harv|Filliozat|2004|pp=143-144}}</ref>  The process was then repeated three more times (with alternating directions) in order to complete the construction.   In the Baudhāyana ''Śulba Sūtra'', this procedure is described in the following words: <blockquote>"II.64. After dividing the quadri-lateral in seven, one divides the transverse [cord] in three.</br>II.65. In another layer one places the [bricks] North-pointing."<ref name=filliozat2004-p143-144/></blockquote>

According to {{Harv|Filliozat|2004|p=144}}, the officiant constructing the altar has only a few tools and materials at his disposal: a cord (Sanskrit, ''rajju'', f.), two pegs (Sanskrit, ''śanku'', m.), and clay to make the bricks (Sanskrit, ''iṣṭakā'', f.).  Concision is achieved in the ''sūtra'', by not explicitly mentioning what the adjective "transverse" qualifies; however, from the feminine form of the (Sanskrit) adjective used, it is easily inferred to qualify "cord."  Similarly, in the second stanza, "bricks"  are not explicitly mentioned, but inferred again by the feminine plural form of "North-pointing."  Finally, the first stanza, never explicitly says that the first layer of bricks are oriented in the East-West direction, but that too is implied by the explicit mention of "North-pointing" in the ''second'' stanza; for, if the orientation was meant to be the same in the two layers, it would either not be mentioned at all or be only mentioned in the first stanza.  All these inferences are made by the officiant as he recalls the formula from his memory.<ref name=filliozat2004-p143-144/>

==Vedic Period (1500 BCE - 400 BCE)==
{{See also|Vedanga|Vedas}}
The religious texts of the [[Vedic Period]] provide evidence for  the use of [[History of large numbers|large numbers]].   By the time of the last Veda, the ''[[Yajurveda|Yajurvedasaṃhitā]]'' (1200-900 BCE), numbers as high as <math>10^{12}</math> were being included in the texts.<ref name=hayashi2005-p360-361>{{Harv|Hayashi|2005|pp=360-361}} </ref>  For example, the [[Mantra|''mantra'']] (sacrificial formula) at the end of the ''annahoma'' ("food-oblation rite") performed during the [[Ashvamedha|''aśvamedha'']] ("horse sacrifice"), and uttered just before-, during-, and just after sunrise,  invokes powers of ten from a hundred to a trillion:<ref name=hayashi2005-p360-361/> <blockquote> "Hail to ''śata'' ("hundred," <math>10^2</math>), hail to ''sahasra'' ("thousand," <math>10^3</math>), hail to ''ayuta'' ("ten thousand," <math>10^4</math>), hail to ''niyuta'' ("hundred thousand," <math>10^4</math>), hail to ''prayuta'' ("million," <math>10^6</math>), hail to ''arbuda'' ("ten million," <math>10^7</math>), hail to ''nyarbuda'' ("hundred million," <math>10^8</math>), hail to ''samudra'' ("billion," <math>10^9</math>, literally "ocean"), hail to ''madhya'' ("ten billion," <math>10^{10}</math>, literally "middle"), hail to ''anta'' ("hundred billion," <math>10^{11}</math>, lit., "end"), hail to ''parārdha'' ("one trillion," <math>10^{12}</math> lit., "beyond parts"), hail to the dawn (''uśas''), hail to the twilight (''vyuṣṭi''), hail to the one which is going to rise (''udeṣyat''), hail to the one which is rising (''udyat''), hail to the one which has just risen (''udita''), hail to the heaven (''svarga''), hail to the world (''loka''), hail to all."<ref name=hayashi2005-p360-361/></blockquote>  The [[Satapatha Brahmana]] (9th century BCE) contains rules for ritual geometric constructions that are similar to the Sulba Sutras.<ref>A. Seidenberg, 1978. The origin of mathematics. Archive for the history of Exact Sciences, vol 18.</ref> 

;<big>Śulba Sūtras</big>

The ''[[Shulba Sutras|Śulba Sūtras]]'' (literally, "Aphorisms of the Chords" in [[Vedic Sanskrit]]) (c. 700-400 BCE) list rules for the construction of sacrificial fire altars.<ref>{{Harv|Staal|1999}}</ref>  Most mathematical problems considered in the  ''Śulba Sūtras'' spring from "a single theological requirement,"<ref name=hayashi2003-p118>{{Harv|Hayashi|2003|p=118}}</ref>  that of constructing fire altars which have different shapes but occupy the same area.  The altars were required to be constructed of five layers of burnt brick, with the further condition that each layer consist of 200 bricks and that no two adjacent layers have congruent arrangements of bricks.<ref name=hayashi2003-p118/>  

According to {{Harv|Hayashi|2005|p=363}}, the ''Śulba Sūtras'' contain  "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the [[Old Babylonian]]s." <blockquote>The diagonal rope (''akṣṇayā-rajju'') of an oblong (rectangle) produces both which the flank (''pārśvamāni'') and the horizontal (''tiryaṇmānī'') <ropes> produce separately."<ref name=hayashi2005-p363>{{Harv|Hayashi|2005|p=363}}</ref></blockquote> Since the statement is a ''sūtra'', it is necessarily compressed and what the ropes ''produce'' is not elaborated on, but the context clearly implies the square areas constructed on their lengths, and would have been explained so by the teacher to the student.<ref name=hayashi2005-p363/>

They contain lists of [[Pythagorean triples]],<ref>Pythagorean triples are triples of integers <math> (a,b,c) </math> with the property: <math>a^2+b^2=c^2</math>.  Thus, <math>3^2+4^2=5^2</math>, <math>8^2+15^2=17^2</math>, <math>12^2+35^2=37^2</math> etc.</ref> which are particular cases of [[Diophantine equations]].<ref name=cooke198>{{Harv|Cooke|2005|p=198}}: "The arithmetic content of the ''Śulva Sūtras'' consists of rules for finding Pythagorean triples such as (3, 4, 5), (5, 12, 13), (8, 15, 17), and (12, 35, 37). It is not certain what practical use these arithmetic rules had. The best conjecture is that they were part of religious ritual. A Hindu home was required to have three fires burning at three different altars. The three altars were to be of different shapes, but all three were to have the same area. These conditions led to certain "Diophantine" problems, a particular case of which is the generation of Pythagorean triples, so as to make one square integer equal to the sum of two others."</ref> 
They also contain statements (that with hindsight we know to be approximate) about [[squaring the circle]] and "circling the square."<ref name=cooke199-200>{{Harv|Cooke|2005|pp=199-200}}: "The requirement of three altars of equal areas but different shapes would explain the interest in transformation of areas. Among other transformation of area problems the Hindus considered in particular the problem of squaring the circle. The ''Bodhayana Sutra'' states the converse problem of constructing a circle equal to a given square. The following approximate construction is given as the solution.... this result is only approximate. The authors, however, made no distinction between the two results. In terms that we can appreciate, this construction gives a value for π of 18&thinsp;(3&nbsp;&minus;&nbsp;2√2), which is about 3.088."</ref>

[[Baudhayana]] (c. 8th century BCE) composed the ''Baudhayana Sulba Sutra'', the best-known ''Sulba Sutra'', which contains examples of simple Pythagorean triples, such as: <math>(3, 4, 5)</math>, <math>(5, 12, 13)</math>, <math>(8, 15, 17)</math>, <math>
(7, 24, 25)</math>, and <math>(12, 35, 37)</math><ref name=joseph229> {{Harv|Joseph|2000|p=229}}</ref>  as well as a statement of the Pythagorean theorem for the sides of a square: "The rope which is stretched across the diagonal of a square produces an area double the size of the original square."<ref name=joseph229/>  It also contains the general statement of the Pythagorean theorem (for the sides of a rectangle): "The rope stretched along the length of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together."<ref name=joseph229/>  Baudhayana gives a formula for the [[square root of two]],<ref name=cooke200>{{Harv|Cooke|2005|p=200}}</ref>
::<math>\sqrt{2} = 1 + \frac{1}{3} + \frac{1}{3\cdot4} - \frac{1}{3\cdot 4\cdot 34} \approx 1.4142156 \cdots</math> 
The formula is accurate up to five decimal places, the true value being <math>1.41421356 \cdots </math><ref>The value of this approximation, 577/408, is the seventh in a sequence of increasingly accurate approximations 3/2, 7/5, 17/12, ... to √2, the numerators and denominators of which were known as "side and diameter numbers" to the ancient Greeks, and in modern mathematics are called the [[Pell numbers]]. If ''x''/''y'' is one term in this sequence of approximations, the next is (''x''+2''y'')/(''x''+''y''). These approximations may also be derived by truncating the [[continued fraction]] representation of √2.</ref> This formula is similar in structure to the formula found on a Mesopotamian tablet<ref> Neugebauer, O. and A. Sachs.  1945.  ''Mathematical Cuneiform Texts'', New Haven, CT, Yale University Press. p. 45.</ref> from the Old Babylonian period (1900-1600 [[BCE]]):<ref name=cooke200/>
:::<math>\sqrt{2} = 1 + \frac{24}{60} + \frac{51}{60^2} + \frac{10}{60^3} = 1.41421297.</math>
which expresses <math>\sqrt{2}</math> in the sexagesimal system, and which too is accurate up to 5 decimal places (after rounding).

According to mathematician S. G. Dani, the Babylonian cuneiform tablet [[Plimpton 322]] written ca. 1850 [[BCE]]<ref>Mathematics Department, University of British Columbia, [http://www.math.ubc.ca/~cass/courses/m446-03/pl322/pl322.html ''The Babylonian tabled Plimpton 322''].</ref> "contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple,<ref>Three positive integers <math>(a, b, c) </math> form a ''primitive'' Pythagorean triple if <math> c^2=a^2+b^2</math> and if the highest common factor of <math> a, b, c </math> is 1. In the particular Plimpton322 example, this means that <math> 13500^2+ 12709^2= 18541^2 </math> and that the three numbers do not have any common factors. However some scholars have disputed the Pythagorean interpretation of this tablet; see [[Plimpton 322]] for details.</ref> indicating, in particular, that there was sophisticated understanding on the topic" in Mesopotamia in 1850 [[BCE]].  "Since these tablets predate the Sulbasutras period by several centuries, taking into account the contextual appearance of some of the triples, it is reasonable to expect that similar understanding would have been there in India."<ref name=dani> {{Harv|Dani|2003}}</ref>  Dani goes on to say:

<blockquote> "As the main objective of the ''Sulvasutras'' was to describe the constructions of altars and the geometric principles involved in them, the subject of Pythagorean triples, even if it had been well understood may still not have featured in the ''Sulvasutras''.  The occurrence of the triples in the ''Sulvasutras'' is comparable to mathematics that one may encounter in an introductory book on architecture or another similar applied area, and
would not correspond directly to the overall knowledge on the topic at that time. Since, unfortunately, no other contemporaneous sources have been found it may never be possible to settle this issue satisfactorily."<ref name=dani/></blockquote>

In all three ''Sulba Sutras'' were composed.  The remaining two, the ''Manava Sulba Sutra'' composed by [[Manava]] (fl. 750-650 BCE) and the ''Apastamba Sulba Sutra'', composed by [[Apastamba]] (c. 600 BCE), contained results similar to the ''Baudhayana Sulba Sutra''.

;Vyakarana

An important landmark of the Vedic period was the work of [[Sanskrit grammarian]], {{IAST|[[Pāṇini]]}} (c. 520-460 BCE). His grammar includes early use of [[Boolean logic]], of the [[null]] operator, and of [[context free grammar]]s, and includes a precursor of the [[Backus–Naur form]] (used in the description [[programming languages]]).

==Jaina Mathematics (400 BCE - 200 CE)==
Although [[Jainism]] as a religion and philosophy predates its most famous exponent, [[Mahavira]] ([[6th century BC]]), who was a contemporary of [[Gautama Buddha]], most Jaina texts on mathematical topcs were composed after the 6th century BCE. [[Jain]]a mathematicians are important historically as crucial links between the mathematics of the Vedic period and that of the "Classical period."

A significant historical contribution of Jaina mathematicians lay in their freeing Indian mathematics from its religious and ritualistic constraints.
In particular, their fascination with the enumeration of very large numbers and [[infinity|infinities]], led them to classify numbers into three classes: enumerable, innumerable and [[infinite]].  Not content with a simple notion of infinity, they went on to define five different types of infinity: the infinite in one direction, the infinite in two directions, the infinite in area, the infinite everywhere, and the infinite perpetually.  In addition, Jaina mathematicians devised notations for simple powers (and exponents) of numbers like squares and cubes, which enabled them to define simple [[algebraic equations]] (''beezganit samikaran'').  Jaina mathematicians were apparently also the first to use the word ''shunya'' (literally ''void'' in [[Sanskrit language|Sanskrit]]) to refer to zero.  More than a millennium later, their appellation  became the English word "zero" after a tortuous journey of translations and transliterations from India to Europe . (See [[0 (number)#Etymology|Zero: Etymology]].)

In addition to ''Surya Prajnapti'', important Jaina works on mathematics included the ''[[Vaishali]] Ganit'' (c. 3rd century BCE); the ''Sthananga Sutra'' (fl. 300 BCE - 200 CE); the ''Anoyogdwar Sutra'' (fl. 200 BCE - 100 CE); and the ''Satkhandagama'' (c. 2nd century CE).  Important Jaina mathematicians included [[Bhadrabahu]] (d. 298 BCE), the author of two astronomical works, the ''Bhadrabahavi-Samhita'' and a commentary on the ''Surya Prajinapti''; Yativrisham Acharya (c. 176 BCE), who authored a mathematical text called ''Tiloyapannati''; and [[Umasvati]] (c. 150 BCE), who, although better known for his influential writings on Jaina philosophy and [[metaphysics]], composed a mathematical work called ''Tattwarthadhigama-Sutra Bhashya''.

;Pingala
Among other scholars of this period who contributed to mathematics, the most notable is [[Pingala]] (''{{IAST|piṅgalá}}'')  ([[Floruit|fl.]] 300-200 BCE), a [[musical theory|musical theorist]] who authored  the '''''[[Chandas]] [[Shastra]]''''' (''{{IAST|chandaḥ-śāstra}}'', also '''''Chandas Sutra''''' ''{{IAST|chandaḥ-sūtra}}''), a [[Sanskrit]] treatise on [[prosody]].  There is evidence that in his work on the enumeration of syllabic combinations, Pingala stumbled upon both the [[Pascal triangle]] and [[Binomial coefficients]], although he did not have knowledge of the [[Binomial theorem]] itself.<ref name=fowler96>{{Harv|Fowler|1996|p=11}}</ref><ref name=singh36>{{Harv|Singh|1936|pp=623-624}}</ref>  Pingala's work also contains the basic ideas of [[Fibonacci number]]s (called ''maatraameru''). Although the ''Chandah sutra'' hasn't survived in its entirety, a 10th century commentary on it by Halāyudha has.  Halāyudha, who refers to the Pascal triangle as ''[[Mount Meru (mythology)|Meru]]-prastāra'' (literally "the staircase to [[Mount Meru (mythology)|Mount Meru]]"), has this to say: <blockquote> "Draw a square. Beginning at half the square, draw two other similar squares below it; below these two, three other squares, and so on. The marking should be started by putting '''1''' in the first square. Put '''1''' in each of the two squares of the second line. In the third line put '''1''' in the two squares at the ends and, in the middle square, the sum of the digits in the two squares lying above it.  In the fourth line put '''1''' in the two squares at the ends. In the middle ones put the sum of the digits in the two squares above each. Proceed in this way. Of these lines, the second gives the combinations with one syllable, the third the combinations with two syllables, ..."<ref name=fowler96/></blockquote>

The text also indicates that Pingala was aware of the [[Combinatorics|combinatorial]] identity:<ref name=singh36/>

<center><math> {n \choose 0} + {n \choose 1} + {n \choose 2} + \cdots + {n \choose n-1} + {n \choose n} = 2^n </math></center>

;Katyayana
Though not a Jaina mathematician, [[Katyayana]] (c. 3rd century BCE) is notable for being the last of the Vedic mathematicians. He wrote the ''Katyayana Sulba Sutra'', which presented much [[geometry]], including the general [[Pythagorean theorem]] and a computation of the square root of 2 correct to five decimal places.

==The Written Tradition: Prose Commentary==
With the increasing complexity of mathematics and other exact sciences, both writing and computation were required. Consequently, many mathematical works began to be written down in manuscripts that were then copied and re-copied from generation to generation. <blockquote> "India today is estimated to have about thirty million manuscripts, the largest body of handwritten reading material anywhere in the world.  The literate culture of Indian science goes back to at least the fifth century B.C. ... as is shown by the elements of Mesopotamian omen literature and astronomy that entered India at that time and (were) definitely not ... preserved orally."<ref name=pingree1988b>{{Harv|Pingree|1988|p=638}}</ref></blockquote> 

The earliest mathematical prose commentary was that on the work, ''[[Aryabhatiya|Āryabhaṭīya]]'' (written 499 CE), a work on astronomy and mathematics.  The mathematical portion of the ''Āryabhaṭīya'' was composed of 33 ''sūtras'' (in verse form) consisting of mathematical statements or rules, but without any proofs.<ref name=hayashi03-p122-123>{{Harv|Hayashi|2003|pp=122-123}}</ref>  However, according to {{Harv|Hayashi|2003|p=123}}, "this does not necessarily mean that their authors did not prove them.  It was probably a matter of style of exposition."  From the time of [[Bhaskara I]] (600 CE onwards), prose commentaries increasingly began to include some derivations (''upapatti'').  Bhaskara I's commentary on the ''Āryabhaṭīya'', had the following structure:<ref name=hayashi03-p122-123/>

*'''Rule''' ('sūtra') in verse by [[Aryabhata|Āryabhaṭa]]
*'''Commentary''' by [[Bhaskara I|Bhāskara I]], consisting of:
**'''Elucidation''' of rule (derivations were still rare then, but became more common later)
**'''Example''' (''uddeśaka'') usually in verse.
**'''Setting''' (''nyāsa/sthāpanā'') of the numerical data.
**'''Working''' (''karana'') of the solution.
**'''Verification''' (''pratyayakaraṇa'', literally "to make conviction") of the answer.  These became rare by the 13th century, derivations or proofs being favored by then.<ref name=hayashi03-p122-123/>

Typically, for any mathematical topic, students in ancient India first memorized the ''sūtras'', which, as explained earlier, were "deliberately inadequate"<ref name=pingree1988b/> in explanatory details (in order to pithily convey the bare-bone mathematical rules).  The students then worked through the topics of the prose commentary by writing (and drawing diagrams) on chalk- and dust-boards (''i.e.'' boards covered with dust).  The latter activity, a staple of mathematical work, was to later prompt mathematician-astronomer, [[Brahmagupta]] ([[floruit|fl.]] 7th century CE), to characterize astronomical computations as "dust work" (Sanskrit: ''dhulikarman'').<ref>{{Harv|Hayashi|2003|p=119}}</ref>

==Numerals and the Decimal Number System==

The earliest extant [[writing system|script]] used in India was the [[Kharoṣṭhī]] script used in the [[Gandhara]] culture of the north-west. It is thought to be of [[Aramaic]] origin and it was in use from the fourth century BCE to the fourth century CE. Almost contemporaneously, another script, the [[Brahmi]], appeared on much of the sub-continent, and would later become the foundation of many scripts of South Asia and South-east Asia. Both scripts had numeral symbols and numeral systems, which were initially ''not'' based on a place-value system.<ref name=hayashi2005-p366>{{Harv|Hayashi|2005|p=366}}</ref> The first datable evidence of the use of the decimal place-value system in India is found in the [[Yavanajataka|''Yavanajātaka'']] ([[circa|ca.]] 270 CE) of Sphujidhvaja, a versification of an earlier (ca. 150 CE) Indian prose adaptation of a lost work of Hellenistic astrology.<ref>{{Harv|Pingree|1978|p=494}}</ref>

==Bakhshali Manuscript==
The oldest extant mathematical manuscript in South Asia is the ''[[Bakhshali Manuscript]]'', a birch bark manuscript written in "Buddhist hybrid Sanskrit"<ref name=plofker-brit6/> in the ''Śāradā'' script, which was used in the northwestern region of the Indian subcontinent between the 8th and 12th centuries CE.<ref name=hayashi2005-371>{{Harv|Hayashi|2005|p=371}}</ref> The manuscript was discovered in 1881 by a farmer while digging in a stone enclosure in the village of Bakhshali, near [[Peshawar]] (then in [[British India]] and now in [[Pakistan]]).  Of unknown authorship and now preserved in the [[Bodleian Library]] in [[Oxford University]], the manuscript has been variously dated&mdash;as early as the "early centuries of the Christian era"<ref name=datta1931>{{Harv|Datta|1931|p=566}}</ref> and as late as between the 9th and 12th century CE.<ref>{{Harv|Ifrah|2000|p=464}} Quote: "To give the second or fourth century CE as the date of this document would be an evident contradiction; it would mean that a northern derivative of ''[[Gupta Empire|Gupta]]'' writing had been developed two or three centuries before the ''Gupta'' writing itself appeared.  ''Gupta'' only began to evolve into ''Shāradā'' style around the ninth century CE.  In other words, the Bak(h)shali manuscript cannot have been written earlier than the ninth century CE.  However, in the light of certain characteristic indications, it could not have been written any later than the twelfth century CE."</ref> The 7th century CE is now considered a plausible date,<ref>{{Harv|Hayashi|2005|p=371}} Quote:"The dates so far proposed for the Bakhshali work vary from the third to the twelfth centuries AD, but a recently made comparative study has shown many similarities, particularly in the style of exposition and terminology, between Bakhshalī work and Bhāskara I's commentary on the ''Āryabhatīya''.  This seems to indicate that both works belong to nearly the same period, although this does not deny the possibility that some of the rules and examples in the Bakhshālī work date from anterior periods."</ref> albeit with the likelihood that the "manuscript in its present-day form constitutes a commentary or a copy of an anterior mathematical work."<ref>{{Harv|Ifrah|2000|p=464}}</ref>

The surviving manuscript has seventy leaves, some of which are in fragments. Its mathematical content consists of rules and examples, written in verse, together with prose commentaries, which include solutions to the examples.<ref name=hayashi2005-371/>   The topics treated include arithmetic (fractions, square roots, profit and loss, simple interest, the [[rule of three]], and ''[[regula falsi]]'') and algebra (simultaneous linear equations and [[quadratic equations]]), and arithmetic progressions.  In addition, there is a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero."<ref name=hayashi2005-371/>  Many of its problems are the so-called equalization problems that lead to systems of linear equations.  One example from Fragment III-5-3v is the following: <blockquote>"One merchant has seven ''asava'' horses, a second has nine ''haya'' horses, and a third has ten camels.  They are equally well off in the value of their animals if each gives two animals, one to each of the others.  Find the price of each animal and the total value for the animals possessed by each merchant."<ref name=anton>Anton, Howard and Chris Rorres.  2005.  ''Elementary Linear Algebra with Applications.'' 9th edition.  New York: John Wiley and Sons.  864 pages.  {{ISBN|0471669598}}.</ref></blockquote>

The prose commentary accompanying the example solves the problem by converting it to three (under-determined) equations in four unknowns and assuming that the prices are all integers.<ref name=anton/>

==Classical Period (400 - 1200)==
This period is often known as the golden age of Indian Mathematics. This period saw mathematicians such as [[Aryabhata]], [[Varahamihira]], [[Brahmagupta]], [[Bhaskara I]], [[Mahavira (mathematician)|Mahavira]], and [[Bhaskara II]] give broader and clearer shape to many branches of mathematics. Their contributions would spread to [[Asia]], the [[Middle East]], and eventually to [[Europe]].  Unlike Vedic mathematics, their works included both astronomical and mathematical contributions.  In fact, mathematics of that period was included in the 'astral science' (''jyotiḥśāstra'') and consisted of three sub-disciplines: mathematical sciences (''gaṇita'' or ''tantra''), horoscope astrology (''horā'' or ''jātaka'') and divination (saṃhitā).<ref name=hayashi2003-p119>{{Harv|Hayashi|2003|p=119}}</ref>  This tripartite division is seen in [[Varahamihira|Varāhamihira's]] sixth century compilation&mdash;''Pancasiddhantika''<ref>{{Harv|Neugebauer|Pingree (eds.)|1970}}</ref> (literally ''panca'', "five," ''siddhānta'', "conclusion of deliberation", dated [[575]] [[Common Era|CE]])&mdash;of five earlier works,  [[Surya Siddhanta]], [[Romaka Siddhanta]], [[Paulisa Siddhanta]], [[Vasishtha Siddhanta]] and [[Paitamaha Siddhanta]], which were adaptations of still earlier works of Mesopotamian, Greek, Egyptian, Roman and Indian astronomy.  As explained earlier, the main texts were composed in Sanskrit verse, and were followed by prose commentaries.<ref name=hayashi2003-p119/>

===Fifth and Sixth Centuries===
;Surya Siddhanta

Though its authorship is unknown, the ''[[Surya Siddhanta]]'' (c. 400) contains the roots of modern [[trigonometry]]. Due to the large number of foreign words in the documents, Historians have concluded that its roots are in Mesopotamia and Greece.<ref name="Origins of Sulva Sutras and Siddhanta">{{cite book|first=Roger|last=Cooke|authorlink=Roger Cooke|title=The History of Mathematics: A Brief Course|publisher=Wiley-Interscience|year=1997|chapter=The Mathematics of the Hindus|pages=197|isbn=0471180823|quote=The word ''Siddhanta'' means ''that which is proved or established''. The ''Sulva Sutras'' are of Hindu origin, but the ''Siddhantas'' contain so many words of foreign origin that they undoubtedly have roots in Mesopotamia and Greece.}}</ref> It uses the following as trigonometric functions for the first time:
*Sine (''Jya'').
*Cosine (''Kojya'').
*[[Inverse sine]] (''Otkram jya'').

It also contains the earliest uses of:
*[[Tangent]].
*[[Secant]].

*The Hindu cosmological time cycles explained in the text, which was copied from an earlier work, gives:
**The average length of the [[sidereal year]] as 365.2563627 days, which is only 1.4 seconds longer than the modern value of 365.2563627 days.
**The average length of the [[tropical year]] as 365.2421756 days, which is only 2 seconds shorter than the modern value of 365.2421988 days.

Later Indian mathematicians such as Aryabhata made references to this text, while later [[Arabic]] and [[Latin]] translations were very influential in Europe and the Middle East.

;Chhedi calendar

This Chhedi calendar (594) contains an early use of the modern [[place-value]] [[Hindu-Arabic numeral system]] now used universally (see also [[Hindu-Arabic numerals]]).

;Aryabhata I

[[Aryabhata]] (476-550) wrote the ''Aryabhatiya.'' He described the important fundamental principles of mathematics in 332 [[shlokas]]. The treatise contained:
*[[Quadratic equation]]s
*[[Trigonometry]]
*The value of [[π]], correct to 4 decimal places.

Aryabhata also wrote the ''Arya Siddhanta'', which is now lost. Aryabhata's contributions include:

Trigonometry:
*Introduced the [[trigonometric function]]s.
*Defined the sine (''jya'') as the modern relationship between half an angle and half a chord.
*Defined the cosine (''kojya'').
*Defined the [[versine]] (''ukramajya'').
*Defined the inverse sine (''otkram jya'').
*Gave methods of calculating their approximate numerical values.
*Contains the earliest tables of sine, cosine and versine values, in 3.75° intervals from 0° to 90°, to 4 decimal places of accuracy.
*Contains the trigonometric formula ''sin (n + 1) x - sin nx = sin nx - sin (n - 1) x - (1/225)sin nx''.
*[[Spherical trigonometry]].

Arithmetic:
*[[Continued fraction]]s.

Algebra:
*Solutions of simultaneous quadratic equations.
*Whole number solutions of [[linear equations]] by a method equivalent to the modern method.
*General solution of the indeterminate linear equation .

Mathematical astronomy:
*Proposed for the first time, a [[heliocentric]] [[solar system]] with the planets spinning on their [[Axis of rotation|axes]] and following an [[ellipse|elliptical]] orbit around the Sun.
*Accurate calculations for astronomical constants, such as the:
**[[Solar eclipse]].
**[[Lunar eclipse]].
**The formula for the sum of the [[Cube (algebra)|cubes]], which was an important step in the development of integral calculus.<ref>Victor J. Katz (1995). "Ideas of Calculus in Islam and India", ''Mathematics Magazine'' '''68''' (3), p. 163-174.</ref>

Calculus:
*[[Infinitesimal]]s:
**In the course of developing a precise mapping of the lunar eclipse, Aryabhatta was obliged to introduce the concept of infinitesimals (''tatkalika gati'') to designate the near instantaneous motion of the moon.<ref name=Joseph-298-300>Joseph (2000), p. 298-300.</ref>
*[[Differential equation]]s:
**He expressed the near instantaneous motion of the moon in the form of a basic differential equation.<ref name=Joseph-298-300/>
*[[Exponential function]]:
**He used the [[exponential function]] ''e'' in his differential equation of the near instantaneous motion of the moon.<ref name=Joseph-298-300/>

;Varahamihira

[[Varahamihira]] (505-587) produced the ''Pancha Siddhanta'' (''The Five Astronomical Canons''). He made important contributions to [[trigonometry]], including sine and cosine tables to 4 decimal places of accuracy and the following formulas relating [[sine]] and [[cosine]] functions:

*<math>\sin^2(x) + \cos^2(x) = 1</math>

*<math>\sin(x)=\cos\left(\frac{\pi}{2}-x\right)</math>

*<math>\frac{1-\cos(2x)}{2}=\sin^2(x)</math>

===Seventh and Eighth Centuries===
[[Image:Brahmaguptra's theorem.svg|thumb|right|200px|Brahmagupta's theorem states that ''AF'' = ''FD''.]]
In the seventh century, two separate fields, [[arithmetic]] (which included [[mensuration]]) and [[algebra]], began to emerge in Indian mathematics.  The two fields would later be called ''pāṭī-gaṇita'' (literally "mathematics of algorithms") and ''bīja-gaṇita'' (lit. "mathematics of seeds," with "seeds"&mdash;like the seeds of plants&mdash;representing unknowns with the potential to generate, in this case, the solutions of equations).<ref name=hayashi2005-p369>{{Harv|Hayashi|2005|p=369}}</ref>  [[Brahmagupta]], in his astronomical work ''[[Brahmasphutasiddhanta|Brāhma Sphuṭa Siddhānta]]'' (628 CE), included two chapters (12 and 18) devoted to these fields.  Chapter 12, containing 66 Sanskrit verses, was divided into two sections:  "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain).<ref name=hayashi2003-p121-122>{{Harv|Hayashi|2003|pp=121-122}}</ref>  In the latter section, he stated his famous theorem on the diagonals of a [[cyclic quadrilateral]]:<ref name=hayashi2003-p121-122/>

'''Brahmagupta's theorem:''' If a [[cyclic quadrilateral]] has diagonals that are [[perpendicular]] to each other, then the perpendicular line drawn from the point of intersection of the diagonals to any side of the quadrilateral always bisects the opposite side.  

Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of [[Heron's formula]]), as well as a complete description of [[rational triangle]]s (''i.e.'' triangles with rational sides and rational areas).

'''Brahmagupta's formula:''' The area, ''A'',  of a [[cyclic quadrilateral]] with sides of lengths ''a'', ''b'', ''c'', ''d'', respectively, is given by

: <math> A = \sqrt{(s-a)(s-b)(s-c)(s-d)}</math>

where ''s'', the [[semiperimeter]], given by: <math> s=\frac{a+b+c+d}{2}.</math>

'''Brahmagupta's Theorem on [[rational triangle]]s:''' A triangle with rational sides <math>a, b, c </math> and rational area is of the form:

:<math>a = \frac{u^2}{v}+v, \ \ b=\frac{u^2}{w}+w, \ \ c=\frac{u^2}{v}+\frac{u^2}{w} - (v+w) </math>
for some rational numbers <math>u, v, </math> and <math> w </math>.<ref>{{Harv|Stillwell|2004|p=77}}</ref>

Chapter 18 contained 103 Sanskrit verses which began with rules for arithmetical operations involving zero and negative numbers<ref name=hayashi2003-p121-122/> and is considered the first systematic treatment of the subject.  The rules (which included <math> a + 0 = \ a</math> and <math> a \times 0 = 0 </math>) were all correct, with one exception: <math> \frac{0}{0} = 0 </math>.<ref name=hayashi2003-p121-122/>  Later in the chapter, he gave the first explicit (although still not completely general) solution of the '''[[quadratic equation]]''': 

:<math>\ ax^2+bx=c</math>

{{cquote|To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value. (''Brahmasphutasiddhanta'' (Colebrook translation, 1817, page 346)<ref name=stillwell2004-p87>{{Harv|Stillwell|2004|p=87}}</ref>}}

This is equivalent to:

:<math>x = \frac{\sqrt{4ac+b^2}-b}{2a} </math>

Also in chapter 18, Brahmagupta was able to make progress in finding (integral) solutions of '''[[Pell's equation]]''',<ref name=stillwell2004-p72-73>{{Harv|Stillwell|2004|pp=72-73}}</ref>
:<math>\ x^2-Ny^2=1, </math>
where <math>N</math> is a nonsquare integer.  He did this by discovering the following identity:<ref name=stillwell2004-p72-73/>

'''Brahmagupta's Identity:''' <math> \ (x^2-Ny^2)(x'^2-Ny'^2) = (xx'+Nyy')^2 - N(xy'+x'y)^2 </math>
which was a generalization of an earlier identity of [[Diophantus]]:<ref name=stillwell2004-p72-73/>  Brahmagupta used his identity to prove the following lemma:<ref name=stillwell2004-p72-73/>

'''Lemma (Brahmagupta):''' If <math>x=x_1,\ \ y=y_1 \ \ </math> is a solution of <math> \ \ x^2 - Ny^2 = k_1, </math> and,
<math> x=x_2, \ \ y=y_2 \ \ </math> is a solution of <math> \ \ x^2 - Ny^2 = k_2, </math>, then:
:<math> x=x_1x_2+Ny_1y_2,\ \ y=x_1y_2+x_2y_1 \ \ </math> is a solution of <math> \ x^2-Ny^2=k_1k_2</math>

He then used this lemma to both generate infinitely many (integral) solutions of Pell's equation, given one solution, and state the following theorem: 

'''Theorem (Brahmagupta):''' If the equation <math> \ x^2 - Ny^2 =k </math> has an integer solution for any one of <math> \ k=\pm 4, \pm 2, -1 </math> then [[Pell's equation]]:
:<math> \ x^2 -Ny^2 = 1 </math>
also has an integer solution.<ref name=stillwell2004-p74-76>{{Harv|Stillwell|2004|pp=74-76}}</ref>  

Brahmagupta did not actually prove the theorem, but rather worked out examples using his method.  The first example he presented was:<ref name=stillwell2004-p72-73/>

'''Example (Brahmagupta):''' Find integers <math>\ x,\ y\ </math> such that:
:<math>\ x^2 - 92y^2=1 </math>
In his commentary, Brahmagupta added, "a person solving this problem within a year is a mathematician."<ref name=stillwell2004-p72-73/>  The solution he provided was: 
:<math>\ x=1151, \ y=120 </math>

;Bhaskara I

[[Bhaskara I]] (c. 600-680) expanded the work of Aryabhata in his books titled ''Mahabhaskariya'', ''Aryabhattiya Bhashya'' and ''Laghu Bhaskariya''. He produced:
*Solutions of indeterminate equations.
*A rational approximation of the [[sine function]].
*A formula for calculating the sine of an acute angle without the use of a table, correct to 2 decimal places.

===Ninth to Twelfth Centuries===

;Virasena

[[Virasena]] (9th century) was a Jaina mathematician in the court of [[Rashtrakuta]] King [[Amoghavarsha]] of [[Manyakheta]], Karnataka. He wrote the ''Dhavala'', a commentary on Jaina mathematics, which:
*Deals with logarithms to base 2 (''ardhaccheda'') and describes its laws.
*First uses logarithms to base 3 (''trakacheda'') and base 4 (''caturthacheda'').

Virasena also gave:
*The derivation of the [[volume]] of a [[frustum]] by a sort of infinite procedure.

;Mahavira

[[Mahavira (mathematician)|Mahavira Acharya]] (c. 800-870) from [[Karnataka]], the last of the notable Jaina mathematicians, lived in the [[9th century]] and was patronised by the [[Rashtrakuta]] king [[Amoghavarsha]]. He wrote a book titled ''Ganit Saar Sangraha'' on numerical mathematics, and also wrote treatises about a wide range of mathematical topics. These include the mathematics of:
*[[0 (number)|Zero]].
*[[Square (algebra)|Squares]].
*[[Cube (arithmetic)|Cubes]].
*[[square root]]s, [[cube root]]s, and the [[series (mathematics)|series]] extending beyond these.
*[[Plane geometry]].
*[[Solid geometry]].
*Problems relating to the casting of [[shadows]].
*Formulae derived to calculate the area of an [[ellipse]] and [[quadrilateral]] inside a [[circle]]

Mahavira also:
*Asserted that the [[square root]] of a [[negative number]] did not exist
*Gave the sum of a [[series (mathematics)|series]] whose terms are [[square (algebra)|square]]s of an [[arithmetical progression]], and gave empirical rules for [[area]] and [[perimeter]] of an [[ellipse]].
*Solved cubic equations.
*Solved quartic equations.
*Solved some [[quintic equation]]s and higher-order [[polynomial]]s.
*Gave the general solutions of the higher order polynomial equations:
**<math>\ ax^n = q</math>
**<math>a \frac{x^n - 1}{x - 1} = p</math>
*Solved indeterminate quadratic equations.
*Solved indeterminate cubic equations.
*Solved indeterminate higher order equations.

;Shridhara

[[Shridhara]] (c. 870-930), who lived in [[Bengal]], wrote the books titled ''Nav Shatika'', ''Tri Shatika'' and ''Pati Ganita''. He gave:
*A good rule for finding the [[volume]] of a [[sphere]].
*The formula for solving [[quadratic equation]]s.

The ''Pati Ganita'' is a work on arithmetic and [[mensuration]]. It deals with various operations, including:
*Elementary operations
*Extracting square and cube roots.
*Fractions.
*Eight rules given for operations involving zero.
*Methods of [[summation]] of different arithmetic and geometric series, which were to become standard references in later works.

;Manjula

Aryabhata's differential equations were elaborated in the 10th century by Manjula (also ''Munjala''), who realised that the expression<ref name=Joseph-298-300/>

<math>\ \sin w' - \sin w</math>

could be approximately expressed as

<math>\ (w' - w)\cos w</math>

He understood the concept of differentiation after solving the differential equation that resulted from substituting this expression into Aryabhata's differential equation.<ref name=Joseph-298-300/>

;Aryabhata II

[[Aryabhata II]] (c. 920-1000) wrote a commentary on Shridhara, and an astronomical treatise ''[[Maha-Siddhanta]]''. The Maha-Siddhanta has 18 chapters, and discusses:
*Numerical mathematics (''Ank Ganit'').
*Algebra.
*Solutions of indeterminate equations (''kuttaka'').

;Shripati

[[Sripati|Shripati Mishra]] (1019-1066) wrote the books ''Siddhanta Shekhara'', a major work on astronomy in 19 chapters, and ''Ganit Tilaka'', an incomplete [[arithmetic]]al treatise in 125 verses based on a work by [[Shridhara]]. He worked mainly on:
*[[Permutations and combinations]].
*General solution of the simultaneous indeterminate linear equation.

He was also the author of ''Dhikotidakarana'', a work of twenty verses on:
*[[Solar eclipse]].
*[[Lunar eclipse]].

The ''Dhruvamanasa'' is a work of 105 verses on:
*Calculating planetary [[longitude]]s
*[[eclipse]]s.
*planetary [[transit]]s.

;Nemichandra Siddhanta Chakravati

Nemichandra Siddhanta Chakravati (c. 1100) authored a mathematical treatise titled ''Gome-mat Saar''.

;Bhaskara II

[[Bhāskara II]] (1114-1185) was a mathematician-astronomer who wrote a number of important treatises, namely the ''Siddhanta Shiromani'', ''[[Lilavati]]'', ''Bijaganita'', ''Gola Addhaya'', ''Griha Ganitam'' and ''Karan Kautoohal''. A number of his contributions were later transmitted to the Middle East and Europe. His contributions include:

Arithmetic:
*Interest computation.
*Arithmetical and geometrical progressions.
*[[Plane geometry]].
*[[Solid geometry]].
*The shadow of the [[gnomon]].
*Solutions of [[combinations]].
*Gave a proof for division by [[0 (number)|zero]] being [[infinity]].

Algebra:
*The recognition of a positive number having two square roots.
*[[Surd]]s.
*Operations with products of several unknowns.
*The solutions of:
**Quadratic equations.
**Cubic equations.
**Quartic equations.
**Equations with more than one unknown.
**Quadratic equations with more than one unknown.
**The general form of [[Pell's equation]] using the [[chakravala method|''chakravala'' method]].
**The general indeterminate quadratic equation using the ''chakravala'' method.
**Indeterminate cubic equations.
**Indeterminate quartic equations.
**Indeterminate higher-order [[polynomial]] equations.

Geometry:
*Gave a proof of the [[Pythagorean theorem]].

Calculus:
*Conceived of [[differential calculus]].
*Discovered the [[derivative]].
*Discovered the [[differential]] coefficient.
*Developed [[differentiation]].
*Stated [[Rolle's theorem]], a special case of the [[mean value theorem]] (one of the most important theorems of calculus and analysis).
*Derived the differential of the sine function.
*Computed [[π]], correct to 5 decimal places.
*Calculated the length of the Earth's revolution around the Sun to 9 decimal places.

Trigonometry:
*Developments of [[spherical trigonometry]]
*The trigonometric formulas: 
**<math>\ \sin(a+b)=\sin(a) \cos(b) + \sin(b) \cos(a)</math>
**<math>\ \sin(a-b)=\sin(a) \cos(b) - \sin(b) \cos(a)</math>

==Kerala Mathematics (1300 - 1600)==
{{Main|Kerala school of astronomy and mathematics}}
The [[Kerala school of astronomy and mathematics]] was founded by [[Madhava of Sangamagrama]] in [[Kerala]], [[South India]] and included among its members: [[Parameshvara]], [[Neelakanta Somayaji]], [[Jyeshtadeva]], [[Achyuta Pisharati]], [[Melpathur Narayana Bhattathiri]] and Achyuta Panikkar. It flourished between the [[14th century|14th]] and [[16th century|16th centuries]] and the original discoveries of the school seems to have ended with [[Melpathur Narayana Bhattathiri|Narayana Bhattathiri]] ([[1559]]-[[1632]]).  In attempting to solve astronomical problems, the Kerala school astronomers ''independently'' created a number of important mathematics concepts.  The most important results, series expansion for [[trigonometric function]]s, were given in [[Sanskrit]] verse in a book by Neelakanta called ''Tantrasangraha'' and a commentary on this work called ''Tantrasangraha-vakhya'' of unknown authorship.  The theorems were stated without proof, but proofs for the series for ''sine'', ''cosine'', and inverse ''tangent'' were provided a century later in the work ''Yuktibhasa'' (c.1500-c.1610), written in [[Malayalam]],  by Jyesthadeva, and also in a commentary on ''Tantrasangraha''.<ref name=roy>{{Harv|Roy|1990}}</ref>  

Their discovery of these three important series expansions of [[calculus]]&mdash;several centuries before calculus was developed in Europe by [[Isaac Newton]] and [[Gottfried Leibniz]]&mdash;was a landmark achievement in mathematics.  However, the Kerala School cannot be said to have invented ''calculus'',<ref name=bressoud/> because, while they were able to develop Taylor series expansions for the important trigonometric functions, they developed neither a comprehensive theory of [[Derivative|differentiation]] or [[Integral|integration]], nor the [[fundamental theorem of calculus]].<ref name=katz/>   The results obtained by the Kerala school include:
 
*The (infinite) [[geometric series]]: <math> \frac{1}{1-x} = 1 + x + x^2 + x^3 + \dots + \infty </math> for <math>|x|<1 </math><ref name =singh>Singh, A. N. Singh.  1936.  "On the Use of Series in Hindu Mathematics." ''Osiris'' 1:606-628.</ref> This formula was already known, for example, in the work of the 10th century Arab mathematician [[Alhazen]] (the Latinized form of the name Ibn Al-Haytham (965-1039)).<ref>Edwards, C. H., Jr.   1979.  ''The Historical Development of the Calculus''.  New York: Springer-Verlag.</ref>
*A semi-rigorous proof (see "induction" remark below) of the result: <math>1^p+ 2^p + \cdots + n^p \approx \frac{n^{p+1}}{p+1}</math> for large ''n''.  This result was also known to Alhazen.<ref name=roy/>
*Intuitive use of [[mathematical induction]], however, the ''[[Inductive hypothesis#Formal description|inductive hypothesis]]'' was not formulated or employed in proofs.<ref name=roy/>
*Applications of ideas from (what was to become) differential and integral [[calculus]] to obtain [[Taylor's theorem|(Taylor-Maclaurin) infinite series]] for <math>\sin x</math>, <math>\cos x</math>, and <math> \arctan x</math><ref name=bressoud>{{Harv|Bressoud|2002}}</ref>  The ''Tantrasangraha-vakhya'' gives the series in verse, which when translated to mathematical notation, can be written as:<ref name=roy/>
:<math>r\arctan(\frac{y}{x}) = \frac{1}{1}\cdot\frac{ry}{x} -\frac{1}{3}\cdot\frac{ry^3}{x^3} + \frac{1}{5}\cdot\frac{ry^5}{x^t} - \cdots , </math> where <math>y/x \leq 1. </math>
:<math>\sin x = x - x\cdot\frac{x^2}{(2^2+2)r^2} + x\cdot \frac{x^2}{(2^2+2)r^2}\cdot\frac{x^2}{(4^2+4)r^2} - \cdot </math>
:<math> r - \cos x = r\cdot \frac{x^2}{(2^2-2)r^2} - r\cdot \frac{x^2}{(2^2-2)r^2}\cdot \frac{x^2}{(4^2-4)r^2} + \cdots , </math> where, for <math> r = 1 </math>, the series reduce to the standard power series for these trigonometric functions, for example: 
**<math>\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots </math> and 
**<math>\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots </math>
*Use of rectification (computation of length) of the arc of a circle to give a proof of these results.  (The later method of Leibniz, using quadrature (''i.e.'' computation of ''area under'' the arc of the circle, was ''not'' used.)<ref name=roy/>
*Use of series expansion of <math>\arctan x</math> to obtain an infinite series expression (later known as Gregory series) for <math>\pi</math>:<ref name=roy/>
:<math>\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \ldots + \infty </math>
*A rational approximation of ''error'' for the finite sum of their series of interest.  For example, the error, <math>f_i(n+1)</math>, (for ''n'' odd, and ''i = 1, 2, 3'') for the series:
:<math>\frac{\pi}{4} \approx 1 - \frac{1}{3}+ \frac{1}{5} - \cdots (-1)^{(n-1)/2}\frac{1}{n} + (-1)^{(n+1)/2}f_i(n+1)</math>
::where <math>f_1(n) = \frac{1}{2n}, \ f_2(n) = \frac{n/2}{n^2+1}, \ f_3(n) = \frac{(n/2)^2+1}{(n^2+5)n/2}.</math>
*Manipulation of error term to derive a faster converging series for <math>\pi</math>:<ref name=roy/>
:<math>\frac{\pi}{4} = \frac{3}{4} + \frac{1}{3^3-3} - \frac{1}{5^3-5} + \frac{1}{7^3-7} - \cdots \infty </math>
*Using the improved series to derive a rational expression,<ref name=roy/> <math>104348/33215</math> for <math>\pi</math> correct up to ''nine'' decimal places, ''i.e.'' <math>3.141592653 </math>
*Use of an intuitive notion of limit to compute these results.<ref name=roy/>
*A semi-rigorous (see remark on limits above) method of differentiation of some trigonometric functions.<ref name=katz>{{Harv|Katz|1995}}</ref> However, they did not formulate the notion of a ''function'', or have knowledge of the exponential or logarithmic functions.

The works of the Kerala school were first written up for the Western world by Englishman C. M. Whish in [[1835]].  According to Whish, the Kerala mathematicians had "''laid the foundation for a complete system of fluxions''" and these works abounded "''with fluxional forms and series to be found in no work of foreign countries.''"<ref name="charles">{{cite book 
 | author =Charles Whish
 | year = 1835
 | title = Transactions of the Royal Asiatic Society of Great Britain and Ireland
 | publisher = 
}}
</ref>
However, Whish's results were almost completely neglected, until over a century later, when the discoveries of the Kerala school were investigated again by C. Rajagopal and his associates.  Their work includes commentaries on the proofs of the arctan series in ''Yuktibhasa'' given in two papers,<ref>Rajagopal, C. and M. S. Rangachari.  1949.  "A Neglected Chapter of Hindu Mathematics."  ''Scripta Mathematica''.  15:201-209.</ref><ref>Rajagopal, C. and M. S. Rangachari.  1951.  "On the Hindu proof of Gregory's series."  ''Ibid.'' 17:65-74.</ref> a commentary on the ''Yuktibhasa'''s proof of the sine and cosine series<ref>Rajagopal, C. and A. Venkataraman.  1949.  "The sine and cosine power series in Hindu mathematics."  ''Journal of the Royal Asiatic Society of Bengal (Science)''. 15:1-13.</ref> and two papers that provide the [[Sanskrit]] verses of the ''Tantrasangrahavakhya'' for the series for arctan, sin, and cosine (with English translation and commentary).<ref>Rajagopal, C. and M. S. Rangachari.  1977.  "On an untapped source of medieval Keralese mathematics." ''Archive for the History of Exact Sciences''. 18:89-102.</ref><ref>Rajagopal, C.  and M. S. Rangachari.  1986.  "On Medieval Kerala Mathematics." ''Archive for the History of Exact Sciences''.  35:91-99.</ref>

The Kerala mathematicians included [[Narayana Pandit]] (c. 1340-1400), who composed two works, an arithmetical treatise, ''Ganita Kaumudi'', and an algebraic treatise, ''Bijganita Vatamsa''. Narayana is also thought to be the author of an elaborate commentary of [[Bhaskara II]]'s [[Lilavati]], titled ''Karmapradipika'' (or ''Karma-Paddhati''). [[Madhava of Sangamagramma]] (c. 1340-1425) was the founder of the [[Kerala school of astronomy and mathematics|Kerala School]].  Although it is possible that he wrote ''Karana Paddhati'' a work written sometime between 1375 and 1475, all we really know of his work comes from works of later scholars.

[[Parameshvara]] (c. 1370-1460) wrote commentaries on the works of [[Bhaskara I]], [[Aryabhata]] and [[Bhaskara II]]. His ''Lilavati Bhasya'', a commentary on Bhaskara II's ''Lilavati'', contains one of his important discoveries: a version of the [[mean value theorem]].  [[Nilakantha Somayaji]] (1444-1544) composed the ''Tantra Samgraha'' (which 'spawned' a later anonymous commentary ''Tantrasangraha-vyakhya'' and a further commentary by the name ''Yuktidipaika'', written in [[1501]]).   He elaborated and extended the contributions of Madhava. 

[[Citrabhanu]] (c. 1530) was a 16th century mathematician from Kerala who gave integer solutions to 21 types of systems of two [[simultaneous equation|simultaneous]] algebraic equations in two unknowns. These types are all the possible pairs of equations of the following seven forms:

<math>\ x + y = a, x - y = b, xy = c, x^2 + y^2 = d, x^2 - y^2 = e, x^3 + y^3 = f, x^3 - y^3 = g</math>

For each case, Citrabhanu gave an explanation and justification of his rule as well as an example. Some of his explanations are algebraic, while others are geometric.  [[Jyesthadeva]] (c. 1500-1575) was another member of the Kerala School.  His key work was the ''Yukti-bhasa'' (written in [[Malayalam]], a regional language of [[Kerala]]).  Jyesthadeva presented proofs of most mathematical theorems and infinite series earlier discovered by Madhava and other Kerala School mathematicians.

==Charges of Eurocentrism==
It has been suggested that Indian contributions to mathematics have not been given due acknowledgement in modern history and that many discoveries and inventions by [[Indian mathematicians]] are presently culturally attributed to their [[Western world|Western]] counterparts, as a result of [[Eurocentrism]].  According to G. G. Joseph:

<blockquote>[Their work] takes on board some of the objections raised about the classical Eurocentric trajectory. The awareness [of Indian and Arabic mathematics] is all too likely to be tempered with dismissive rejections of their importance compared to Greek mathematics. The contributions from other civilizations - most notably China and India, are perceived either as borrowers from Greek sources or having made only minor contributions to mainstream mathematical development. An openness to more recent research findings, especially in the case of Indian and Chinese mathematics, is sadly missing"<ref> Joseph, G. G.  1997.  "Foundations of Eurocentrism in Mathematics."  In ''Ethnomathematics: Challenging Eurocentrism in Mathematics Education'' (Eds. Powell, A. B. et al.).  SUNY Press. {{ISBN|0791433528}}.  p.67-68.</ref></blockquote>

The historian of mathematics, [[Florian Cajori]], suggested that he "suspect[s] that [[Diophantus]] got his first glimpse of algebraic knowledge from India."<ref> {{cite book
 | last = Cajori
 | first = Florian
 | authorlink = Florian Cajori
 | title = A History of Mathematics P 86
 | edition = 
 | year = 1893
 | publisher = Macmillan & Co.
 | language = English
 | chapter = The Hindoos 
 | quote = "In algebra, there was probably a mutual giving and receiving [between Greece and India]. We suspect that Diophantus got his first glimpse of algebraic knowledge from India"
 }}
</ref> 

More recently, as discussed in the above section, the infinite series of [[calculus]] for trigonometric functions (rediscovered by Gregory, Taylor, and Maclaurin in the late 17th century) were described (with proofs) in India, by mathematicians of the [[Kerala School]], remarkably some two centuries earlier. Some scholars have recently suggested that knowledge of these results might have been transmitted to Europe through the trade route from [[Kerala]] by traders and [[Jesuit]] missionaries.<ref name=almeida/>  Kerala was in continuous contact with [[China]] and [[Arabia]], and, from around [[1500]], with Europe.  The existence of communication routes and a suitable chronology certainly make such a transmission a possibility. However, there is no direct evidence by way of relevant manuscripts that such a transmission actually took place.<ref name=almeida>Almeida, D. F., J. K. John, and A. Zadorozhnyy. 2001. "Keralese Mathematics: Its Possible Transmission to Europe and the Consequential Educational Implications." ''Journal of Natural Geometry'', 20:77-104.</ref> Indeed, according to David Bressoud, "there is no evidence that the Indian work of series was known beyond India, or even outside of Kerala, until the nineteenth century."<ref name=bressoud/><ref name=gold> Gold, D. and D. Pingree.  1991.  "A hitherto unknown Sanskrit work concerning Madhava's derivation of the power series for sine and cosine."  ''Historia Scientiarum''.  42:49-65.</ref>

Both Arab and Indian scholars made discoveries before the 17th century that are now considered a part of calculus.<ref name=katz/>  However, they were not able to, as [[Isaac Newton|Newton]] and [[Gottfried Leibniz|Leibniz]] were, to "combine many differing ideas under the two unifying themes of the [[derivative]] and the [[integral]], show the connection between the two, and turn calculus into the great problem-solving tool we have today."<ref name=katz/>  The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own;<ref name=katz/> however, it is not known with certainty whether the immediate ''predecessors'' of Newton and Leibniz, "including, in particular, Fermat and Roberval, learned of some of the ideas of the Islamic and Indian mathematicians through sources we are not now aware."<ref name=katz/>  This is an active area of current research, especially in the manuscripts collections of [[Spain]] and [[Maghreb]], research that is now being pursued, among other places, at the Centre National de Recherche Scientifique in [[Paris]].<ref name=katz/>

== See also ==
*[[List of Indian mathematicians]]
*[[Indian science and technology]]
*[[Indian logic]]
*[[Indian astronomy]]
*[[History of mathematics]]

==Notes==
{{reflist|2}}

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*{{citation
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 | year=1978
 | title=The Yavanajātaka of Sphujidhvaja
 | place=edited, translated and commented by D. Pingree, Cambridge, MA
 | publisher=Harvard Oriental Series 48 (2 vols.)
 | isbn=
 | url=
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*{{citation
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 | first1=K. V. (ed)
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 | place=critically edited with Introduction and Appendices, New Delhi
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*{{citation
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*{{citation
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==External links==
*[http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Indian_mathematics.html An overview of Indian mathematics], ''[[MacTutor History of Mathematics Archive]]'', [[St Andrews University]], 2000.
*[http://www-groups.dcs.st-and.ac.uk/~history/Indexes/Indians.html 'Index of Ancient Indian mathematics'], ''MacTutor History of Mathematics Archive'', St Andrews University, 2004.
*[http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce Indian Mathematics: Redressing the balance], [http://www-history.mcs.st-and.ac.uk/Projects Student Projects in the History of Mathematics].  Ian Pearce.  ''MacTutor History of Mathematics Archive'', St Andrews University, 2002.
* [http://cs.annauniv.edu/insight/insight/maths/history/index.htm Online course material for InSIGHT] {{Webarchive|url=https://web.archive.org/web/20090822050837/http://cs.annauniv.edu/insight/insight/maths/history/index.htm |date=2009-08-22 }}, a workshop on traditional Indian sciences for school children conducted by the Computer Science department of Anna University, Chennai, India.

{{Indian mathematics}}

<nowiki>[[Category:Indian mathematics| ]]
[[Category:Science and technology in India]]
</nowiki>
[[de:Indische Mathematik]]
[[es:Matemática en la India]]
[[fr:Mathématiques indiennes]]
[[hi:भारतीय विद्वान सूची]]
[[kn:ಭಾರತೀಯ ಗಣಿತಜ್ಞರು]]
[[new:भारतीय गणितिज्ञतेगु धलः]]
[[ja:インドの数学]]