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			{{Short description\|Development of mathematics in South Asia}}
	{{histOfScience}}
			{{redirect\|Mathematics in India\|the 2009 monograph by Kim Plofker\|Mathematics in India (book)}}
	The chronology of '''Indian mathematics''' spans from the ] (3300-1500 BCE) and ] (1500-500 BCE) to modern ] (]).
			{{Use Indian English\|date=June 2020}}
			{{Use dmy dates\|date=May 2022}}

			'''Indian mathematics''' emerged and developed in the ]<ref name="plofker" /> from about 1200 BCE<ref name=hayashi2005-p360-361>{{Harv\|Hayashi\|2005\|pp=360–361}}</ref> until roughly the end of the 18th century CE (approximately 1800 CE). In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like ], ], ], ], and ]. The ] in use today<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> was first recorded in Indian mathematics.<ref>{{Harv\|Plofker\|2009\|pp=44–47}}</ref> Indian mathematicians made early contributions to the study of the concept of ] as a number,<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> ],<ref name=bourbaki49>{{Harv\|Bourbaki\|1998\|p=49}}: Modern arithmetic was known during medieval times as "Modus Indorum" or method of the Indians. ] wrote that compared to method of the Indians all other methods is a mistake. This method of the Indians is none other than our very simple arithmetic of addition, subtraction, multiplication and division. Rules for these four simple procedures was first written down by ] during 7th century AD. "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> ], and ].<ref name=concise-britannica/> In addition, ]<ref>{{Harv\|Pingree\|2003\|p=45}} Quote: "Geometry, and its branch trigonometry, was the mathematics Indian astronomers used most frequently. Greek mathematicians used the full chord and never imagined the half chord that we use today. Half chord was first used by Aryabhata which made trigonometry much more simple. 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>
	] have made major contributions to the development of mathematics as we know it today. One of the biggest contributions of Indian mathematics is the modern arithmetic and ] of numbers used universally throughout the world (known as the ]). ], the famous ] mathematician published a dissertation titled "Remarks on the astronomy of Brahmins" in 1790. His following quotation shows the appreciation of the then European Scientific community on the achievements of ancient Indian mathematicians and scientists.
			was further advanced in India, and, in particular, the modern definitions of ] and ] 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 (], ], ]) 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. {{Webarchive\|url=https://web.archive.org/web/20070929134632/http://www.britannica.com/ebc/article-231064 \|date=29 September 2007 }}. 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 ], usually consisted of a section of '']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 so important as the ideas involved.<ref name="plofker">{{Harv\|Kim Plofker\|2007\|p=1}}</ref><ref name=filliozat-p140to143>{{Harv\|Filliozat\|2004\|pp=140–143}}</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 ], discovered in 1881 in the village of ], near ] (modern day ]) and is likely from the 7th century CE.<ref name=hayashi95>{{Harv\|Hayashi\|1995}}</ref><ref name=plofker-brit6>{{Harv\|Kim Plofker\|2007\|p=6}}</ref>
	"''The Constructions and these tables imply a great knowledge of geometry,arithmetic and even of the theoretical part of astronomy.But what, without doubt is to be accounted,the greatest refinement in this system, is the hypothesis employed in calculating the equation of the centre for the Sun,Moon and the planets that of a circular orbit having a double eccentricity or having its centre in the middle between the earth and the point about which the angular motion is uniform.If to this we add the great extent of the geometrical knowledge required to combine this and the other principles of their astronomy together and to deduce from them the just conclusion;the possession of a calculus equivalent to trigonometry and lastly their approximation to the quadrature of the circle, we shall be astonished at the magnitude of that body of science which must have enlightened the inhabitants of India in some remote age and which whatever it may have communicated to the Western nations appears to have received another from them....''"

			A later landmark in Indian mathematics was the development of the ] expansions for ]s (sine, cosine, and ]) by mathematicians of the ] in the 15th century CE. Their work, completed two centuries before the invention of ] in Europe, provided what is now considered the first example of a ] (apart from geometric series).<ref>{{Harv\|Stillwell\|2004\|p=173}}</ref> However, they did not formulate a systematic theory of ] and ], nor is there any evidence of their results being transmitted outside ].<ref>{{Harv\|Bressoud\|2002\|p=12}} Quote: "There is no evidence that the Indian work on series was known beyond India, or even outside Kerala, until the nineteenth century. Gold and Pingree assert that by the time these series were rediscovered in Europe, they had, for all practical purposes, been lost to India. The expansions of the sine, cosine, and arc tangent had been passed down through several generations of disciples, but they remained sterile observations for which no one could find much use."</ref><ref>{{Harv\|Plofker\|2001\|p=293}} Quote: "It is not unusual to encounter in discussions of Indian mathematics such assertions as that "the concept of differentiation was understood from the time of Manjula (... in the 10th century)" , or that "we may consider Madhava to have been the founder of mathematical analysis" (Joseph 1991, 293), or that Bhaskara II may claim to be "the precursor of Newton and Leibniz in the discovery of the principle of the differential calculus" (Bag 1979, 294). ... The points of resemblance, particularly between early European calculus and the Keralese work on power series, have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world (e.g., in (Bag 1979, 285)). ... It should be borne in mind, however, that such an emphasis on the similarity of Sanskrit (or Malayalam) and Latin mathematics risks diminishing our ability fully to see and comprehend the former. To speak of the Indian "discovery of the principle of the differential calculus" somewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa, as in the examples we have seen, remained within that specific trigonometric context. The differential "principle" was not generalised to arbitrary functions—in fact, the explicit notion of an arbitrary function, not to mention that of its
	] in the ] also comments on the importance of Indian arithmetic: "''We owe a lot to the Indians, who taught us how to count, without which no worthwhile scientific discovery could have been made.''"
			derivative or an algorithm for taking the derivative, is irrelevant here"</ref><ref>{{Harv\|Pingree\|1992\|p=562}} Quote:"One example I can give you relates to the Indian Mādhava's demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by ], in the 1830s, it was heralded as the Indians' discovery of the calculus. This claim and Mādhava's achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the ''Transactions of the Royal Asiatic Society'', in which Whish's article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the series ''without'' the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Mādhava's mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution."</ref><ref>{{Harv\|Katz\|1995\|pp=173–174}} Quote:"How close did Islamic and Indian scholars come to inventing the calculus? Islamic scholars nearly developed a general formula for finding integrals of polynomials by A.D. 1000—and evidently could find such a formula for any polynomial in which they were interested. But, it appears, they were not interested in any polynomial of degree higher than four, at least in any of the material that has come down to us. Indian scholars, on the other hand, were by 1600 able to use ibn al-Haytham's sum formula for arbitrary integral powers in calculating power series for the functions in which they were interested. By the same time, they also knew how to calculate the differentials of these functions. So some of the basic ideas of calculus were known in Egypt and India many centuries before Newton. It does not appear, however, that either Islamic or Indian mathematicians saw the necessity of connecting some of the disparate ideas that we include under the name calculus. They were apparently only interested in specific cases in which these ideas were needed. ... There is no danger, therefore, that we will have to rewrite the history texts to remove the statement that Newton and Leibniz invented calculus. They were certainly the ones who were able to combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between them, and turn the calculus into the great problem-solving tool we have today."</ref>

			==Prehistory==
	Said ]: "''The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. The importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of antiquity, ] and ].''"<ref></ref>
			]
			Excavations at ], ] and other sites of the ] have uncovered evidence of the use of "practical mathematics". The people of the Indus Valley Civilization manufactured bricks whose dimensions were in the proportion 4:2:1, considered favourable for the stability of a brick structure. They used a standardised 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 ] shapes, which included ], ]s, ]s, and ]s, thereby demonstrating knowledge of basic ].<ref>{{Citation\|last=Sergent\|first=Bernard\|title=Genèse de l'Inde\|year=1997\|page=113\|language=fr\|isbn=978-2-228-89116-5\|publisher=Payot\|location=Paris}}</ref>

			The inhabitants of Indus civilisation also tried to standardise measurement of length to a high degree of accuracy. They designed a ruler—the ''Mohenjo-daro ruler''—whose unit of length (approximately 1.32 inches or 3.4 centimetres) was divided into ten equal parts. Bricks manufactured in ancient Mohenjo-daro often had dimensions that were integral multiples of this unit of length.<ref>{{Citation\|last=Coppa\|first=A.\|title=Early Neolithic tradition of dentistry: Flint tips were surprisingly effective for drilling tooth enamel in a prehistoric population\|journal=Nature\|volume=440\|date=6 April 2006\|doi=10.1038/440755a\|postscript=.\|pmid=16598247\|issue=7085\|pages=755–6\|display-authors=etal\|bibcode = 2006Natur.440..755C \|s2cid=6787162}}</ref><ref>{{Citation\|last=Bisht\|first=R. S.\|year=1982\|chapter=Excavations at Banawali: 1974–77\|editor=Possehl, Gregory L.\|title=Harappan Civilisation: A Contemporary Perspective\|pages=113–124\|location=New Delhi\|publisher=Oxford and IBH Publishing Co.}}</ref>
	Other examples include ], ], and the ]s of ] and ], which have all provided some of the biggest impetuses to advances in the field. Concepts from ] and ] India were carried to ] and the ], where they were studied extensively. From there they made their way to ] and other parts of the world.

			Hollow cylindrical objects made of shell and found at ] (2200 BCE) and ] are demonstrated to have the ability to measure angles in a plane, as well as to determine the position of stars for navigation.<ref>{{cite journal \|first=S. R. \|last=Rao \|date=July 1992 \|journal=Marine Archaeology \|volume=3 \|pages=61–62 \|url=http://drs.nio.org/drs/bitstream/handle/2264/3082/J_Mar_Archaeol_3_61.pdf?sequence=2 \|title=A Navigational Instrument of the Harappan Sailors \|url-status=dead \|archive-url=https://web.archive.org/web/20170808011822/http://drs.nio.org/drs/bitstream/handle/2264/3082/J_Mar_Archaeol_3_61.pdf?sequence=2 \|archive-date=2017-08-08}}</ref>
	==Fields of Indian mathematics==
	Some of the areas of mathematics studied in ancient and medieval India include the following:

			==Vedic period==
	*]
			{{Science and technology in India}}
	**] — decimal units go back as far as the ]
			{{See also\|Vedanga\|Vedas}}
	**]s — see ]
	**] — see ]
	**], the modern ] ] now used universally
	**] numbers — see ]
	*]
	**] — see ]
	**]s — see
	**]s — see ]
	*]
	**]s — see ], ]
	**]s — see ]
	**] — see ]; ] and ] gave ] of the ]{{Fact\|date=February 2007}}
	**] — see ]
	**] — see ]
	*]
	**]s — see ], ], ]
	**]s — see ], ]
	**]s (biquadratic equations) — see ], ]
	*]
	**]s, ], the ] — see ]
	**] — see ]
	*]s — see ]
	*Earliest forms of ] — see ]
	*]s, ] — see
	*]
	**]s — see ], ]
	**] — see ], ]
	*]s
	**] — see ], ]
	*]
	**] — see ], ], ], ]
	**] — see ], ]
	*], including discoveries foundational to the development of ]
	**] — see ]
	**], ] — see ], ]

			===Samhitas and Brahmanas===
	==Harappan Mathematics (2600 BCE - 1700 BCE)==
			The texts of the ] provide evidence for the use of ]. By the time of the '']'' (1200–900 BCE), numbers as high as {{math\|10<sup>12</sup>}} were being included in the texts.<ref name="hayashi2005-p360-361"/> For example, the '']'' (sacred recitation) at the end of the ''annahoma'' ("food-oblation rite") performed during the ], and uttered just before-, during-, and just after sunrise, invokes powers of ten from a hundred to a trillion:<ref name=hayashi2005-p360-361/>
	{{See also\|Indus Valley Civilization}}
	The first appearance of evidence of the use of mathematics in the ] was in the ], which dates back to around 3300 BC. Excavations at ], ] and the surrounding area of the ], have uncovered much evidence of the use of basic mathematics. The mathematics used by this early Harappan civilisation was very much for practical means, and was primarily concerned with:

			{{blockquote\|Hail to ''śata'' ("hundred," {{math\|10<sup>2</sup>}}), hail to ''sahasra'' ("thousand," {{math\|10<sup>3</sup>}}), hail to ''ayuta'' ("ten thousand," {{math\|10<sup>4</sup>}}), hail to ''niyuta'' ("hundred thousand," {{math\|10<sup>5</sup>}}), hail to ''prayuta'' ("million," {{math\|10<sup>6</sup>}}), hail to ''arbuda'' ("ten million," {{math\|10<sup>7</sup>}}), hail to ''nyarbuda'' ("hundred million," {{math\|10<sup>8</sup>}}), hail to ''samudra'' ("billion," {{math\|10<sup>9</sup>}}, literally "ocean"), hail to ''madhya'' ("ten billion," {{math\|10<sup>10</sup>}}, literally "middle"), hail to ''anta'' ("hundred billion," {{math\|10<sup>11</sup>}}, lit., "end"), hail to ''parārdha'' ("one trillion," {{math\|10<sup>12</sup>}} lit., "beyond parts"), hail to the ''{{IAST\|uṣas}}'' (dawn) , hail to the ''{{IAST\|vyuṣṭi}}'' (twilight), hail to ''{{IAST\|udeṣyat}}'' (the one which is going to rise), hail to ''udyat'' (the one which is rising), hail ''udita'' (to the one which has just risen), hail to ''svarga'' (the heaven), hail to ''martya'' (the world), hail to all.<ref name=hayashi2005-p360-361/>}}
	*Weights and measuring scales
	*A surprisingly advanced ''brick technology'', which utilised ratios. The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding.

			The solution to partial fraction was known to the Rigvedic People as states in the purush Sukta (RV 10.90.4):
	The achievements of the Harappan people of the ] include:

			{{blockquote\|With three-fourths Puruṣa went up: one-fourth of him again was here.}}
	*Great accuracy in measuring length, mass, and time.

			The ] ({{circa}} 7th 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 History of Exact Sciences, vol 18.</ref>
	*The first system of uniform weights and measures.

			===Śulba Sūtras===
	*Extremely precise measurements. Their smallest division, which is marked on an ivory scale found in ], was approximately 1.704mm, the smallest division ever recorded on a scale of the ].
			{{Main\|Shulba Sutras}}
			The '']'' (literally, "Aphorisms of the Chords" in ]) (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 Hayashi, the ''Śulba Sūtras'' contain "the earliest extant verbal expression of the ] in the world, although it had already been known to the ]." <blockquote>The diagonal rope (''{{IAST\|akṣṇayā-rajju}}'') of an oblong (rectangle) produces both which the flank (''pārśvamāni'') and the horizontal (''{{IAST\|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/>
	*The decimal division of measurement for all practical purposes, including the measurement of mass as revealed by their hexahedron weights.

			They contain lists of ],<ref>Pythagorean triples are triples of integers {{math\|(a, b, c)}} with the property: {{math\|1=a<sup>2</sup>+b<sup>2</sup> = c<sup>2</sup>}}. Thus, {{math\|1=3<sup>2</sup>+4<sup>2</sup> = 5<sup>2</sup>}}, {{math\|1=8<sup>2</sup>+15<sup>2</sup> = 17<sup>2</sup>}}, {{math\|1=12<sup>2</sup>+35<sup>2</sup> = 37<sup>2</sup>}}, etc.</ref> which are particular cases of ].<ref name=cooke198>{{Harv\|Cooke\|2005\|p=198}}: "The arithmetic content of the ''Śulva Sūtras'' consists of rules for finding Pythagorean triples such as {{math\|(3, 4, 5)}}, {{math\|(5, 12, 13)}}, {{math\|(8, 15, 17)}}, and {{math\|(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 ] 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 {{math\|π}} of 18 (3 − 2{{radic\|2}}), which is about 3.088."</ref>
	*Decimal weights based on ratios of 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, with each unit weighing approximately 28 grams, similar to the English ounce or Greek uncia.

			] (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\|(5, 12, 13)}}, {{math\|(8, 15, 17)}}, {{math\|(7, 24, 25)}}, and {{math\|(12, 35, 37)}},<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/><ref>{{Cite web\|url=https://www.theintellibrain.com/vedicmaths/\|title=Vedic Maths Complete Detail\|website= ALLEN IntelliBrain\|access-date=22 October 2022}}</ref> 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 an expression for the ]:<ref name=cooke200>{{Harv\|Cooke\|2005\|p=200}}</ref>
	*Many of the weights uncovered have been produced in definite ] shapes, including ]s, ]s, ]s, and ]s to name a few, which present knowledge of basic ], including the circle.
			::<math>\sqrt{2} \approx 1 + \frac{1}{3} + \frac{1}{3\cdot4} - \frac{1}{3\cdot 4\cdot 34} = 1.4142156 \ldots</math>
			The expression is accurate up to five decimal places, the true value being 1.41421356...<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 {{radic\|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 ]. 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 ] representation of {{radic\|2}}.</ref> This expression is similar in structure to the expression 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 ]):<ref name=cooke200/>
			::<math>\sqrt{2} \approx 1 + \frac{24}{60} + \frac{51}{60^2} + \frac{10}{60^3} = 1.41421297 \ldots</math>
			which expresses {{radic\|2}} in the sexagesimal system, and which is also accurate up to 5 decimal places.

			According to mathematician S. G. Dani, the Babylonian cuneiform tablet ] written c. 1850 BCE<ref>Mathematics Department, University of British Columbia, {{Webarchive\|url=https://web.archive.org/web/20200617151320/http://www.math.ubc.ca/~cass/courses/m446-03/pl322/pl322.html \|date=17 June 2020 }}.</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\|1=c<sup>2</sup> = a<sup>2</sup>+b<sup>2</sup>}} and if the highest common factor of {{math\|a, b, c}} is 1. In the particular Plimpton322 example, this means that {{math\|1=13500<sup>2</sup>+12709<sup>2</sup> = 18541<sup>2</sup>}} 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:
	*This culture produced artistic designs of a mathematical nature and there is evidence on carvings that these people could draw concentric and intersecting circles and triangles.

			{{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/>}}
	*Further to the use of circles in ''decorative'' design there is indication of the use of bullock carts, the wheels of which may have had a metallic band wrapped round the rim. This clearly points to the possession of knowledge of the ratio of the length of the circumference of the circle and its diameter, and thus values of ].

			In all, three ''Sulba Sutras'' were composed. The remaining two, the ''Manava Sulba Sutra'' composed by ] (fl. 750–650 BCE) and the ''Apastamba Sulba Sutra'', composed by ] (c. 600 BCE), contained results similar to the ''Baudhayana Sulba Sutra''.
	*Also of great interest is a remarkably accurate decimal ruler known as the Mohenjo-daro ruler. Subdivisions on the ruler have a maximum error of just 0.005 inches and, at a length of 1.32 inches, have been named the Indus inch.

			;Vyakarana
	*A correspondence has been noted between the Indus scale and brick size. Bricks (found in various locations) were found to have dimensions that were integral multiples of the graduations of their respective scales, which suggests advanced mathematical thinking.
			{{main\|Vyakarana}}

			The Vedic period saw the work of Sanskrit grammarian {{IAST\|]}} (c. 520–460 BCE). His grammar includes a precursor of the ] (used in the description ]).<ref>{{cite journal\|last1=Ingerman\|first1=Peter Zilahy\|title="Pānini-Backus Form" suggested\|journal=Communications of the ACM\|date=1 March 1967\|volume=10\|issue=3\|pages=137\|doi=10.1145/363162.363165\|s2cid=52817672\|issn=0001-0782\|doi-access=free}}</ref>
	*Some historians believe the Harappan civilization may have used a ] ].

			==Pingala (300 BCE – 200 BCE)==
	*Unique Harappan inventions include an instrument which was used to measure whole sections of the horizon and the tidal dock. The engineering skill of the Harappans was remarkable, especially in building docks after a careful study of tides, waves, and currents.
			Among the scholars of the post-Vedic period who contributed to mathematics, the most notable is ] (''{{IAST\|piṅgalá}}'') (] 300–200 BCE), a ] who authored the ] ] (''{{IAST\|chandaḥ-śāstra}}'', also Chhandas Sutra ''{{IAST\|chhandaḥ-sūtra}}''), a ] treatise on ]. Pingala's work also contains the basic ideas of ]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 '']-prastāra'' (literally "the staircase to 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>{{Harv\|Fowler\|1996\|p=11}}</ref>}}
	*In ], a thick ring-like shell object found with four slits each in two margins served as a ] to measure angles on plane surfaces or in horizon in multiples of 40–360 degrees. Such shell instruments were probably invented to measure 8–12 whole sections of the horizon and sky, explaining the slits on the lower and upper margins. Archaeologists consider this as evidence the Lothal experts had achieved something 2,000 years before the Greeks are credited with doing: an 8–12 fold division of horizon and sky, as well as an instrument to measure angles and perhaps the position of stars, and for navigation purposes.

			The text also indicates that Pingala was aware of the ] identity:<ref name=singh36>{{Harv\|Singh\|1936\|pp=623–624}}</ref>
	*Lothal contributes one of three measurement scales that are integrated and linear (others found in Harappa and Mohenjodaro). An ivory scale from Lothal has the smallest-known decimal divisions in Indus civilization. The scale is 6mm thick, 15 ] broad and the available length is 128 mm, but only 27 graduations are visible over 146 mm, the distance between graduation lines being 1.704 mm (the small size indicate use for finer purposes). The sum total of ten graduations from Lothal is approximate to the ''angula'' in the '']''.

			::<math> {n \choose 0} + {n \choose 1} + {n \choose 2} + \cdots + {n \choose n-1} + {n \choose n} = 2^n </math>
	*The Lothal craftsmen took care to ensure durability and accuracy of stone weights by blunting edges before polishing. The Lothal weight of 12.184 gm is almost equal to the Egyptian ''Oedet'' of 13.792 gm.

			;Kātyāyana
	==Vedic Mathematics (1500 BCE - 400 BCE)==
			] (c. 3rd century BCE) is notable for being the last of the Vedic mathematicians. He wrote the ''Katyayana Sulba Sutra'', which presented much ], including the general ] and a computation of the ] correct to five decimal places.
	{{Unreferencedsect\|date=January 2007}}
	{{See also\|#Assessment of Mathematics of the Vedic Period\|Vedic science\|Ancient Vedic weights and measures}}
	:''Note: The section is unrelated to the article on ], based on a system of mental calculation developed by ], which, it is claimed, is based on a lost appendix of ]

			==Jain mathematics (400 BCE – 200 CE)==
	===Vedic Mathematics Overview===
			Although ] as a religion and philosophy predates its most famous exponent, the great ]swami (6th century BCE), most Jain texts on mathematical topics were composed after the 6th century BCE. ] 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 Jain 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 ] led them to classify numbers into three classes: enumerable, innumerable and ]. Not content with a simple notion of infinity, their texts 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, Jain mathematicians devised notations for simple powers (and exponents) of numbers like squares and cubes, which enabled them to define simple ] ({{IAST\|bījagaṇita samīkaraṇa}}). Jain mathematicians were apparently also the first to use the word ''shunya'' (literally ''void'' in ]) to refer to zero. This word is the ultimate ], as it was ]d into Arabic as {{lang\|ar\|ṣifr}} and then subsequently borrowed into ] as {{lang\|la\|zephirum}}, finally arriving at English after passing through one or more ] (c.f. French {{lang\|fr\|zéro}}, Italian {{lang\|it\|zero}}).<ref>{{multiref2
	The geometry in Vedic mathematics was used for elaborate construction of religious and ] sites. Many aspects of practical mathematics are found in Vedic mathematics.<ref>Mathematical Expeditions: Chronicles by the Explorers by David Pengelley, Reinhard C. Laubenbacher</ref>{{page number}}
			\| {{cite encyclopedia\| first= Douglas\| last= Harper \|date=2011\| entry-url=https://www.etymonline.com/index.php?allowed_in_frame=0&search=zero&searchmode=none \| entry= Zero \|archive-url=https://web.archive.org/web/20170703014638/http://www.etymonline.com/index.php?allowed_in_frame=0&search=zero&searchmode=none \|archive-date=3 July 2017 \| title= Etymonline Etymology Dictionary\| quote="figure which stands for naught in the Arabic notation," also "the absence of all quantity considered as quantity", {{circa}} 1600, from French zéro or directly from Italian zero, from Medieval Latin zephirum, from Arabic sifr "cipher", translation of Sanskrit sunya-m "empty place, desert, naught"}}
			\|{{Cite book \|last=Menninger \|first=Karl \|url=https://books.google.com/books?id=BFJHzSIj2u0C \|title=Number Words and Number Symbols: A cultural history of numbers \|publisher=Courier Dover Publications \|year=1992 \|isbn=978-0-486-27096-8 \|pages=399–404 \|access-date=5 January 2016 }}
			\| {{Cite web \|date=December 2011 \|title=zero, n. \|url=http://www.oed.com/view/Entry/232803?rskey=zGcSoq&result=1&isAdvanced=false \|url-status=live \|archive-url=https://www.webcitation.org/65yd7ur9u?url=http://www.oed.com/view/Entry/232803?rskey=zGcSoq&result=1&isAdvanced=false \|archive-date=7 March 2012 \|access-date=4 March 2012 \|website=] Online \|publisher=] \|quote="French zéro (1515 in Hatzfeld & Darmesteter) or its source Italian zero, for *zefiro, < Arabic çifr" }} }}</ref>

			In addition to ''Surya Prajnapti'', important Jain works on mathematics included the '']'' (c. 300 BCE – 200 CE); the ''Anuyogadwara Sutra'' (c. 200 BCE – 100 CE), which includes the earliest known description of ]s in Indian mathematics;<ref>{{cite book \| last1 = Datta \| first1 = Bibhutibhusan \| last2 = Singh \| first2 = Awadhesh Narayan \| editor1-last = Kolachana \| editor1-first = Aditya \| editor2-last = Mahesh \| editor2-first = K. \| editor3-last = Ramasubramanian \| editor3-first = K. \| contribution = Use of permutations and combinations in India \| doi = 10.1007/978-981-13-7326-8_18 \| pages = 356–376 \| publisher = Springer Singapore \| series = Sources and Studies in the History of Mathematics and Physical Sciences \| title = Studies in Indian Mathematics and Astronomy: Selected Articles of Kripa Shankar Shukla \| year = 2019\| isbn = 978-981-13-7325-1 \| s2cid = 191141516 }}. Revised by K. S. Shukla from a paper in ''Indian Journal of History of Science'' 27 (3): 231–249, 1992, {{MR \| MR1189487}}. See p. 363.</ref> and the '']'' (c. 2nd century CE). Important Jain mathematicians included ] (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 '']''; and ] (c. 150 BCE), who, although better known for his influential writings on Jain philosophy and ], composed a mathematical work called the '']''.
	*Extensive use of geometric shapes, including triangles, rectangles, squares, trapezia and circles. <ref>Vedic mathematics for schools. Book 2: By James T. Glover</ref>{{page number}}
	*Equivalence through numbers and area.
	*]{{Fact\|date=January 2007}}
	*Circling the square. {{Fact\|date=February 2007}}
	*A list of ] discovered ]ically.<ref name = smith>Smith, David Eugene and Louis Charles Karpinski. 1911. ''The Hindu-Arabic Numerals''. Boston and London: Ginn and Company Publishers. 160 pages. Page 13</ref>
	*Statement and numerical proof of the ].<ref name=smith/>
	*Computations of ].{{Fact\|date=January 2007}}

			==Oral tradition==
	Vedic works also contain:
			Mathematicians of ancient and early medieval India were almost all ] ]s (''{{IAST\|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 (]), ] (]) and logic (])."<ref name=filliozat-p137/> Memorisation of "what is heard" ('']'' in Sanskrit) through recitation played a major role in the transmission of sacred texts in ancient India. Memorisation 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>
	*All four arithmetical operators (addition, subtraction, multiplication and division).{{Fact\|date=February 2007}}
	*A definite system for denoting any number up to 10<sup>55</sup>.{{Fact\|date=February 2007}}
	*The existence of zero.{{Fact\|date=February 2007}}
	*Prime numbers.{{Fact\|date=February 2007}}
	*The ].{{Fact\|date=February 2007}}
	*A number of other discoveries.{{Fact\|date=February 2007}}

			===Styles of memorisation===
	Of all the mathematics contained in the Vedic works, it is the definite appearance of ] symbols for numerals and a ] system that should perhaps be considered the most phenomenal.{{Fact\|date=January 2007}}
			Prodigious 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, memorisation of the sacred '']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 ''{{IAST\|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 in the original order.<ref name=filliozat-p139>{{Harv\|Filliozat\|2004\|p=139}}</ref> The recitation thus proceeded as:
			<div style="text-align: center;"> '''word1word2, word2word1, word1word2; word2word3, word3word2, word2word3; ...'''</div>
			In another form of recitation, ''{{IAST\|dhvaja-pāṭha}}''<ref name=filliozat-p139/> (literally "flag recitation") a sequence of ''N'' words were recited (and memorised) by pairing the first two and last two words and then proceeding as:
			<div style="text-align: center;"> '''word<sub>1</sub>word<sub>2</sub>, word<sub>''N'' − 1</sub>word<sub>''N''</sub>; word<sub>2</sub>word<sub>3</sub>, word<sub>''N'' − 2</sub>word<sub>''N'' − 1</sub>; ..; word<sub>''N'' − 1</sub>word<sub>''N''</sub>, word<sub>1</sub>word<sub>2</sub>;'''</div>
			The most complex form of recitation, ''{{IAST\|ghana-pāṭha}}'' (literally "dense recitation"), according to Filliozat,<ref name="filliozat-p139"/> took the form:
			<div style="text-align: center;">'''word1word2, word2word1, word1word2word3, word3word2word1, word1word2word3; word2word3, word3word2, word2word3word4, word4word3word2, word2word3word4; ... '''</div>
			That these methods have been effective is testified to by the preservation of the most ancient Indian religious text, the '']'' (c. 1500 BCE), as a single text, without any variant readings.<ref name=filliozat-p139/> Similar methods were used for memorising mathematical texts, whose transmission remained exclusively oral until the end of the ] (c. 500 BCE).

	===~~Vedas~~===		===The ''Sutra'' genre===
			Mathematical activity in ancient India began as a part of a "methodological reflexion" on the sacred ]s, which took the form of works called ], 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 ] (]) and '']'' (]s); to conserve its meaning by use of ] (]) and '']'' (]); and to correctly perform the rites at the correct time by the use of '']'' (]) and ] (]), gave rise to the six disciplines of the ''{{IAST\|Vedāṇgas}}''.<ref name=filliozat2004-p140-141/> Mathematics arose as a part of the last two disciplines, ritual and astronomy (which also included astrology).
	{{see also\|Vedas}}
			Since the ''{{IAST\|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 ] (literally, "thread"):
	The '']'' (c. 1500-1200 BCE) contains some rules for the construction of great fire altars.

			{{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/>}}
	The '']'' (c. 1200-900 BCE) contains:
	*Sacrificial formulae for ceremonial occasions.
	*] ] ] (recognizably the ancestor of ]){{Fact\|date=January 2007}}
	*The earliest known use of numbers up to a trillion (''parardha'') and numbers even larger up to 10<sup>55</sup>.{{Fact\|date=January 2007}}
	*The earliest evidence of numeric ] (''purna'' "fullness"), stating that if you subtract ''purna'' from ''purna'', you are still left with ''purna''.

			Extreme brevity was achieved through multiple means, which included using ] "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 ], '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).
	The '']'' (c. 1200-900 BCE) contains arithmetical sequences and a collection of magical formulae and spells. According to Shri Bharati Krishna Tirthaji, his system of mental calculation also known as ] is based on a lost appendix of the ''Atharva-Veda''.{{Fact\|date=February 2007}}
			]
			The domestic fire-altar in the ] 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 in three.<br/>II.65. In another layer one places the North-pointing.<ref name=filliozat2004-p143-144/>}}
	===Lagadha===
	] (fl. 1350-1000 BCE) composed the ''] ]'', a work consisting of 49 verses, which contains:
	*Descriptions of rules for tracking the motions of the Sun and the Moon.
	*Procedures for calculating the time and position of the Sun and Moon in various ''naksatras'' (signs of the zodiac).
	*The earliest known use of ] and ] for astronomy.{{Fact\|date=February 2007}}

			According to Filliozat,<ref>{{Harv\|Filliozat\|2004\|p=144}}</ref> 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, ''{{IAST\|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/>
	Much of Lagadha's works were later destroyed by foreign invaders of India.{{Fact\|date=February 2007}}

			==The written tradition: prose commentary==
	===Kalpa Vedanga===
			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.
	The ''] Vedanga'' (c. 1200-900 BCE) contains mathematical rules for rituals and ceremonials.

			{{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>}}
	===Samhitas===
	The ''] Samhita'' (c. 1200-900 BCE) contains:
	*Rules for the construction of great fire altars.
	*A rule implying knowledge of the Pythagorean theorem.{{Fact\|date=February 2007}}

			The earliest mathematical prose commentary was that on the work, '']'' (written 499 CE), a work on astronomy and mathematics. The mathematical portion of the ''{{IAST\|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 Hayashi,<ref>{{Harv\|Hayashi\|2003\|p=123}}</ref> "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 ] (600 CE onwards), prose commentaries increasingly began to include some derivations (''upapatti''). Bhaskara I's commentary on the ''{{IAST\|Āryabhaṭīya}}'', had the following structure:<ref name=hayashi03-p122-123/>
	The other '']'' (c. 1200-500 BCE) contain:
	*Fractions.
	*Equations, such as 972x<sup>2</sup> = 972 + m for example.{{Fact\|date=February 2007}}

			*'''Rule''' ('sūtra') in verse by ]
	===Yajnavalkya===
			*'''Commentary''' by Bhāskara I, consisting of:
	] (fl. 900-700 BCE) composed the astronomical text '']'', which contains:
			**'''Elucidation''' of rule (derivations were still rare then, but became more common later)
	*Geometric, constructional, algebraic and computational aspects.{{Fact\|date=February 2007}}
			**'''Example''' (''uddeśaka'') usually in verse.
	*A rule implying knowledge of the Pythagorean theorem.{{Fact\|date=February 2007}}
			**'''Setting''' (''nyāsa/sthāpanā'') of the numerical data.
	*Several computations of ], with the closest being correct to 2 decimal places, which remained the most accurate approximation of π anywhere in the world for another seven centuries.{{Fact\|date=February 2007}}
			**'''Working''' (''karana'') of the solution.
	*References to the motions of the Sun and the Moon.
			**'''Verification''' (''{{IAST\|pratyayakaraṇa}}'', literally "to make conviction") of the answer. These became rare by the 13th century, derivations or proofs being favoured by then.<ref name=hayashi03-p122-123/>
	*A 95-year cycle to synchronize the motions of the Sun and the Moon, which gives the average length of the ] as 365.24675 days, which is only 6 minutes longer than the modern value of 365.24220 days. This estimate for the length of the ] remained the most accurate anywhere in the world for over a thousand years.{{Fact\|date=February 2007}}
	*The distances of the Moon and the Sun from the Earth expressed as 108 times the diameters of these heavenly bodies. These are very close to the modern values of 110.6 for the Moon and 107.6 for the Sun, which were obtained using modern instruments.{{Fact\|date=February 2007}}

			Typically, for any mathematical topic, students in ancient India first memorised 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, ] (] 7th century CE), to characterise astronomical computations as "dust work" (Sanskrit: ''dhulikarman'').<ref name="hayashi2003-p119"/>
	=== Satapatha Brahmana (ca. 800 BCE) ===
	The ] contains rules on geometry 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> The geometry of the Satapatha Brahmana predates Greek geometry.<ref>A. Seidenberg, 1978. The origin of mathematics. Archive for the history of Exact Sciences, vol 18. (cited in Subhash Kak: From Vedic Science to Vedanta, Adyar Library Bulletin, 1995</ref>

			==Numerals and the decimal number system==
	===Sulba Geometry (ca. 800-500 BCE)===
			It is well known that the decimal place-value system ''in use today'' was first recorded in India, then transmitted to the Islamic world, and eventually to Europe.<ref name=plofker2007-p395>{{Harv\|Plofker\|2007\|p=395}}</ref> The Syrian bishop ] wrote in the mid-7th century CE about the "nine signs" of the Indians for expressing numbers.<ref name=plofker2007-p395/> However, how, when, and where the first decimal place value system was invented is not so clear.<ref>{{Harv\|Plofker\|2007\|p=395}}; {{Harv\|Plofker\|2009\|pp=47–48}}</ref>
	{{See\|Sulba Sutras}}
	''Sulba Sutra'' means "''Rule of Chords''" in ], and is another name for geometry. The Sulba Sutras were appendices to the ] giving rules for the construction of religious altars. The following discoveries found in these texts are mostly a result of altar construction:
	*The first use of ].{{Fact\|date=January 2007}}
	*The first use of ] of the form ax<sup>2</sup> = c and ax<sup>2</sup> + bx = c.
	*]s.{{Fact\|date=February 2007}}
	*]s{{Fact\|date=January 2007}}
	*A list of ] discovered algebraically{{Fact\|date=January 2007}}
	*A statement and numerical proof of the ] predating ] (572 BC - 497 BC){{Fact\|date=January 2007}}
	*Evidence of a number of geometrical proofs.{{Fact\|date=February 2007}}
	*].
	*Circling the square.
	*Calculations for the ] of 2 found in three of the Sulba Sutras, which were correct to a remarkable five decimal places.{{Fact\|date=January 2007}}
	*The earliest use of sine.{{Fact\|date=February 2007}}
	*The sine of ]/4 (45]) correctly computed as 1/√2 in a procedure for circling the square. {{inote\|Crest of the Peacock, page 232}}{{Fact\|date=February 2007}}

			The earliest extant ] used in India was the ] script used in the ] culture of the north-west. It is thought to be of ] origin and it was in use from the 4th century BCE to the 4th century CE. Almost contemporaneously, another script, the ], 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>
	It has been suggested by some scholars that the ''Sulba Sutras'' were written during the Harappan period. This is based on the evidence of advanced brick technology found in these texts, which was developed to a higher degree in the Harappan period than in the Vedic period (where it was limited to the bulding of religious altars). If the ''Sulba Sutras'' were not written during the Harappan period however, it is still possible that Harappan mathematics was at least as advanced as the ''Sulba Sutras'', based on the evidence of superior brick technology in the Harappan period.{{Fact\|date=January 2007}}

			The earliest surviving evidence of decimal place value numerals in India and southeast Asia is from the middle of the first millennium CE.<ref name=plofker2009-p45>{{Harv\|Plofker\|2009\|p=45}}</ref> A copper plate from Gujarat, India mentions the date 595 CE, written in a decimal place value notation, although there is some doubt as to the authenticity of the plate.<ref name=plofker2009-p45/> Decimal numerals recording the years 683 CE have also been found in stone inscriptions in Indonesia and Cambodia, where Indian cultural influence was substantial.<ref name=plofker2009-p45/>
	====Baudhayana====
	] (c. 8th century BCE) composed the ''Baudhayana 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=joseph> Joseph, G. G. 2000. ''The Crest of the Peacock: The Non-European Roots of Mathematics''. Princeton University Press. 416 pages. ISBN 0691006598. page 229.</ref> (Note: 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.)
	*A statement of the Pythagorean theorem in terms of 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=joseph/>
	*All three ''Sulbasutras'' have a statement of the Pythagorean theorem in terms of 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=joseph/>
	*Geometrical proof of the Pythagorean theorem for a 45° ] (the earliest proof of the Pythagorean theorem).{{Fact\|date=February 2007}}
	*Geometric solutions of a linear equation in a single unknown.{{Fact\|date=February 2007}}
	*Several approximations of ], with the closest value being 3.114.{{Fact\|date=February 2007}}
	*Approximations for irrational numbers. All three ''Sulbasutras'' give a formula for <math>\sqrt{2}</math> given by:<ref name=cooke> Cooke, R. 2005. ''The History of Mathematics: A Brief Course''. Wiley-Interscience. 632 pages. ISBN 0471444596. page 200.</ref>
	::<math>\sqrt{2} = 1 + \frac{1}{3} + \frac{1}{3\cdot4} - \frac{1}{3\cdot 4\cdot 34} \approx 1.4142156 ...</math> The true value is <math>1.414213 ... </math>
	:Although this remarkable formula arose as a result of geometric measurements, with hindsight, it can be seen as a first order Taylor expansion in calculus:
	:<math>\sqrt{a^2+r} = \sqrt{\left(a+\frac{r}{2a}\right)^2-\left(\frac{r}{2a}\right)^2} </math> <math> \approx a + \frac{r}{2a} - \frac{(r/2a)^2}{2(a+\frac{r}{2a})}, </math> with <math> a = 4/3 </math> and <math> r = 2/9 </math>.<ref name=cooke/>
	*The earliest use of quadratic equations of the forms ax<sup>2</sup> = c and ax<sup>2</sup> + bx = c.{{Fact\|date=February 2007}}
	*Indeterminate equations.
	*Two sets of positive integral solutions to a set of simultaneous Diophantine equations.{{Fact\|date=February 2007}}
	*Uses simultaneous Diophantine equations with up to four unknowns.{{Fact\|date=February 2007}}

			There are older textual sources, although the extant manuscript copies of these texts are from much later dates.<ref name=plofker2009-p46>{{Harv\|Plofker\|2009\|p=46}}</ref> Probably the earliest such source is the work of the Buddhist philosopher Vasumitra dated likely to the 1st century CE.<ref name=plofker2009-p46/> Discussing the counting pits of merchants, Vasumitra remarks, "When clay counting-piece is in the place of units, it is denoted as one, when in hundreds, one hundred."<ref name=plofker2009-p46/> Although such references seem to imply that his readers had knowledge of a decimal place value representation, the "brevity of their allusions and the ambiguity of their dates, however, do not solidly establish the chronology of the development of this concept."<ref name=plofker2009-p46/>
	====Manava====
	] (fl. 750-650 BCE) composed the ''Manava Sulba Sutra'', which contains:
	*Approximate constructions of circles from rectangles.{{Fact\|date=February 2007}}
	*Squaring the circle.{{Fact\|date=February 2007}}
	*Approximation of ], with the closest value being 3.125.

			A third decimal representation was employed in a verse composition technique, later labelled '']'' (literally, "object numbers") used by early Sanskrit authors of technical books.<ref name=plofker2009-p47>{{Harv\|Plofker\|2009\|p=47}}</ref> Since many early technical works were composed in verse, numbers were often represented by objects in the natural or religious world that correspondence to them; this allowed a many-to-one correspondence for each number and made verse composition easier.<ref name=plofker2009-p47/> According to Plofker,<ref name="Plofker 2009">{{Harv\|Plofker\|2009}}</ref> the number 4, for example, could be represented by the word "]" (since there were four of these religious texts), the number 32 by the word "teeth" (since a full set consists of 32), and the number 1 by "moon" (since there is only one moon).<ref name=plofker2009-p47/> So, Veda/teeth/moon would correspond to the decimal numeral 1324, as the convention for numbers was to enumerate their digits from right to left.<ref name=plofker2009-p47/> The earliest reference employing object numbers is a c. 269 CE Sanskrit text, ] (literally "Greek horoscopy") of Sphujidhvaja, a versification of an earlier (c. 150 CE) Indian prose adaptation of a lost work of Hellenistic astrology.<ref>{{Harv\|Pingree\|1978\|p=494}}</ref> Such use seems to make the case that by the mid-3rd century CE, the decimal place value system was familiar, at least to readers of astronomical and astrological texts in India.<ref name=plofker2009-p47/>
	====Apastamba====
	] (c. 600 BCE) composed the ''Apastamba Sulba Sutra'', which:
	*Gives methods for ] and also considers the problem of dividing a segment into 7 equal parts.
	*Calculates the square root of 2 correct to five decimal places.{{Fact\|date=January 2007}}
	*Solves the general ].{{Fact\|date=February 2007}}
	*Contains indeterminate equations and simultaneous Diophantine equations with up to five unknowns.{{Fact\|date=January 2007}}
	*The general numerical proof of the ], using an area computation (the earliest general proof of the Pythagorean theorem). According to historian Albert Burk, this is the original proof of the theorem, and ] copied it on his visit to India.{{Fact\|date=January 2007}}

			It has been hypothesized that the Indian decimal place value system was based on the symbols used on Chinese counting boards from as early as the middle of the first millennium BCE.<ref name=plofker2009-p48>{{Harv\|Plofker\|2009\|p=48}}</ref> According to Plofker,<ref name="Plofker 2009"/> <blockquote>These counting boards, like the Indian counting pits, ..., had a decimal place value structure ... Indians may well have learned of these decimal place value "rod numerals" from Chinese Buddhist pilgrims or other travelers, or they may have developed the concept independently from their earlier non-place-value system; no documentary evidence survives to confirm either conclusion."<ref name=plofker2009-p48/></blockquote>
	===Panini===
	{{Unicode\|]}} (c. 520-460 BCE) was a ] ] and is the world's earliest known ], and often considered the founder of ]. He also made contributions to mathematics, which include:
	*The earliest comprehensive and scientific theory of ], ], and ].
	*The formulation of the 3959 rules of ] morphology known as the '']''. The construction of sentences, compound nouns etc. is explained as ordered rules operating on underlying structures in a manner similar to modern theory. In many ways, {{Unicode\|Pāṇini}}'s constructions are similar to the way that a mathematical function is defined today.
	*The earliest use of ].
	*The earliest use of the ] operator.
	*The earliest use of metarules, ]s and ]s, which were used with such sophistication that his ] had the computing power equivalent to a ]. In this sense Panini may be considered the father of ].
	*He conceived of ].
	*He conceived of ]s.
	*The ] used to describe most modern ] is significantly similar to Panini's grammar rules.
	*Paninian grammars have also been devised for non-Sanskrit languages.

			==Bakhshali Manuscript==
	{{Unicode\|Pāṇini}}'s grammar of Sanskrit was responsible the transition from Vedic Sanskrit to classical Sanskrit, hence marking the end of the Vedic period.
			The oldest extant mathematical manuscript in India is the '']'', 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 ] (then in ] and now in ]). Of unknown authorship and now preserved in the ] in the ], the manuscript has been dated recently as 224 AD- 383 AD.<ref>{{cite web \| url=https://blog.sciencemuseum.org.uk/illuminating-india-starring-oldest-recorded-origins-zero-bakhshali-manuscript/ \| title=Illuminating India: Starring the oldest recorded origins of 'zero', the Bakhshali manuscript \| date=14 September 2017 }}</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 ], and '']'') and algebra (simultaneous linear equations and ]), 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 of a category known as 'equalisation problems' that lead to systems of linear equations. One example from Fragment III-5-3v is the following:
	==Assessment of Mathematics of the Vedic Period==

			{{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\|0-471-66959-8}}.</ref>}}
	According to ], Emeritus Professor of Philosophy and South & Southeast Asian Studies at the ] and an expert on Greek and Vedic Geometry,<ref>Staal, J. F. 1999. "Greek and Vedic Geometry." ''Journal of Indian Philosophy'' 27(1-2):105-127.</ref> Vedic mathematics consisted entirely of geometry, similar in some respects to Greek geometry, but entirely devoted to ''rituals''. The geometry was replace a millennium later by trigonometry and algebra (in the works of ], ] and others).<ref>Staal, J. F. 2003. "The Future of Asian Studies" in ''Imagining Asian Studies'', IAAS Newsletter, Number 34, November 2003</ref>

			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/>
	According to J.J. O'Connor and E. F. Robertson,<ref> O'Connor, J. J. and E. F. Robertson, , School of Mathematics, University of St Andrew, Scotland.</ref> the ''Sulbasutras'' were appendices to the Vedas giving ''rules for constructing altars''. "They contained quite an amount of geometrical knowledge, but the mathematics was being developed, not for its own sake, but purely for practical religious purposes."

			In 2017, three samples from the manuscript were shown by ] to come from three different centuries: from 224 to 383 AD, 680-779 AD, and 885-993 AD. It is not known how fragments from different centuries came to be packaged together.<ref name="Devlin">{{Cite news\|url=https://www.theguardian.com/science/2017/sep/14/much-ado-about-nothing-ancient-indian-text-contains-earliest-zero-symbol\|title=Much ado about nothing: ancient Indian text contains earliest zero symbol\|last=Devlin\|first=Hannah \|date=2017-09-13\|work=The Guardian\|access-date=2017-09-14\|issn=0261-3077}}</ref><ref name="Mason">{{cite news\|url=http://www.arch.ox.ac.uk/reader/items/oxford-radiocarbon-accelerator-unit-dates-the-worlds-oldest-recorded-origin-of-the-zero-symbol.html\|title=Oxford Radiocarbon Accelerator Unit dates the world's oldest recorded origin of the zero symbol\|last=Mason\|first=Robyn\|date=2017-09-14\|work=School of Archaeology, University of Oxford\|access-date=2017-09-14\|archive-url=https://web.archive.org/web/20170914215605/http://www.arch.ox.ac.uk/reader/items/oxford-radiocarbon-accelerator-unit-dates-the-worlds-oldest-recorded-origin-of-the-zero-symbol.html\|archive-date=14 September 2017\|url-status=dead}}</ref><ref name="Bodleian Library">{{cite news\|url=http://www.bodleian.ox.ac.uk/bodley/news/2017/sep-14\|title=Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol 'zero'\|date=2017-09-14\|work=Bodleian Library\|access-date=2017-09-14}}</ref>
	And, also, according to O'Connor and Robertson:<ref>O'Connor, J. J. and E. F. Robertson. 2001. . History of Mathematics Project, School of Mathematics, University of St Andrews, Scotland.</ref> the ''Sulbasutras'' do not contain any proofs of the rules they state. Some of the rules are exact, while others are approximations, however, the ''Sulbasutras'' make no distinction between the two. "Did the writers of the Sulbasutras know which methods were exact and which were approximations? ... If we follow the suggestion of some historians that the writers of the ''Sulbasutras'' were merely copying an approximation already known to the Babylonians then we might come to the conclusion that Indian mathematics of this period was far less advanced than if we follow Datta's suggestion."<ref>Dutta, B. 1932. ''The Science of Sulba''. Calcutta.</ref>

			==Classical period (400–1300)==
	According to S. G. Dani, Professor of Mathematics, ], ], the Babylonian cuneiform tablet Plimpton 322 written ca. 1850 ]<ref>Mathematics Department, University of British Columbia, .</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.</ref> indicating, in particular, that there was sophisticated understanding on the topic" in Mesopotamia in 1850 ]. "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> Dani, S. G. 2003. ''Current Science'', 85(2) 25 JULY 2003.</ref> Dani goes on to say:
			This period is often known as the golden age of Indian Mathematics. This period saw mathematicians such as ], ], ], ], ], ], ] and ] 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 Varāhamihira's 6th century compilation—''Pancasiddhantika''<ref>{{Harv\|Neugebauer\|Pingree\|1970}}</ref> (literally ''panca'', "five," ''siddhānta'', "conclusion of deliberation", dated 575 ])—of five earlier works, ], ], ], ] and ], 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/>

			===Fourth to sixth centuries===
	<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 to mathematics that one may encounter in an introductory book on architecture or another similar applied area, and
			;Surya Siddhanta
	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>

			Though its authorship is unknown, the '']'' (c. 400) contains the roots of modern ].{{Citation needed\|date=March 2011}} Because it contains many words of foreign origin, some authors consider that it was written under the influence of ] and Greece.<ref name="Origins of Sulva Sutras and Siddhanta">{{Citation\|first=Roger\|last=Cooke\|author-link=Roger Cooke (mathematician)\|title=The History of Mathematics: A Brief Course\|publisher=Wiley-Interscience\|year=1997\|chapter=The Mathematics of the Hindus\|isbn=978-0-471-18082-1\|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 ] and Greece.\|page=\|chapter-url=https://archive.org/details/historyofmathema0000cook/page/197}}</ref>{{Better source needed\|date=April 2017}}
	Finally, according to C. B. Boyer and U. C. Merzback, in their book, ''History of Mathematics''<ref name=boyer>Boyer, C. B. and U. C. Merzback (with forward by Issac Asimov). 1991. . John Wiley and Sons. 736 pages. ISBN 0471543977. p 207-208.</ref> <blockquote>Three versions, all in verse, of the work referred to as the ''Sulvasutras'' are extant, the best-known being that bearing the name of ''Apastamba''. In this primitive account, dating back perhaps as far as the time of Pythagoras, we find rules for the construction of right angles by means of triples of cords the lengths of which form Pythagorean triads, such as 3, 4, and 5, or 5, 12, and 13, or 8, 15, and 17, or 12, 35, and 37. However, all of these triads are easily derived from the old Babylonian rule; hence, Mesopotamian influence in the ''Sulvasutras'' is not unlikely.</blockquote>

			This ancient text uses the following as trigonometric functions for the first time:{{Citation needed\|date=March 2011}}
	As for the remarkable approximation formula for <math>\sqrt{2}</math> and other approximation formulas for irrationals in the ''Sulbasutras'', R. Cooke in ''A History of Mathematics: a brief course'' says: <blockquote> It is not certain just how the early Hindu mathematicians concieved of irrational numbers, whether they had a name for them, or were merely content to find a number that would serve for practical purposes. ... Here we see an instance in which the Greek insistence on logical correctness was a hindrance. The Greek did not regard <math> \sqrt{2}</math> as a number since they could not express it exactly as a ratio and they knew that they could not (''i.e.'' after Euclid's proof that <math>\sqrt{2}</math> is irrational). The Hindus may or may not have known of the impossibility of a rational expression for this number (they certainly knew that they did not ''have'' any rational expression for it); but, undeterred by the incompleteness of their knowledge, they proceeded to make what use they could of this number. This same "reckless" spirit served them well in the use of infinity and the invention of zero and negative numbers. They saw the usefulness of such numbers and either chose to live with, or did not notice, certain difficulties of a metaphysical character.<ref name=cooke/></blockquote>
			*Sine ('']'').
			*Cosine ('']'').
			*] (''Otkram jya'').

			Later Indian mathematicians such as Aryabhata made references to this text, while later ] and ] translations were very influential in Europe and the Middle East.
	==Jaina Mathematics (400 BCE - 200 CE)==
	] is a religion and philosophy that predates ] (]) a contemprory of ] who founded ]. Followers of these religions played an important role in the future development of India. As most of the Jaina texts were composed after Mahavira, not much information is available prior to ]. ]a mathematicians were particularly important in bridging the gap between earlier Indian mathematics and the 'Classical period', which was heralded by the work of ] from the 5th century CE.

			;Chhedi calendar
	Regrettably there are few extant Jaina works, but in the limited material that exists, an incredible level of originality is demonstrated. Perhaps the most historically important Jaina contribution to mathematics as a subject is the progression of the subject from purely practical or religious requirements. During the Jaina period, mathematics became an abstract discipline to be cultivated "for its own sake".

			This Chhedi calendar (594) contains an early use of the modern ] ] now used universally.
	The important developments of the Jainas include:
	*The theory of numbers.
	*The ].
	*Their fascination with the enumeration of very large numbers and ].
	*All numbers were classified into three sets: enumerable, innumerable and ].
	*Five different types of infinity are recognised in Jaina works: infinite in one and two directions, infinite in area, infinite everywhere and infinite perpetually. This theory was not realised in Europe until the late 19th century (usually attributed to ]).
	*Notations for squares, cubes and other exponents of numbers.
	*Giving shape to ''beezganit samikaran'' (]).
	*Using the word ''shunya'' meaning ''void'' to refer to zero. This word eventually became zero after a series of translations and transliterations. (See ].)

			;Aryabhata I
	Jaina works also contained:
	*The fundamental laws of ].
	*Arithmetical operations.
	*Geometry.
	*Operations with fractions.
	*Simple equations.
	*]s.
	*]s (the Jaina contribution to algebra has been severely neglected).
	*Formula for ] (root 10, comes up almost inadvertently in a problem about infinity).
	*Operations with logarithms (to ]).
	*] and progressions.
	*Of interest is the appearance of ] in Jaina works, which was used in the formation of a ], called ''Meru-prastara'', used by ] many centuries before ] used it.

			] (476–550) wrote the ''Aryabhatiya.'' He described the important fundamental principles of mathematics in 332 ]. The treatise contained:
	The Jaina work on ] included:
			*]s
	*The earliest concept of infinite ]s.
			*]
	*The earliest concept of ]s.
			*The value of ], correct to 4 decimal places.
	*A classification of all numbers into three groups: enumerable, innumerable and infinite.
	*Each of these was in turn, subdivided into three orders:
	**Enumerable: lowest, intermediate and highest.
	**Innumerable: nearly innumerable, truly innumerable and innumerably innumerable.
	**Infinite: nearly infinite, truly infinite, infinitely infinite.
	*The idea that all infinites were not the same or equal.
	*The recognition of five different types of infinity:
	**Infinite in one direction (one ]).
	**Infinite in two directions (one dimension).
	**Infinite in area (two dimensions).
	**Infinite everywhere (three dimensions)
	**Infinite perpetually (infinite number of dimenstions).
	*The highest enumerable number (''N'') of the Jains corresponds to the modern concept of ] <math>\aleph_0</math> (the cardinal number of the infinite set of integers 1, 2, ..., ''N''), the smallest transfinite cardinal number.
	*A whole system of transfinite numbers, of which aleph-null is the smallest.

			Aryabhata also wrote the ''Arya Siddhanta'', which is now lost. Aryabhata's contributions include:
	In the Jaina work on ]:
	*Two basic types of transfinite numbers are distinguished. On both physical and ] grounds, a distinction was made between:
	**Rigidly bounded infinities (''Asmkhyata'').
	**Loosely bounded infinities (''Ananata'').
	*With this distinction, the way was open for the Jains to develop a detailed classification of transfinite numbers and mathematical operations for handling transfinite numbers of different kinds. However, further research needs to be done on Jaina mathematics to understand more about their system of transfinite numbers.

			Trigonometry:
	===Surya Prajnapti===
	''Surya Prajnapti'' (c. 400 BCE) is a mathematical and astronomical text which:
	*Classifies all numbers into three sets: enumerable, innumerable and ].
	*Recognises five different types of ]: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.
	*First uses transfinite numbers.
	*Measures the length of the ] (the ] of the Moon around the Earth) as 29.5161290 days, which is only 20 minutes longer than the modern measurement of 29.5305888 days.

			(See also : ])
	===Pingala===
	] (fl. 400-200 BCE) was a scholar and ] who authored of the ''Chhandah-shastra''. His contributions to mathematics include:
	*The formation of a ].
	*Invention of the ] (while he was forming a matrix for musical purposes).
	*The concept of a binary code, similar to ].
	*First use of the ]
	*First use of ], which he refers to as ''Meru-prastaara''.
	*Used a dot (.) to denote ]
	*His work, along with Panini's work, was foundational to the development of computing.

			*Introduced the ]s.
	===Bhadrabahu===
			*Defined the sine ('']'') as the modern relationship between half an angle and half a chord.
	] (d. 298 BCE) was the author of two astronomical works, the ''Bhadrabahavi-Samhita'' and a commentary on the ''Surya Prajinapti''.
			*Defined the cosine ('']'').
			*Defined the ] ('']'').
			*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''.
			*].

			Arithmetic:
	===Vaishali Ganit===
			*]s.
	The ''] Ganit'' (c. 3rd century BCE) is a book that discusses the following in detail:
	*The basic calculations of mathematics.
	*The numbers based on 10.
	*Fractions.
	*Square and cubes.
	*Rules of the ].
	*Interest methods.
	*Questions on purchase and sale.

			Algebra:
	The book has given the answers of the problems and also described testing methods.
			*Solutions of simultaneous quadratic equations.
			*Whole number solutions of ] by a method equivalent to the modern method.
			*General solution of the indeterminate linear equation .

			Mathematical astronomy:
	===Sthananga Sutra===
			*Accurate calculations for astronomical constants, such as the:
	The ''Sthananga Sutra'' (fl. 300 BCE - 200 CE) gave classifications of:
			**].
	*The five types of ].
			**].
	*Linear equation (''yavat-tavat'').
			**The formula for the sum of the ], which was an important step in the development of integral calculus.<ref name=katz>{{Harv\|Katz\|1995}}</ref>
	*Quadratic equation (''varga'').
	*Cubic equation (''ghana'').
	*Quartic equation (''varga-varga'' or biquadratic).

			;Varahamihira
	===Katyayana===
	Though not a Jaina mathematician, Katyayana (c. 3rd century BCE) is notable for being the last of the Vedic mathematicians. He wrote the ''] Sulba Sutra'', which presented much ], including:
	*The general ].
	*An accurate computation of the square root of 2 correct to five decimal places.

			] (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 ] and ] functions:
	===Anoyogdwar Sutra===
	The ''Anoyogdwar Sutra'' (fl. 200 BCE - 100 CE) described:
	*Four types of ''Pramaan'' (''Measure'').
	*], which were termed as ''Bhang'' and ''Vikalp''.
	*The law of ].
	*The first use of ]s.

			*<math>\sin^2(x) + \cos^2(x) = 1</math>
	===Yativrisham Acharya===
			*<math>\sin(x)=\cos\left(\frac{\pi}{2}-x\right)</math>
	Yativrisham Acharya (c. 176 BCE) wrote a famous mathematical text called ''Tiloyapannati''.
			*<math>\frac{1-\cos(2x)}{2}=\sin^2(x)</math>

			===Seventh and eighth centuries===
	===Umasvati===
			] states that ''AF'' = ''FD''.]]
	] (c. 150 BCE) was famous for his influential writings on Jaina philosophy and ] but also wrote a work called ''Tattwarthadhigama-Sutra Bhashya'', which contains mathematics. This book contains mathematical formulae and two methods of multiplication and division:
			In the 7th century, two separate fields, ] (which included ]) and ], began to emerge in Indian mathematics. The two fields would later be called ''{{IAST\|pāṭī-gaṇita}}'' (literally "mathematics of algorithms") and ''{{IAST\|bīja-gaṇita}}'' (lit. "mathematics of seeds," with "seeds"—like the seeds of plants—representing unknowns with the potential to generate, in this case, the solutions of equations).<ref name=hayashi2005-p369>{{Harv\|Hayashi\|2005\|p=369}}</ref> ], in his astronomical work '']'' (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 ]:<ref name=hayashi2003-p121-122/>
	*Multiplication by factor (later mentioned by ]).
	*Division by factor (later found in the ''Trisatika'' of ]).

			'''Brahmagupta's theorem:''' If a cyclic quadrilateral has diagonals that are ] 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.
	===Satkhandagama===
	The ''Satkhandagama'' (c. 2nd century) contains:
	*Operations with ].
	*A theory of ]s.

			Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalisation of ]), as well as a complete description of ]s (''i.e.'' triangles with rational sides and rational areas).
	Various sets are operated upon by:
	*Logarithmic functions to ]
	*Squaring and extracting square roots.
	*Raising to finite or infinite powers.

			'''Brahmagupta's formula:''' The area, ''A'', of a cyclic quadrilateral with sides of lengths ''a'', ''b'', ''c'', ''d'', respectively, is given by
	These operations are repeated to produce new sets.

			: <math> A = \sqrt{(s-a)(s-b)(s-c)(s-d)} \, </math>
	==Bakhshali Manuscript (200 BCE - 400 CE)==
	The ] is a text that bridged the gap between the earlier Jaina mathematics and the 'Classical period' of Indian mathematics, though the authorship of this text is unknown. Perhaps the most important developments found in this manuscript are:

			where ''s'', the ], given by <math> s=\frac{a+b+c+d}{2}.</math>
	*The use of zero as a number.
	*The use of negative numbers.
	*The earliest use of the modern ] ] now used universally (see also ]).
	*The development of syncopated algebra, evident in its algebraic notation, which using letters of the alphabet, and the . and + signs to represent zero and negative numbers respectively.

			'''Brahmagupta's Theorem on rational triangles:''' A triangle with rational sides <math>a, b, c </math> and rational area is of the form:
	There are eight principal topics discussed in the ''Bakhshali Manuscript'':

			:<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>
	*Examples of the ] (and profit, loss and interest).
			for some rational numbers <math>u, v, </math> and <math> w </math>.<ref>{{Harv\|Stillwell\|2004\|p=77}}</ref>
	*Solutions of ] with as many as five unknowns.
	*The solution of the ] (a development of remarkable quality).
	*Arithmetic and geometric progressions.
	*] (some evidence that work begun by Jainas continued).
	*Quadratic ]s (origin of type ''ax/c = y'').
	*]s.
	*Fractions.
	*Other advances in notation including the use of ] and ] sign.
	*An improved method for calculating square roots allowing extremely accurate approximations for irrational numbers to be calculated, and can compute square roots of numbers as large as a million correct to at least 11 decimal places. (See ].)

			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 ''']''':
	==Classical Period (400 - 1200)==
	This period is often known as the golden age of Indian Mathematics. Although earlier Indian mathematics was also very significant, this period saw great mathematicians such as ], ], ], ] and ] give a broader and clearer shape to almost all the branches of mathematics. The system of Indian mathematics used in this period was far superior to ], in everything except geometry. Their important contributions to mathematics would spread throughout ] and the ], and eventually ] and other parts of the world.

			:<math>\ ax^2+bx=c</math>
	===Surya Siddhanta===
	Though its authorship is unknown, the '']'' (c. 400) contains the roots of modern ]. It uses the following as trigonometric functions for the first time:
	*Sine (''Jya'').
	*Cosine (''Kojya'').
	*] (''Otkram jya'').

			{{blockquote\|To the absolute number multiplied by four times the square, add the square of the middle term; the square root of the same, less the middle term, being divided by twice the square is the value.<ref name=stillwell2004-p87>{{Harv\|Stillwell\|2004\|p=87}}</ref>}}
	It also contains the earliest uses of:
	*].
	*].

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

			:<math>x = \frac{\sqrt{4ac+b^2}-b}{2a} </math>
	Later Indian mathematicians such as Aryabhata made references to this text, while later ] and ] translations were very influential in Europe and the Middle East.

			Also in chapter 18, Brahmagupta was able to make progress in finding (integral) solutions of ''']''',<ref name=stillwell2004-p72-73>{{Harv\|Stillwell\|2004\|pp=72–73}}</ref>
	===Aryabhata I===
			:<math>\ x^2-Ny^2=1, </math>
	] (476-550) was a resident of ] in the Indian state of ]. He described the important fundamental principles of mathematics in 332 ]. He produced the ''Aryabhatiya'', a treatise on:
			where <math>N</math> is a nonsquare integer. He did this by discovering the following identity:<ref name=stillwell2004-p72-73/>
	*]s
	*]
	*The value of ], correct to 4 decimal places.

			'''Brahmagupta's Identity:''' <math> \ (x^2-Ny^2)(x'^2-Ny'^2) = (xx'+Nyy')^2 - N(xy'+x'y)^2 </math>
	Aryabhata also wrote the ''Arya Siddhanta'', which is now lost. Aryabhata's contributions include:
			which was a generalisation of an earlier identity of ]:<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,
	Trigonometry:
			<math> x=x_2, \ \ y=y_2 \ \ </math> is a solution of <math> \ \ x^2 - Ny^2 = k_2, </math>, then:
	*Introduced the ]s.
			:<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>
	*Defined the sine (''jya'') as the modern relationship between half an angle and half a chord.
	*Defined the cosine (''kojya'').
	*Defined the ] (''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''.
	*].

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

			'''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:
	Algebra:
			:<math> \ x^2 -Ny^2 = 1 </math>
	*Solutions of simultaneous quadratic equations.
			also has an integer solution.<ref name=stillwell2004-p74-76>{{Harv\|Stillwell\|2004\|pp=74–76}}</ref>
	*Whole number solutions of ] by a method equivalent to the modern method.
	*General solution of the indeterminate linear equation .

			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/>
	Mathematical astronomy:
	*Proposed for the first time, a ] ] with the planets spinning on their ] and following an ] orbit around the Sun.
	*Accurate calculations for astronomical constants, such as the:
	**].
	**].
	**The length of a day using ].

			'''Example (Brahmagupta):''' Find integers <math>\ x,\ y\ </math> such that:
	Calculus:
			:<math>\ x^2 - 92y^2=1 </math>
	*]s:
			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:
	**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.
			:<math>\ x=1151, \ y=120 </math>
	*]s:
	**He expressed the near instantaneous motion of the moon in the form of a basic differential equation.
	*]:
	**He used the exponential function ''e'' in his differential equation of the near instantaneous motion of the moon.

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

			;Bhaskara I
	===Chhedi calendar===
	This Chhedi calendar (594) contains an early use of the modern ] ] now used universally (see also ]).

			] (c. 600–680) expanded the work of Aryabhata in his books titled ''Mahabhaskariya'', ''Aryabhatiya-bhashya'' and ''Laghu-bhaskariya''. He produced:
	===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.		*Solutions of indeterminate equations.
	*A rational approximation of the ].		*A rational approximation of the ].
	*A formula for calculating the sine of an acute angle without the use of a table, correct to 2 decimal places.		*A formula for calculating the sine of an acute angle without the use of a table, correct to two decimal places.

			===Ninth to twelfth centuries===
	===Brahmagupta===
			;Virasena
	]'s (598-668) famous work is his book titled '']'', which contributed:
	*The first lucid explanation of zero as both a place-holder and a decimal digit.
	*The integration of zero into the Indian numeral system.
	*A method of calculating the volume of ]s and ]s.
	*Description of how to sum a ].
	*The method of solving indeterminate equations of the second degree.
	*the first use of ] to solve astronomical problems.

	Other contributions in the ''Brahma Sphuta Siddhanta'':
	*] is clearly explained for the first time.
	*The modern ] ] is fully developed.
	*Rules are given for manipulating both ].
	*Methods are given for computing ]s.
	*methods are given for solving ] and ]s.
	*Contains rules for summing ].
	*].
	*].
	*].

	===Virasena===
	] (8th century) was a Jaina mathematician who 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'').

			] (8th century) was a Jain mathematician in the court of ] King ] of ], Karnataka. He wrote the ''Dhavala'', a commentary on Jain mathematics, which:
			*Deals with the concept of ''ardhaccheda'', the number of times a number could be halved, and lists various rules involving this operation. This coincides with the ] when applied to ],<ref>{{citation\| contribution=History of Mathematics in India\|title=Students' Britannica India: Select essays\|editor1-first=Dale\|editor1-last=Hoiberg\|editor2-first=Indu\|editor2-last=Ramchandani\|first=R. C.\|last=Gupta \|page=329\|publisher=Popular Prakashan\|year=2000\| contribution-url=https://books.google.com/books?id=-xzljvnQ1vAC&q=Virasena+logarithm&pg=PA329}}</ref><ref name="Dhavala">{{Citation\|first=A. N.\|last=Singh\|place=Lucknow University \|title=Mathematics of Dhavala\|url=http://www.jainworld.com/JWHindi/Books/shatkhandagama-4/02.htm\|access-date=31 July 2010\|archive-url=https://web.archive.org/web/20110511032215/http://www.jainworld.com/JWHindi/Books/shatkhandagama-4/02.htm \|archive-date=11 May 2011\|url-status=dead}}</ref> but differs on other numbers, more closely resembling the ].
	Virasena also gave:		Virasena also gave:
	*The derivation of the ] of a ] by a sort of infinite procedure.		*The derivation of the ] of a ] by a sort of infinite procedure.
			It is thought that much of the mathematical material in the ''Dhavala'' can attributed to previous writers, especially Kundakunda, Shamakunda, Tumbulura, Samantabhadra and Bappadeva and date who wrote between 200 and 600 CE.<ref name="Dhavala"/>

	~~===~~Mahavira~~===~~		;Mahavira

	] (c. 800-870) from ], the last of the notable Jaina mathematicians, lived in the ] and was patronised by the ] king ]. 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:
			] (c. 800–870) from ], the last of the notable Jain 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:
	*].
	*].		*]
	*].		*]
			*]
	*]s, ]s, and the ] extending beyond these.
			*]s, ]s, and the ] extending beyond these
	*].
	*].		* Plane geometry
			*]
	*Problems relating to the casting of ].
	*Formulae derived to calculate the area of an ] and ] inside a ]		*Problems relating to the casting of ]
			*Formulae derived to calculate the area of an ] and ] inside a ].

	Mahavira also:		Mahavira also:
	*Asserted that the ] of a ] did not exist		*Asserted that the square root of a ] did not exist
	*Gave the sum of a ] whose terms are ]s of an ], and gave empirical rules for ] and ] of an ].		*Gave the sum of a series whose terms are ]s of an ], and gave empirical rules for area and ] of an ellipse.
	*Solved cubic equations.		*Solved cubic equations.
	*Solved quartic equations.		*Solved quartic equations.
	*Solved ]s.		*Solved some ]s and higher-order ]s.
	*Solved higher order ] equations.
	*Gave the general solutions of the higher order polynomial equations:		*Gave the general solutions of the higher order polynomial equations:
	**<math>\ ax^n = q</math>		**<math>\ ax^n = q</math>
Line 521:		Line 307:
	*Solved indeterminate higher order equations.		*Solved indeterminate higher order equations.

	~~===~~Shridhara~~===~~		;Shridhara

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

	The ''Pati Ganita'' is a work on arithmetic and ]. It deals with various operations, including:		The ''Pati Ganita'' is a work on arithmetic and ]. It deals with various operations, including:
	*Elementary operations		*Elementary operations
	*Extracting square and cube roots.		*Extracting square and cube roots.
Line 533:		Line 320:
	*Methods of ] of different arithmetic and geometric series, which were to become standard references in later works.		*Methods of ] of different arithmetic and geometric series, which were to become standard references in later works.

	~~===~~Manjula~~===~~		;Manjula
	Aryabhata's differential equations were elaborated on by Manjula (10th century), who realised that the expression

			Aryabhata's equations were elaborated in the 10th century by Manjula (also ''Munjala''), who realised that the expression<ref name=Joseph-298-300>Joseph (2000), p. 298–300.</ref>
	<math>\ \sin w' - \sin w</math>
			: <math>\ \sin w' - \sin w</math>

	could be expressed as		could be approximately expressed as

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

			This was elaborated by his later successor Bhaskara ii thereby finding the derivative of sine.<ref name=Joseph-298-300/>
	He understood the concept of differentiation after solving the differential equation that resulted from substituting this expression into Aryabhata's differential equation.

	~~===~~Aryabhata II~~===~~		;Aryabhata II

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

	~~===~~Shripati~~===~~		;Shripati

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

Line 562:		Line 351:
	*Calculating planetary ]s		*Calculating planetary ]s
	*]s.		*]s.
	*planetary ]s.		*planetary ].

			;Nemichandra Siddhanta Chakravati

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

	~~===~~Bhaskara II~~===~~		;Bhaskara II

	] Acharya (1114-1185) was a mathematician-astronomer who wrote a number of important treatises, namely the ''Siddhanta Shiromani'', '']'', ''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:
			] (1114–1185) was a mathematician-astronomer who wrote a number of important treatises, namely the ''Siddhanta Shiromani'', '']'', '']'', ''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:		Arithmetic:
	*Interest computation.		*Interest computation
	*Arithmetical and geometrical progressions.		*Arithmetical and geometrical progressions
	*].		*Plane geometry
	*].		*Solid geometry
	*The shadow of the ].		*The shadow of the ]
	*Solutions of ].		*Solutions of ]
	*Gave a proof for division by ] being ].		*Gave a proof for division by zero being ].

	Algebra:		Algebra:
	*The recognition of a positive number having two square roots.		*The recognition of a positive number having two square roots.
	*]s.		*].
	*Operations with products of several unknowns.		*Operations with products of several unknowns.
	*The solutions of:		*The solutions of:
Line 593:		Line 384:
	**Indeterminate cubic equations.		**Indeterminate cubic equations.
	**Indeterminate quartic equations.		**Indeterminate quartic equations.
	**Indeterminate higher-order ] equations.		**Indeterminate higher-order polynomial equations.

	Geometry:		Geometry:
Line 599:		Line 390:

	Calculus:		Calculus:
			*Preliminary concept of differentiation
	*Conceived of ].
	*Discovered the ].		*Discovered the ].
			*Stated early form of ], a special case of the ] (one of the most important theorems of calculus and analysis).
	*Discovered the ] coefficient.
			*Derived the differential of the sine function although didn't perceive the notion of derivative.
	*Developed ].
			*Computed ], correct to five decimal places.
	*Stated ], a special case of the ] (one of the most important theorems of calculus and analysis).
			*Calculated the length of the Earth's revolution around the Sun to 9 decimal places.<ref>{{Cite book \|last=Cooke \|first=Roger \|url=http://archive.org/details/historyofmathema0000cook \|title=The history of mathematics : a brief course \|date=1997 \|publisher=New York : Wiley \|others=Internet Archive \|isbn=978-0-471-18082-1}}</ref>
	*Derived the differential of the sine function.
	*Computed ], correct to 5 decimal places.
	*Calculated the length of the Earth's revolution around the Sun to 9 decimal places.

	Trigonometry:		Trigonometry:
	*Developments of ]		*Developments of ]
	*The trigonometric formulas:		*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>
	**<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>

			==Medieval and early modern mathematics (1300–1800)==
	==Kerala Mathematics (1300 - 1600)==
	{{Main\|Kerala School}}
	The ] was a school of mathematics and astronomy founded by ] in ] (in ]) which included as its prominent members ], ], ], ], ] and Achyuta Panikkar. It flourished between the ] and ] and has its intellectual roots with ] who lived in the ]. The lineage continues down to modern times but the original research seems to have ended with ] (]-]) mostly due to subsequent political upheaval in Kerala. These astronomers, in attempting to solve problems, invented a number of important concepts including:
	*Revolutionary ideas of ].
	*The theory of ].
	*Infinite series expansions of functions.
	*].
	*].
	*].
	*Tests of ] (often attributed to ]).
	*Methods of ].
	*].
	*Term by term integration.
	*Numerical integration by means of infinite series.
	*The theory that the area under a curve is its ].
	*Used their intuitive understanding of integration in deriving the areas of curved surfaces and the volumes enclosed by them.
	*] for solution of ] equations.
	*Decimal ] numbers, and using this system of numbers, they were able to investigate and rationalise about the ] of ].

			=== Navya-Nyaya ===
	They achieved most of these results several centuries before European mathematicians. Jyeshtadeva consolidated the Kerala School's discoveries in the ''Yuktibhasa'', the world's first calculus text. In many ways, the Kerala School represents the peak of mathematical knowledge in the ].
			{{Main\|Navya-Nyāya}}
			The Navya-Nyāya or Neo-Logical darśana (school) of Indian philosophy was founded in the 13th century by the philosopher ] of ].<ref>{{Cite book \|last=Vidyabhusana \|first=Satis Chandra \|url=https://books.google.com/books?id=0lG85RD9YZoC \|title=A History of Indian Logic: Ancient, Mediaeval and Modern Schools \|publisher=Motilal Banarsidass \|year=1920 \|isbn=9788120805651 \|location=Delhi \|pages=405–6 \|language=en}}</ref> It was a development of the classical Nyāya darśana. Other influences on Navya-Nyāya were the work of earlier philosophers ] (900–980 CE) and ] (late 10th century).

			Gangeśa's book ] ("Thought-Jewel of Reality") was written partly in response to Śrīharśa's Khandanakhandakhādya, a defence of ], which had offered a set of thorough criticisms of Nyāya theories of thought and language.<ref>{{cite book \|last1=Satis Chandra Vidyabhusana \|url=https://archive.org/details/historyindianlog00vidy \|title=A History of Indian Logic: Ancient, Mediaeval and Modern Schools \|date=1920 \|publisher=Motilal Banarsidas \|isbn=9788120805651 \|location=Delhi \|page=}}</ref> Navya-Nyāya developed a sophisticated language and conceptual scheme that allowed it to raise, analyze, and solve problems in logic and epistemology. It involves naming each object to be analyzed, identifying a distinguishing characteristic for the named object, and verifying the appropriateness of the defining characteristic using ''pramanas''.<ref>{{Citation \|last=Ganeri \|first=Jonardon \|title=Analytic Philosophy in Early Modern India \|date=2023 \|editor-last=Zalta \|editor-first=Edward N. \|url=https://plato.stanford.edu/archives/win2023/entries/early-modern-india/ \|access-date=2024-01-23 \|edition=Winter 2023 \|publisher=Metaphysics Research Lab, Stanford University \|editor2-last=Nodelman \|editor2-first=Uri \|encyclopedia=The Stanford Encyclopedia of Philosophy}}</ref>
	===Narayana Pandit===
	] (c. 1340-1400), the earliest of the notable Kerala mathematicians, had written two works, an arithmetical treatise called ''Ganita Kaumudi'' and an algebraic treatise called ''Bijganita Vatamsa''. Narayana is also thought to be the author of an elaborate commentary of ]'s ], titled ''Karmapradipika'' (or ''Karma-Paddhati'').

			=== Kerala School ===
	Although the ''Karmapradipika'' contains little original work, the following are found within it:
			{{Main\|Kerala school of astronomy and mathematics}}
	*Seven different methods for squaring numbers, a contribution that is wholly original to the author.
			]]]
			] c.1530]]
			The ] was founded by ] in Kerala, ] and included among its members: ], ], ], ], ] and Achyuta Panikkar. It flourished between the 14th and 16th centuries and the original discoveries of the school seems to have ended with 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 ]s, were given in ] 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 '']'' (c.1500–c.1610), written in ], by ].<ref name="roy">{{Harv\|Roy\|1990}}</ref>

			Their discovery of these three important series expansions of ]—several centuries before calculus was developed in Europe by ] and ]—was an achievement. However, the Kerala School did not invent ''calculus'',<ref name=bressoud/> because, while they were able to develop ] expansions for the important ], they developed neither a theory of ] or ], nor the ].<ref name=katz/> The results obtained by the Kerala school include:
	Narayana's other major works contain a variety of mathematical developments, including:
	*A rule to calculate approximate values of square roots.
	*The second order indeterminate equation ''nq''<sup>2</sup> + 1 = ''p''<sup>2</sup> (]).
	*Solutions of indeterminate higher-order equations.
	*Mathematical operations with ].
	*Several geometrical rules.
	*Discussion of ]s and similar figures.
	*Evidence also exists that Narayana made minor contributions to the ideas of ] found in Bhaskara II's work.
	*Narayana has also made contributions to the topic of ]s.

			*The (infinite) ]: <math> \frac{1}{1-x} = 1 + x + x^2 + x^3 + x^4+ \cdots\text{ for }\|x\|<1 </math><ref name=singh>{{Harv\|Singh\|1936}}</ref>
	===Madhava of Sangamagramma===
			*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''.<ref name=roy/>
	] (c. 1340-1425) was the founder of the ] and considered to be one of the greatest mathematician-astronomers of the ]. It is vaguely possible that he may have written ''Karana Paddhati'' a work written sometime between 1375 and 1475 but all we really know of Madhava comes from works of later scholars.
			*Intuitive use of ], however, the '']'' was not formulated or employed in proofs.<ref name=roy/>
			*Applications of ideas from (what was to become) differential and integral calculus to obtain ] for sin x, cos x, and arctan x.<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\left(\frac{y}{x}\right) = \frac{1}{1}\cdot\frac{ry}{x} -\frac{1}{3}\cdot\frac{ry^3}{x^3} + \frac{1}{5}\cdot\frac{ry^5}{x^5} - \cdots ,\text{ where }y/x \leq 1. </math>
			:: <math>r\sin x = x - x \frac{x^2}{(2^2+2)r^2} + x \frac{x^2}{(2^2+2)r^2}\cdot\frac{x^2}{(4^2+4)r^2} - \cdots </math>
			:: <math> r - \cos x = r \frac{x^2}{(2^2-2)r^2} - r \frac{x^2}{(2^2-2)r^2} \frac{x^2}{(4^2-4)r^2} + \cdots, </math>
			: where, for ''r'' = 1, the series reduces 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 the series expansion of <math>\arctan x</math> to obtain the ]:<ref name=roy/>
			:: <math>\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots </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>
			:: <math>\text{where }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 </math>
			*Using the improved series to derive a rational expression,<ref name=roy/> 104348/33215 for ''π'' correct up to ''nine'' decimal places, ''i.e.'' 3.141592653.
			*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/> 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 ] 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="whish">{{Harv\|Whish\|1835}}</ref>
	Perhaps his most significant contribution was in:
	*Moving on from the finite procedures of ancient mathematics to ''treat their limit passage to infinity'', which is considered to be the essence of modern classical analysis, and thus he is considered the father of ].

			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 ''Yuktibhāṣā'' given in two papers,<ref>{{Citation \| last1 = Rajagopal \| first1 = C. \| last2 = Rangachari \| first2 = M. S. \| year = 1949 \| title = A Neglected Chapter of Hindu Mathematics \| journal = ] \| volume = 15 \| pages = 201–209 \| postscript = . }}</ref><ref>{{Citation \| last1 = Rajagopal \| first1 = C. \| last2 = Rangachari \| first2 = M. S. \| year = 1951 \| title = On the Hindu proof of Gregory's series \| journal = ] \| volume = 17 \| pages = 65–74 \| postscript = . }}</ref> a commentary on the ''Yuktibhāṣā'''s proof of the sine and cosine series<ref>{{Citation \| last1 = Rajagopal \| first1 = C. \| last2 = Venkataraman \| first2 = A. \| year = 1949 \| title = The sine and cosine power series in Hindu mathematics \| journal = Journal of the Royal Asiatic Society of Bengal (Science) \| volume = 15 \| pages = 1–13 \| postscript = . }}</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>{{Citation \| last1 = Rajagopal \| first1 = C. \| last2 = Rangachari \| first2 = M. S. \| year = 1977 \| title = On an untapped source of medieval Keralese mathematics \| doi = 10.1007/BF00348142 \| journal = Archive for History of Exact Sciences \| volume = 18 \| issue = 2 \| pages = 89–102 \| s2cid = 51861422 \| postscript = . }}</ref><ref>{{Citation \| last1 = Rajagopal \| first1 = C. \| last2 = Rangachari \| first2 = M. S. \| year = 1986 \| title = On Medieval Kerala Mathematics \| journal = Archive for History of Exact Sciences \| volume = 35 \| issue = 2\| pages = 91–99 \| doi = 10.1007/BF00357622 \| s2cid = 121678430 \| postscript = . }}</ref>
	Madhava was also responsible for many other significant and original discoveries, including:
	*Infinite series expansions of functions.
	*The power series.
	*The Taylor series.
	*Trigonometric series.
	*Rational approximations of infinite series.
	*Taylor series of the sine and cosine functions (Madhava-] power series).
	*Taylor series of the ] function.
	*Taylor series of the ] function (Madhava-] series).
	*Second-order Taylor series approximations of the sine and cosine functions.
	*Third-order Taylor series approximation of the sine function.
	*Power series of ] (usually attributed to ]).
	*Power series of π/4 (]'s series).
	*Power series of the ].
	*Power series of the ].
	*Power series of the ].
	*Power series of angle ] (equivalent to the Gregory series).
	*Infinite ]s.
	*].
	*Term by term integration.
	*The solution of ] by ].
	*Approximation of ]s by continued fractions.
	*] of infinite series.
	*Correctly computed the value of π to 11 decimal places, the most accurate value of π after almost a thousand years.
	*Sine and cosine tables to 9 decimal places of accuracy, which would remain the most accurate up to the ].
	*Laying the foundations for the development of ], which was then further developed by his successors at the Kerala School.

			Parameshvara (c. 1370–1460) wrote commentaries on the works of ], ] and Bhaskara II. His ''Lilavati Bhasya'', a commentary on Bhaskara II's ''Lilavati'', contains one of his important discoveries: a version of the ]. ] (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.
	He also extended some results found in earlier works, including those of ].

			] (c. 1530) was a 16th-century mathematician from Kerala who gave integer solutions to 21 types of systems of two ] algebraic equations in two unknowns. These types are all the possible pairs of equations of the following seven forms:
	===Parameshvara===
			: <math>
	] (c. 1370-1460) wrote commentaries on the works of ], ] and ]. His ''Lilavati Bhasya'', a commentary on Bhaskara II's ''Lilavati'', contains one of his most important discoveries:
			\begin{align}
	*An outstanding version of the ], which is the most important result in differential calculus and one of the most important theorems in mathematical analysis. This result was later essential in proving the ].
			& 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
			\end{align}
			</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. ] (c. 1500–1575) was another member of the Kerala School. His key work was the ''Yukti-bhāṣā'' (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.
	The ''Siddhanta-dipika'' by Paramesvara is a commentary on the commentary of ] on ]'s ''Maha-bhaskariya''. It contains:
	*Some of his eclipse observations in this work including one made at Navaksetra in 1422 and two made at Gokarna in 1425 and 1430.
	*A mean value type formula for inverse interpolation of the sine.
	*It presents a one-point iterative technique for calculating the sine of a given angle.
	*A more efficient approximation that works using a two-point iterative algorithm, which is essentially the same as the modern ] method.

			=== Others ===
	He was also the first mathematician to:
			] was a 14th century mathematician who composed two important mathematical works, an arithmetical treatise, ''Ganita Kaumudi'', and an algebraic treatise, ''Bijganita Vatamsa''. ''Ganita Kaumudi'' is one of the most revolutionary works in the field of combinatorics with developing a method for ] of a given sequence.
	*Give the radius of circle with inscribed ], an expression that is normally attributed to Lhuilier (]).
			In his ''Ganita Kaumudi'' Narayana proposed the following problem on a herd of cows and calves:
			{{blockquote\|A cow produces one calf every year. Beginning in its fourth year, each calf produces one calf at the beginning of each year. How many cows and calves are there altogether after 20 years?}}
			Translated into the modern mathematical language of ]:
			:{{math\|1= N<sub>n</sub> = N<sub>n-1</sub> + N<sub>n-3</sub>}} for {{math\|n > 2}},
			with initial values
			:{{math\|1= N<sub>0</sub> = N<sub>1</sub> = N<sub>2</sub> = 1}}.

			The first few terms are 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88,... {{OEIS\|A000930}}.
	===Nilakantha Somayaji===
			The limit ratio between consecutive terms is the ].
	In ]'s (1444-1544) most notable work ''Tantra Samgraha'' (which 'spawned' a later anonymous commentary ''Tantrasangraha-vyakhya'' and a further commentary by the name ''Yuktidipaika'', written in ]) he elaborates and extends the contributions of Madhava. Sadly none of his mathematical works are extant, however it can be determined that he was a mathematician of some note. Nilakantha was also the author of ''Aryabhatiya-bhasa'' a commentary of the ''Aryabhatiya''. Of great significance in Nilakantha's work includes:
			. Narayana is also thought to be the author of an elaborate commentary of ]'s ], titled ](or ''Karma-Paddhati'').<ref>{{Citation \|last=Divakaran \|first=P. P. \|title=The Mathematics of India \|chapter=From 500 BCE to 500 CE \|date=2018 \|series=Sources and Studies in the History of Mathematics and Physical Sciences \|pages=143–173 \|chapter-url=http://dx.doi.org/10.1007/978-981-13-1774-3_6 \|access-date=2024-06-18 \|place=Singapore \|publisher=Springer Singapore \|doi=10.1007/978-981-13-1774-3_6 \|isbn=978-981-13-1773-6}}</ref>
	*The presence of ] mathematical proof.
	*Proof of the Madhava-Gregory series of the arctangent.
	*Improvements and proofs of other infinite series expansions by Madhava.
	*An improved series expansion of π/4 that converges more rapidly.
	*The relationship between the power series of π/4 and arctangent.

	===Citrabhanu===
	] (c. 1530) was a 16th century mathematician from Kerala who gave integer solutions to 21 types of systems of two ] 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 ], a regional language of ]), the world's first calculus text. It contained most of the developments of earlier Kerala School mathematicians, particularly Madhava. Similarly to the work of Nilakantha, it is almost unique in the history of Indian mathematics, in that it contains:
	*Proofs of theorems.
	*Derivations of rules and series.
	*Proofs of most mathematical theorems and infinite series earlier discovered by Madhava and other Kerala School mathematicians.
	*Proof of the series expansion of the arctangent function (equivalent to Gregory's proof), and the sine and cosine functions.

	He also studied various topics found in many previous Indian works, including:
	*Integer solutions of systems of first degree equations solved using ''kuttaka''.
	*Rules of finding the sines and the cosines of the sum and difference of two angles.

	Jyesthadeva also gave:
	*The earliest statement of Wallis' theorem.
	*Geometric derivations of series.

	==Charges of Eurocentrism==		==Charges of Eurocentrism==
	Indian contributions have not been given due acknowledgement in modern history. ~~Many~~ discoveries and inventions by ] are presently culturally attributed to their ~~western~~ counterparts, as a result of ].		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 ] are presently culturally attributed to their ] counterparts, as a result of ]. According to G. G. Joseph's take on "]":

			<blockquote> takes on board some of the objections raised about the classical Eurocentric trajectory. The awareness is all too likely to be tempered with dismissive rejections of their importance compared to Greek mathematics. The contributions from other civilisations – 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\|0-7914-3352-8}}. p.67-68.</ref></blockquote>
	The historian ], one of the most celebrated historians of mathematics in the early ], suggested that "], the father of Greek ], got the first algebraic knowledge from India." This theory is supported by evidence of continuous contact between ] and the ] from the late ], and earlier evidence that the eminent ] ] studied in India, which further 'throws open' the Eurocentric ideal.

			Historian of mathematics ] wrote that he and others "suspect that ] got his first glimpse of algebraic knowledge from India".<ref>{{Citation
	More recently, evidence has been unearthed that reveals that the foundations of ] were laid in India, at the ]. Some scholars have suggested that calculus and other mathematics of India were through the trade route from ] by traders and ] missionaries. Kerala was in continuous contact with ], ], and from around ], Europe as well, thus transmission would have been possible. There is no direct evidence by way of relevant manuscripts but the evidence of methodological similarities, communication routes and a suitable chronology for transmission is hard to dismiss.
			\| last = Cajori
	:''Further information: ]''
			\| first = Florian
			\| author-link = Florian Cajori
			\| title = A History of Mathematics P 86
			\| year = 1893
			\| publisher = Macmillan & Co.
			\| chapter = The Hindoos
			\| quote = In algebra, there was probably a mutual giving and receiving . We suspect that Diophantus got his first glimpse of algebraic knowledge from India
			}}
			</ref> He also wrote that "it is certain that portions of Hindu mathematics are of Greek origin".<ref>Florian Cajori (2010). "''''". p.94. {{ISBN\|1-4460-2221-8}}</ref>

			More recently, as discussed in the above section, the infinite series of ] for trigonometric functions (rediscovered by Gregory, Taylor, and Maclaurin in the late 17th century) were described in India, by mathematicians of the ], 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 ] by traders and ] missionaries.<ref name=almeida/> Kerala was in continuous contact with China and ], and, from around 1500, with Europe. The fact that the communication routes existed and the chronology is suitable certainly make such transmission a possibility. However, no evidence of transmission has been found.<ref name=almeida>{{Citation \| last1 = Almeida \| first1 = D. F. \| last2 = John \| first2 = J. K. \| last3 = Zadorozhnyy \| first3 = A. \| year = 2001 \| title = Keralese Mathematics: Its Possible Transmission to Europe and the Consequential Educational Implications \| journal = Journal of Natural Geometry \| volume = 20 \| pages = 77–104 \| postscript = . }}</ref> According to ], "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>{{Citation \| last1 = Gold \| first1 = D. \| last2 = Pingree \| first2 = D. \| year = 1991 \| title = A hitherto unknown Sanskrit work concerning Madhava's derivation of the power series for sine and cosine \| journal = Historia Scientiarum \| volume = 42 \| pages = 49–65 \| postscript = . }}</ref>
	==Notes==
	<references/>

			Both Arab and Indian scholars made discoveries before the 17th century that are now considered a part of calculus.<ref name=katz/> However, they did not (as ] and ] did) "combine many differing ideas under the two unifying themes of the ] and the ], 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 area of current research, especially in the manuscript collections of Spain and ], and is being pursued, among other places, at the ].<ref name=katz/>
	== Bibliography ==
	*Bibhutibhusan Datta and Avadhesh Narayan Singh. ''History of Hindu Mathematics: A Source Book'', Asia Publishing House, 1962.
	*D. F. Almeida, J. K. John and A. Zadorozhnyy. 'Keralese Mathematics: Its Possible Transmission to Europe and the Consequential Educational Implications', ''Journal of Natural Geometry'' '''20''' (pages 77-104), 2001.
	*D. P. Agrawal. '''', , 2001.
	*D. P. Agrawal. '''', Infinity Foundation, 2001.
	*David Gray. '''', Infinity Foundation, 2000.
	*Dr. John J. O'Connor and Professor Edmund F. Robertson. , '']'', ], 2000.
	*Dr. John J. O'Connor and Professor Edmund F. Robertson. , ''MacTutor History of Mathematics Archive'', St Andrews University, 2004.
	*Dwight William Johnson. '''', 2003.
	*Ebenezer Burgess. 'Surya Siddhanta: A Text Book of Hindu Astronomy', ''Journal of the American Oriental Society'' '''6''', ], 1860.
	*F. Nau. 'Notes d'astronomie indienne', ''Journal Asiatique'' '''10''' (pages 209 - 228), 1910.
	*George Gheverghese Joseph. ''The Crest of the Peacock: Non-European Roots of Mathematics'', 2nd Edition, ], 2000.
	*Ian G. Pearce. , ''MacTutor History of Mathematics Archive'', St Andrews University, 2002.
	*Kalakriti. '''', 2002.
	*R. C. Gupta. 'Indian Mathematics Abroad up to the tenth Century A.D.', ''Ganita-Bharati'' '''4''' (pages 10-16), 1982.
	*Sarvesh Srivastava. '''', 2001.
	*Sherry Hix. '''', ], 1998.
	*Victor J. Katz. ''A History of Mathematics: An Introduction'', 2nd Edition, ], 1998.

	== ~~External links~~ ==		==See also==
			{{div col\|colwidth=27em}}
	* , a workshop on traditional Indian sciences for school children conducted by the Computer Science department of Anna University, Chennai, India.
			*]
	*
			*]
	*
			*]
	*
			*]
	*
			*]
	== See also ==
			*]
	*]		*]
	*]		*]
	*]		*]
			*]
	*]		*]
			*]
			{{div col end}}

			==Notes==
	]
			{{Reflist}}
	]

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			==Further reading==
			===Source books in Sanskrit===
			*{{Citation
			\| last1=Keller
			\| first1=Agathe
			\| year=2006
			\| title=Expounding the Mathematical Seed. Vol. 1: The Translation: A Translation of Bhaskara I on the Mathematical Chapter of the Aryabhatiya
			\| publisher=Basel, Boston, and Berlin: Birkhäuser Verlag, 172 pages
			\| isbn=978-3-7643-7291-0
			}}.
			*{{Citation
			\| last1=Keller
			\| first1=Agathe
			\| year=2006
			\| title=Expounding the Mathematical Seed. Vol. 2: The Supplements: A Translation of Bhaskara I on the Mathematical Chapter of the Aryabhatiya
			\| publisher=Basel, Boston, and Berlin: Birkhäuser Verlag, 206 pages
			\| isbn=978-3-7643-7292-7
			}}.
			*{{Citation
			\| editor1-last=Sarma
			\| editor1-first=K. V.
			\| editor1-link=K. V. Sarma
			\| year=1976
			\| title={{IAST\|Āryabhaṭīya}} of {{IAST\|Āryabhaṭa}} with the commentary of Sūryadeva Yajvan
			\| publisher=critically edited with Introduction and Appendices, New Delhi: Indian National Science Academy
			}}.
			*{{Citation
			\| editor1-last=Sen
			\| editor1-first=S. N.
			\| editor2-last=Bag
			\| editor2-first=A. K.
			\| year=1983
			\| title=The Śulbasūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava
			\| publisher=with Text, English Translation and Commentary, New Delhi: Indian National Science Academy
			}}.
			*{{Citation
			\| editor1-last=Shukla
			\| editor1-first=K. S.
			\| year=1976
			\| title={{IAST\|Āryabhaṭīya}} of {{IAST\|Āryabhaṭa}} with the commentary of Bhāskara I and Someśvara
			\| publisher=critically edited with Introduction, English Translation, Notes, Comments and Indexes, New Delhi: Indian National Science Academy
			}}.
			*{{Citation
			\| editor1-last=Shukla
			\| editor1-first=K. S.
			\| year=1988
			\| title={{IAST\|Āryabhaṭīya of Āryabhaṭa}}
			\| publisher=critically edited with Introduction, English Translation, Notes, Comments and Indexes, in collaboration with ], New Delhi: Indian National Science Academy
			}}.

			==External links==
			{{Wikiquote}}
			*
			*, '']'', ], 2000.
			*
			*, ''MacTutor History of Mathematics Archive'', St Andrews University, 2004.
			*, . Ian Pearce. ''MacTutor History of Mathematics Archive'', St Andrews University, 2002.
			* {{In Our Time\|Indian Mathematics\|p0038xb0}}
			* , a workshop on traditional Indian sciences for school children conducted by the Computer Science department of Anna University, Chennai, India.
			*
			*
			*
			{{India topics}}
			{{History of mathematics}}
			{{histOfScience}}

			{{DEFAULTSORT:Indian Mathematics}}
	]
			]
	]
			]
	]
	]
	]

Latest revision as of 03:27, 14 January 2025

Development of mathematics in South Asia "Mathematics in India" redirects here. For the 2009 monograph by Kim Plofker, see Mathematics in India (book).

Indian mathematics emerged and developed in the Indian subcontinent from about 1200 BCE until roughly the end of the 18th century CE (approximately 1800 CE). In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, Bhaskara II, Varāhamihira, and Madhava. The decimal number system in use today was first recorded in Indian mathematics. Indian mathematicians made early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, and algebra. In addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East, China, and Europe 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 sutras 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 so important as the ideas involved. 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) and is likely from the 7th century CE.

A later landmark in Indian mathematics was the development of the series expansions for trigonometric functions (sine, cosine, and arc tangent) by mathematicians of the Kerala school in the 15th century CE. Their 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). However, they did not formulate a systematic theory of differentiation and integration, nor is there any evidence of their results being transmitted outside Kerala.

Prehistory

Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley civilisation have uncovered evidence of the use of "practical mathematics". The people of the Indus Valley Civilization manufactured bricks whose dimensions were in the proportion 4:2:1, considered favourable for the stability of a brick structure. They used a standardised 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 hexahedra, barrels, cones, and cylinders, thereby demonstrating knowledge of basic geometry.

The inhabitants of Indus civilisation also tried to standardise measurement of length to a high degree of accuracy. They designed a ruler—the Mohenjo-daro ruler—whose unit of length (approximately 1.32 inches or 3.4 centimetres) was divided into ten equal parts. Bricks manufactured in ancient Mohenjo-daro often had dimensions that were integral multiples of this unit of length.

Hollow cylindrical objects made of shell and found at Lothal (2200 BCE) and Dholavira are demonstrated to have the ability to measure angles in a plane, as well as to determine the position of stars for navigation.

Vedic period

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Samhitas and Brahmanas

The texts of the Vedic Period provide evidence for the use of large numbers. By the time of the Yajurvedasaṃhitā- (1200–900 BCE), numbers as high as 10 were being included in the texts. For example, the mantra (sacred recitation) at the end of the annahoma ("food-oblation rite") performed during the aśvamedha, and uttered just before-, during-, and just after sunrise, invokes powers of ten from a hundred to a trillion:

Hail to śata ("hundred," 10), hail to sahasra ("thousand," 10), hail to ayuta ("ten thousand," 10), hail to niyuta ("hundred thousand," 10), hail to prayuta ("million," 10), hail to arbuda ("ten million," 10), hail to nyarbuda ("hundred million," 10), hail to samudra ("billion," 10, literally "ocean"), hail to madhya ("ten billion," 10, literally "middle"), hail to anta ("hundred billion," 10, lit., "end"), hail to parārdha ("one trillion," 10 lit., "beyond parts"), hail to the uṣas (dawn) , hail to the vyuṣṭi (twilight), hail to udeṣyat (the one which is going to rise), hail to udyat (the one which is rising), hail udita (to the one which has just risen), hail to svarga (the heaven), hail to martya (the world), hail to all.

The solution to partial fraction was known to the Rigvedic People as states in the purush Sukta (RV 10.90.4):

With three-fourths Puruṣa went up: one-fourth of him again was here.

The Satapatha Brahmana (c. 7th century BCE) contains rules for ritual geometric constructions that are similar to the Sulba Sutras.

Śulba Sūtras

Main article: Shulba Sutras

The Śulba Sūtras (literally, "Aphorisms of the Chords" in Vedic Sanskrit) (c. 700–400 BCE) list rules for the construction of sacrificial fire altars. Most mathematical problems considered in the Śulba Sūtras spring from "a single theological requirement," 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.

According to Hayashi, 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 Babylonians."

The diagonal rope (akṣṇayā-rajju) of an oblong (rectangle) produces both which the flank (pārśvamāni) and the horizontal (tiryaṇmānī) <ropes> produce separately."

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.

They contain lists of Pythagorean triples, which are particular cases of Diophantine equations. They also contain statements (that with hindsight we know to be approximate) about squaring the circle and "circling the square."

Baudhayana (c. 8th century BCE) composed the Baudhayana Sulba Sutra, the best-known Sulba Sutra, which contains examples of simple Pythagorean triples, such as: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), and (12, 35, 37), 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." 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." Baudhayana gives an expression for the square root of two:

{\sqrt {2}}\approx 1+{\frac {1}{3}}+{\frac {1}{3\cdot 4}}-{\frac {1}{3\cdot 4\cdot 34}}=1.4142156\ldots

The expression is accurate up to five decimal places, the true value being 1.41421356... This expression is similar in structure to the expression found on a Mesopotamian tablet from the Old Babylonian period (1900–1600 BCE):

{\sqrt {2}}\approx 1+{\frac {24}{60}}+{\frac {51}{60^{2}}}+{\frac {10}{60^{3}}}=1.41421297\ldots

which expresses √2 in the sexagesimal system, and which is also accurate up to 5 decimal places.

According to mathematician S. G. Dani, the Babylonian cuneiform tablet Plimpton 322 written c. 1850 BCE "contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple, 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." Dani goes on to say:

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.

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

Main article: Vyakarana

The Vedic period saw the work of Sanskrit grammarian Pāṇini (c. 520–460 BCE). His grammar includes a precursor of the Backus–Naur form (used in the description programming languages).

Pingala (300 BCE – 200 BCE)

Among the scholars of the post-Vedic period who contributed to mathematics, the most notable is Pingala (piṅgalá) (fl. 300–200 BCE), a music theorist who authored the Chhandas Shastra (chandaḥ-śāstra, also Chhandas Sutra chhandaḥ-sūtra), a Sanskrit treatise on prosody. Pingala's work also contains the basic ideas of Fibonacci numbers (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 Meru-prastāra (literally "the staircase to Mount Meru"), has this to say:

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, ...

The text also indicates that Pingala was aware of the combinatorial identity:

{n \choose 0}+{n \choose 1}+{n \choose 2}+\cdots +{n \choose n-1}+{n \choose n}=2^{n}

Kātyāyana

Kātyāyana (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.

Jain mathematics (400 BCE – 200 CE)

Although Jainism as a religion and philosophy predates its most famous exponent, the great Mahaviraswami (6th century BCE), most Jain texts on mathematical topics were composed after the 6th century BCE. Jain 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 Jain 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 infinities led them to classify numbers into three classes: enumerable, innumerable and infinite. Not content with a simple notion of infinity, their texts 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, Jain mathematicians devised notations for simple powers (and exponents) of numbers like squares and cubes, which enabled them to define simple algebraic equations (bījagaṇita samīkaraṇa). Jain mathematicians were apparently also the first to use the word shunya (literally void in Sanskrit) to refer to zero. This word is the ultimate etymological origin of the English word "zero", as it was calqued into Arabic as ṣifr and then subsequently borrowed into Medieval Latin as zephirum, finally arriving at English after passing through one or more Romance languages (c.f. French zéro, Italian zero).

In addition to Surya Prajnapti, important Jain works on mathematics included the Sthānāṅga Sūtra (c. 300 BCE – 200 CE); the Anuyogadwara Sutra (c. 200 BCE – 100 CE), which includes the earliest known description of factorials in Indian mathematics; and the Ṣaṭkhaṅḍāgama (c. 2nd century CE). Important Jain 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 Jain philosophy and metaphysics, composed a mathematical work called the Tattvārtha Sūtra.

Oral tradition

Mathematicians of ancient and early medieval India were almost all Sanskrit pandits (paṇḍita "learned man"), who were trained in Sanskrit language and literature, and possessed "a common stock of knowledge in grammar (vyākaraṇa), exegesis (mīmāṃsā) and logic (nyāya)." Memorisation of "what is heard" (śruti in Sanskrit) through recitation played a major role in the transmission of sacred texts in ancient India. Memorisation 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."

Styles of memorisation

Prodigious energy was expended by ancient Indian culture in ensuring that these texts were transmitted from generation to generation with inordinate fidelity. For example, memorisation of the sacred Vedas 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 in the original order. The recitation thus proceeded as:

word1word2, word2word1, word1word2; word2word3, word3word2, word2word3; ...

In another form of recitation, dhvaja-pāṭha (literally "flag recitation") a sequence of N words were recited (and memorised) by pairing the first two and last two words and then proceeding as:

word₁word₂, word_{N − 1}word_N; word₂word₃, word_{N − 2}word_{N − 1}; ..; word_{N − 1}word_N, word₁word₂;

The most complex form of recitation, ghana-pāṭha (literally "dense recitation"), according to Filliozat, took the form:

word1word2, word2word1, word1word2word3, word3word2word1, word1word2word3; word2word3, word3word2, word2word3word4, word4word3word2, word2word3word4; ...

That these methods have been effective is testified to by the preservation of the most ancient Indian religious text, the Ṛgveda (c. 1500 BCE), as a single text, without any variant readings. Similar methods were used for memorising mathematical texts, whose transmission remained exclusively oral until the end of the Vedic period (c. 500 BCE).

The Sutra genre

Mathematical activity in ancient India began as a part of a "methodological reflexion" on the sacred Vedas, which took the form of works called Vedāṇgas, or, "Ancillaries of the Veda" (7th–4th century BCE). The need to conserve the sound of sacred text by use of śikṣā (phonetics) and chhandas (metrics); to conserve its meaning by use of vyākaraṇa (grammar) and nirukta (etymology); and to correctly perform the rites at the correct time by the use of kalpa (ritual) and jyotiṣa (astrology), gave rise to the six disciplines of the Vedāṇgas. 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 sūtra (literally, "thread"):

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.

Extreme brevity was achieved through multiple means, which included using ellipsis "beyond the tolerance of natural language," using technical names instead of longer descriptive names, abridging lists by only mentioning the first and last entries, and using markers and variables. 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 parampara, 'uninterrupted succession from teacher (guru) to the student (śisya),' and it was not open to the general public" and perhaps even kept secret. The brevity achieved in a sūtra is demonstrated in the following example from the Baudhāyana Śulba Sūtra (700 BCE).

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. 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:

II.64. After dividing the quadri-lateral in seven, one divides the transverse in three.
II.65. In another layer one places the North-pointing.

According to Filliozat, 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.

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.

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.

The earliest mathematical prose commentary was that on the work, 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. However, according to Hayashi, "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:

Rule ('sūtra') in verse by Āryabhaṭa
Commentary by 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 favoured by then.

Typically, for any mathematical topic, students in ancient India first memorised the sūtras, which, as explained earlier, were "deliberately inadequate" 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 (fl. 7th century CE), to characterise astronomical computations as "dust work" (Sanskrit: dhulikarman).

Numerals and the decimal number system

It is well known that the decimal place-value system in use today was first recorded in India, then transmitted to the Islamic world, and eventually to Europe. The Syrian bishop Severus Sebokht wrote in the mid-7th century CE about the "nine signs" of the Indians for expressing numbers. However, how, when, and where the first decimal place value system was invented is not so clear.

The earliest extant 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 4th century BCE to the 4th century CE. Almost contemporaneously, another script, the Brāhmī script, 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.

The earliest surviving evidence of decimal place value numerals in India and southeast Asia is from the middle of the first millennium CE. A copper plate from Gujarat, India mentions the date 595 CE, written in a decimal place value notation, although there is some doubt as to the authenticity of the plate. Decimal numerals recording the years 683 CE have also been found in stone inscriptions in Indonesia and Cambodia, where Indian cultural influence was substantial.

There are older textual sources, although the extant manuscript copies of these texts are from much later dates. Probably the earliest such source is the work of the Buddhist philosopher Vasumitra dated likely to the 1st century CE. Discussing the counting pits of merchants, Vasumitra remarks, "When clay counting-piece is in the place of units, it is denoted as one, when in hundreds, one hundred." Although such references seem to imply that his readers had knowledge of a decimal place value representation, the "brevity of their allusions and the ambiguity of their dates, however, do not solidly establish the chronology of the development of this concept."

A third decimal representation was employed in a verse composition technique, later labelled Bhuta-sankhya (literally, "object numbers") used by early Sanskrit authors of technical books. Since many early technical works were composed in verse, numbers were often represented by objects in the natural or religious world that correspondence to them; this allowed a many-to-one correspondence for each number and made verse composition easier. According to Plofker, the number 4, for example, could be represented by the word "Veda" (since there were four of these religious texts), the number 32 by the word "teeth" (since a full set consists of 32), and the number 1 by "moon" (since there is only one moon). So, Veda/teeth/moon would correspond to the decimal numeral 1324, as the convention for numbers was to enumerate their digits from right to left. The earliest reference employing object numbers is a c. 269 CE Sanskrit text, Yavanajātaka (literally "Greek horoscopy") of Sphujidhvaja, a versification of an earlier (c. 150 CE) Indian prose adaptation of a lost work of Hellenistic astrology. Such use seems to make the case that by the mid-3rd century CE, the decimal place value system was familiar, at least to readers of astronomical and astrological texts in India.

It has been hypothesized that the Indian decimal place value system was based on the symbols used on Chinese counting boards from as early as the middle of the first millennium BCE. According to Plofker,

These counting boards, like the Indian counting pits, ..., had a decimal place value structure ... Indians may well have learned of these decimal place value "rod numerals" from Chinese Buddhist pilgrims or other travelers, or they may have developed the concept independently from their earlier non-place-value system; no documentary evidence survives to confirm either conclusion."

Bakhshali Manuscript

The oldest extant mathematical manuscript in India is the Bakhshali Manuscript, a birch bark manuscript written in "Buddhist hybrid Sanskrit" in the Śāradā script, which was used in the northwestern region of the Indian subcontinent between the 8th and 12th centuries CE. 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 the University of Oxford, the manuscript has been dated recently as 224 AD- 383 AD.

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. 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." Many of its problems are of a category known as 'equalisation problems' that lead to systems of linear equations. One example from Fragment III-5-3v is the following:

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.

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.

In 2017, three samples from the manuscript were shown by radiocarbon dating to come from three different centuries: from 224 to 383 AD, 680-779 AD, and 885-993 AD. It is not known how fragments from different centuries came to be packaged together.

Classical period (400–1300)

This period is often known as the golden age of Indian Mathematics. This period saw mathematicians such as Aryabhata, Varahamihira, Brahmagupta, Bhaskara I, Mahavira, Bhaskara II, Madhava of Sangamagrama and Nilakantha Somayaji 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). This tripartite division is seen in Varāhamihira's 6th century compilation—Pancasiddhantika (literally panca, "five," siddhānta, "conclusion of deliberation", dated 575 CE)—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.

Fourth to sixth centuries

Surya Siddhanta

Though its authorship is unknown, the Surya Siddhanta (c. 400) contains the roots of modern trigonometry. Because it contains many words of foreign origin, some authors consider that it was written under the influence of Mesopotamia and Greece.

This ancient text uses the following as trigonometric functions for the first time:

Sine (Jya).
Cosine (Kojya).
Inverse sine (Otkram jya).

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.

Aryabhata I

Aryabhata (476–550) wrote the Aryabhatiya. He described the important fundamental principles of mathematics in 332 shlokas. The treatise contained:

Quadratic equations
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:

(See also : Aryabhata's sine table)

Introduced the trigonometric functions.
Defined the sine (jya) as the modern relationship between half an angle and half a chord.
Defined the cosine (kojya).
Defined the versine (utkrama-jya).
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:

Simple continued fractions.

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:

Accurate calculations for astronomical constants, such as the:
- Solar eclipse.
- Lunar eclipse.
- The formula for the sum of the cubes, which was an important step in the development of integral calculus.

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:

$\sin ^{2}(x)+\cos ^{2}(x)=1$
$\sin(x)=\cos \left({\frac {\pi }{2}}-x\right)$
${\frac {1-\cos(2x)}{2}}=\sin ^{2}(x)$

Seventh and eighth centuries

In the 7th century, two separate fields, arithmetic (which included measurement) 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"—like the seeds of plants—representing unknowns with the potential to generate, in this case, the solutions of equations). Brahmagupta, in his astronomical work 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). In the latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral:

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 generalisation of Heron's formula), as well as a complete description of rational triangles (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

A={\sqrt {(s-a)(s-b)(s-c)(s-d)}}\,

where s, the semiperimeter, given by $s={\frac {a+b+c+d}{2}}.$

Brahmagupta's Theorem on rational triangles: A triangle with rational sides $a,b,c$ and rational area is of the form:

a={\frac {u^{2}}{v}}+v,\ \ b={\frac {u^{2}}{w}}+w,\ \ c={\frac {u^{2}}{v}}+{\frac {u^{2}}{w}}-(v+w)

for some rational numbers $u,v,$ and $w$ .

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

\ ax^{2}+bx=c

To the absolute number multiplied by four times the square, add the square of the middle term; the square root of the same, less the middle term, being divided by twice the square is the value.

This is equivalent to:

x={\frac {{\sqrt {4ac+b^{2}}}-b}{2a}}

Also in chapter 18, Brahmagupta was able to make progress in finding (integral) solutions of Pell's equation,

\ x^{2}-Ny^{2}=1,

where $N$ is a nonsquare integer. He did this by discovering the following identity:

Brahmagupta's Identity: $\ (x^{2}-Ny^{2})(x'^{2}-Ny'^{2})=(xx'+Nyy')^{2}-N(xy'+x'y)^{2}$ which was a generalisation of an earlier identity of Diophantus: Brahmagupta used his identity to prove the following lemma:

Lemma (Brahmagupta): If $x=x_{1},\ \ y=y_{1}\ \$ is a solution of $\ \ x^{2}-Ny^{2}=k_{1},$ and, $x=x_{2},\ \ y=y_{2}\ \$ is a solution of $\ \ x^{2}-Ny^{2}=k_{2},$ , then:

x=x_{1}x_{2}+Ny_{1}y_{2},\ \ y=x_{1}y_{2}+x_{2}y_{1}\ \

is a solution of

\ x^{2}-Ny^{2}=k_{1}k_{2}

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 $\ x^{2}-Ny^{2}=k$ has an integer solution for any one of $\ k=\pm 4,\pm 2,-1$ then Pell's equation:

\ x^{2}-Ny^{2}=1

also has an integer solution.

Brahmagupta did not actually prove the theorem, but rather worked out examples using his method. The first example he presented was:

Example (Brahmagupta): Find integers $\ x,\ y\$ such that:

\ x^{2}-92y^{2}=1

In his commentary, Brahmagupta added, "a person solving this problem within a year is a mathematician." The solution he provided was:

\ x=1151,\ y=120

Bhaskara I

Bhaskara I (c. 600–680) expanded the work of Aryabhata in his books titled Mahabhaskariya, Aryabhatiya-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 two decimal places.

Ninth to twelfth centuries

Virasena

Virasena (8th century) was a Jain mathematician in the court of Rashtrakuta King Amoghavarsha of Manyakheta, Karnataka. He wrote the Dhavala, a commentary on Jain mathematics, which:

Deals with the concept of ardhaccheda, the number of times a number could be halved, and lists various rules involving this operation. This coincides with the binary logarithm when applied to powers of two, but differs on other numbers, more closely resembling the 2-adic order.

Virasena also gave:

The derivation of the volume of a frustum by a sort of infinite procedure.

It is thought that much of the mathematical material in the Dhavala can attributed to previous writers, especially Kundakunda, Shamakunda, Tumbulura, Samantabhadra and Bappadeva and date who wrote between 200 and 600 CE.

Mahavira

Mahavira Acharya (c. 800–870) from Karnataka, the last of the notable Jain 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:

Zero
Squares
Cubes
square roots, cube roots, and the 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 whose terms are squares of an arithmetical progression, and gave empirical rules for area and perimeter of an ellipse.
Solved cubic equations.
Solved quartic equations.
Solved some quintic equations and higher-order polynomials.
Gave the general solutions of the higher order polynomial equations:
- $\ ax^{n}=q$
- $a{\frac {x^{n}-1}{x-1}}=p$
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 equations.

The Pati Ganita is a work on arithmetic and measurement. 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 equations were elaborated in the 10th century by Manjula (also Munjala), who realised that the expression

\ \sin w'-\sin w

could be approximately expressed as

\ (w'-w)\cos w

This was elaborated by his later successor Bhaskara ii thereby finding the derivative of sine.

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

Shripati Mishra (1019–1066) wrote the books Siddhanta Shekhara, a major work on astronomy in 19 chapters, and Ganit Tilaka, an incomplete arithmetical 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:

The Dhruvamanasa is a work of 105 verses on:

Calculating planetary longitudes
eclipses.
planetary transits.

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 zero being infinity.

Algebra:

The recognition of a positive number having two square roots.
Surds.
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.
- 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:

Preliminary concept of differentiation
Discovered the differential coefficient.
Stated early form of 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 although didn't perceive the notion of derivative.
Computed π, correct to five 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:
- $\ \sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$
- $\ \sin(a-b)=\sin(a)\cos(b)-\sin(b)\cos(a)$

Medieval and early modern mathematics (1300–1800)

Navya-Nyaya

Main article: Navya-Nyāya

The Navya-Nyāya or Neo-Logical darśana (school) of Indian philosophy was founded in the 13th century by the philosopher Gangesha Upadhyaya of Mithila. It was a development of the classical Nyāya darśana. Other influences on Navya-Nyāya were the work of earlier philosophers Vācaspati Miśra (900–980 CE) and Udayana (late 10th century).

Gangeśa's book Tattvacintāmaṇi ("Thought-Jewel of Reality") was written partly in response to Śrīharśa's Khandanakhandakhādya, a defence of Advaita Vedānta, which had offered a set of thorough criticisms of Nyāya theories of thought and language. Navya-Nyāya developed a sophisticated language and conceptual scheme that allowed it to raise, analyze, and solve problems in logic and epistemology. It involves naming each object to be analyzed, identifying a distinguishing characteristic for the named object, and verifying the appropriateness of the defining characteristic using pramanas.

Kerala School

Main article: 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 and 16th centuries and the original discoveries of the school seems to have ended with 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 functions, 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 Yuktibhāṣā (c.1500–c.1610), written in Malayalam, by Jyesthadeva.

Their discovery of these three important series expansions of calculus—several centuries before calculus was developed in Europe by Isaac Newton and Gottfried Leibniz—was an achievement. However, the Kerala School did not invent calculus, because, while they were able to develop Taylor series expansions for the important trigonometric functions, they developed neither a theory of differentiation or integration, nor the fundamental theorem of calculus. The results obtained by the Kerala school include:

The (infinite) geometric series: ${\frac {1}{1-x}}=1+x+x^{2}+x^{3}+x^{4}+\cdots {\text{ for }}|x|<1$
A semi-rigorous proof (see "induction" remark below) of the result: $1^{p}+2^{p}+\cdots +n^{p}\approx {\frac {n^{p+1}}{p+1}}$ for large n.
Intuitive use of mathematical induction, however, the inductive hypothesis was not formulated or employed in proofs.
Applications of ideas from (what was to become) differential and integral calculus to obtain (Taylor–Maclaurin) infinite series for sin x, cos x, and arctan x. The Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as:

r\arctan \left({\frac {y}{x}}\right)={\frac {1}{1}}\cdot {\frac {ry}{x}}-{\frac {1}{3}}\cdot {\frac {ry^{3}}{x^{3}}}+{\frac {1}{5}}\cdot {\frac {ry^{5}}{x^{5}}}-\cdots ,{\text{ where }}y/x\leq 1.

r\sin x=x-x{\frac {x^{2}}{(2^{2}+2)r^{2}}}+x{\frac {x^{2}}{(2^{2}+2)r^{2}}}\cdot {\frac {x^{2}}{(4^{2}+4)r^{2}}}-\cdots

r-\cos x=r{\frac {x^{2}}{(2^{2}-2)r^{2}}}-r{\frac {x^{2}}{(2^{2}-2)r^{2}}}{\frac {x^{2}}{(4^{2}-4)r^{2}}}+\cdots ,

where, for r = 1, the series reduces to the standard power series for these trigonometric functions, for example:

\sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots

and

\cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots

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.)
Use of the series expansion of $\arctan x$ to obtain the Leibniz formula for π:

{\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots

A rational approximation of error for the finite sum of their series of interest. For example, the error, $f_{i}(n+1)$ , (for n odd, and i = 1, 2, 3) for the series:

{\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)

{\text{where }}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}}.

Manipulation of error term to derive a faster converging series for $\pi$ :

{\frac {\pi }{4}}={\frac {3}{4}}+{\frac {1}{3^{3}-3}}-{\frac {1}{5^{3}-5}}+{\frac {1}{7^{3}-7}}-\cdots

Using the improved series to derive a rational expression, 104348/33215 for π correct up to nine decimal places, i.e. 3.141592653.
Use of an intuitive notion of limit to compute these results.
A semi-rigorous (see remark on limits above) method of differentiation of some trigonometric functions. 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."

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 Yuktibhāṣā given in two papers, a commentary on the Yuktibhāṣā's proof of the sine and cosine series and two papers that provide the Sanskrit verses of the Tantrasangrahavakhya for the series for arctan, sin, and cosine (with English translation and commentary).

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 algebraic equations in two unknowns. These types are all the possible pairs of equations of the following seven forms:

{\begin{aligned}&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\end{aligned}}

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-bhāṣā (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.

Others

Narayana Pandit was a 14th century mathematician who composed two important mathematical works, an arithmetical treatise, Ganita Kaumudi, and an algebraic treatise, Bijganita Vatamsa. Ganita Kaumudi is one of the most revolutionary works in the field of combinatorics with developing a method for systematic generation of all permutations of a given sequence. In his Ganita Kaumudi Narayana proposed the following problem on a herd of cows and calves:

A cow produces one calf every year. Beginning in its fourth year, each calf produces one calf at the beginning of each year. How many cows and calves are there altogether after 20 years?

Translated into the modern mathematical language of recurrence sequences:

N_n = N_n-1 + N_n-3 for n > 2,

with initial values

N₀ = N₁ = N₂ = 1.

The first few terms are 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88,... (sequence A000930 in the OEIS). The limit ratio between consecutive terms is the supergolden ratio. . Narayana is also thought to be the author of an elaborate commentary of Bhaskara II's Lilavati, titled Ganita Kaumudia(or Karma-Paddhati).

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 counterparts, as a result of Eurocentrism. According to G. G. Joseph's take on "Ethnomathematics":

takes on board some of the objections raised about the classical Eurocentric trajectory. The awareness is all too likely to be tempered with dismissive rejections of their importance compared to Greek mathematics. The contributions from other civilisations – 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"

Historian of mathematics Florian Cajori wrote that he and others "suspect that Diophantus got his first glimpse of algebraic knowledge from India". He also wrote that "it is certain that portions of Hindu mathematics are of Greek origin".

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 in India, by mathematicians of the Kerala school, 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. Kerala was in continuous contact with China and Arabia, and, from around 1500, with Europe. The fact that the communication routes existed and the chronology is suitable certainly make such transmission a possibility. However, no evidence of transmission has been found. 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".

Both Arab and Indian scholars made discoveries before the 17th century that are now considered a part of calculus. However, they did not (as Newton and Leibniz did) "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". The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own; 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." This is an area of current research, especially in the manuscript collections of Spain and Maghreb, and is being pursued, among other places, at the CNRS.

Notes

^ (Kim Plofker 2007, p. 1)
^ (Hayashi 2005, pp. 360–361)
(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."
(Plofker 2009, pp. 44–47)
(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."
(Bourbaki 1998, p. 49): Modern arithmetic was known during medieval times as "Modus Indorum" or method of the Indians. Leonardo of Pisa wrote that compared to method of the Indians all other methods is a mistake. This method of the Indians is none other than our very simple arithmetic of addition, subtraction, multiplication and division. Rules for these four simple procedures was first written down by Brahmagupta during 7th century AD. "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."
^ "algebra" 2007. Britannica Concise Encyclopedia Archived 29 September 2007 at the Wayback Machine. 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."
(Pingree 2003, p. 45) Quote: "Geometry, and its branch trigonometry, was the mathematics Indian astronomers used most frequently. Greek mathematicians used the full chord and never imagined the half chord that we use today. Half chord was first used by Aryabhata which made trigonometry much more simple. 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."
(Bourbaki 1998, p. 126): "As for trigonometry, it is disdained by geometers and abandoned to surveyors and astronomers; it is these latter (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 $\theta <\pi$ on a circle of radius r, in other words the number $2r\sin \left(\theta /2\right)$ ; the introduction of the sine, more easily handled, is due to Hindu mathematicians of the Middle Ages)."
(Filliozat 2004, pp. 140–143)
(Hayashi 1995)
^ (Kim Plofker 2007, p. 6)
(Stillwell 2004, p. 173)
(Bressoud 2002, p. 12) Quote: "There is no evidence that the Indian work on series was known beyond India, or even outside Kerala, until the nineteenth century. Gold and Pingree assert that by the time these series were rediscovered in Europe, they had, for all practical purposes, been lost to India. The expansions of the sine, cosine, and arc tangent had been passed down through several generations of disciples, but they remained sterile observations for which no one could find much use."
(Plofker 2001, p. 293) Quote: "It is not unusual to encounter in discussions of Indian mathematics such assertions as that "the concept of differentiation was understood from the time of Manjula (... in the 10th century)" , or that "we may consider Madhava to have been the founder of mathematical analysis" (Joseph 1991, 293), or that Bhaskara II may claim to be "the precursor of Newton and Leibniz in the discovery of the principle of the differential calculus" (Bag 1979, 294). ... The points of resemblance, particularly between early European calculus and the Keralese work on power series, have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world (e.g., in (Bag 1979, 285)). ... It should be borne in mind, however, that such an emphasis on the similarity of Sanskrit (or Malayalam) and Latin mathematics risks diminishing our ability fully to see and comprehend the former. To speak of the Indian "discovery of the principle of the differential calculus" somewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa, as in the examples we have seen, remained within that specific trigonometric context. The differential "principle" was not generalised to arbitrary functions—in fact, the explicit notion of an arbitrary function, not to mention that of its derivative or an algorithm for taking the derivative, is irrelevant here"
(Pingree 1992, p. 562) Quote:"One example I can give you relates to the Indian Mādhava's demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by Charles Matthew Whish, in the 1830s, it was heralded as the Indians' discovery of the calculus. This claim and Mādhava's achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the Transactions of the Royal Asiatic Society, in which Whish's article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the series without the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Mādhava's mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution."
(Katz 1995, pp. 173–174) Quote:"How close did Islamic and Indian scholars come to inventing the calculus? Islamic scholars nearly developed a general formula for finding integrals of polynomials by A.D. 1000—and evidently could find such a formula for any polynomial in which they were interested. But, it appears, they were not interested in any polynomial of degree higher than four, at least in any of the material that has come down to us. Indian scholars, on the other hand, were by 1600 able to use ibn al-Haytham's sum formula for arbitrary integral powers in calculating power series for the functions in which they were interested. By the same time, they also knew how to calculate the differentials of these functions. So some of the basic ideas of calculus were known in Egypt and India many centuries before Newton. It does not appear, however, that either Islamic or Indian mathematicians saw the necessity of connecting some of the disparate ideas that we include under the name calculus. They were apparently only interested in specific cases in which these ideas were needed. ... There is no danger, therefore, that we will have to rewrite the history texts to remove the statement that Newton and Leibniz invented calculus. They were certainly the ones who were able to combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between them, and turn the calculus into the great problem-solving tool we have today."
Sergent, Bernard (1997), Genèse de l'Inde (in French), Paris: Payot, p. 113, ISBN 978-2-228-89116-5
Coppa, A.; et al. (6 April 2006), "Early Neolithic tradition of dentistry: Flint tips were surprisingly effective for drilling tooth enamel in a prehistoric population", Nature, 440 (7085): 755–6, Bibcode:2006Natur.440..755C, doi:10.1038/440755a, PMID 16598247, S2CID 6787162.
Bisht, R. S. (1982), "Excavations at Banawali: 1974–77", in Possehl, Gregory L. (ed.), Harappan Civilisation: A Contemporary Perspective, New Delhi: Oxford and IBH Publishing Co., pp. 113–124
Rao, S. R. (July 1992). "A Navigational Instrument of the Harappan Sailors" (PDF). Marine Archaeology. 3: 61–62. Archived from the original (PDF) on 8 August 2017.
A. Seidenberg, 1978. The origin of mathematics. Archive for History of Exact Sciences, vol 18.
(Staal 1999)
^ (Hayashi 2003, p. 118)
^ (Hayashi 2005, p. 363)
Pythagorean triples are triples of integers (a, b, c) with the property: a+b = c. Thus, 3+4 = 5, 8+15 = 17, 12+35 = 37, etc.
(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."
(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 (3 − 2√2), which is about 3.088."
^ (Joseph 2000, p. 229)
"Vedic Maths Complete Detail". ALLEN IntelliBrain. Retrieved 22 October 2022.
^ (Cooke 2005, p. 200)
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 + 2y)/(x + y). These approximations may also be derived by truncating the continued fraction representation of √2.
Neugebauer, O. and A. Sachs. 1945. Mathematical Cuneiform Texts, New Haven, CT, Yale University Press. p. 45.
Mathematics Department, University of British Columbia, The Babylonian tabled Plimpton 322 Archived 17 June 2020 at the Wayback Machine.
Three positive integers $(a,b,c)$ form a primitive Pythagorean triple if c = a+b and if the highest common factor of a, b, c is 1. In the particular Plimpton322 example, this means that 13500+12709 = 18541 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.
^ (Dani 2003)
Ingerman, Peter Zilahy (1 March 1967). ""Pānini-Backus Form" suggested". Communications of the ACM. 10 (3): 137. doi:10.1145/363162.363165. ISSN 0001-0782. S2CID 52817672.
(Fowler 1996, p. 11)
(Singh 1936, pp. 623–624)
- Harper, Douglas (2011). "Zero". Etymonline Etymology Dictionary. Archived from the original on 3 July 2017. figure which stands for naught in the Arabic notation," also "the absence of all quantity considered as quantity", c. 1600, from French zéro or directly from Italian zero, from Medieval Latin zephirum, from Arabic sifr "cipher", translation of Sanskrit sunya-m "empty place, desert, naught
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- "zero, n." OED Online. Oxford University Press. December 2011. Archived from the original on 7 March 2012. Retrieved 4 March 2012. French zéro (1515 in Hatzfeld & Darmesteter) or its source Italian zero, for *zefiro, < Arabic çifr
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External links

Science and Mathematics in India
An overview of Indian mathematics, MacTutor History of Mathematics Archive, St Andrews University, 2000.
Indian Mathematicians
Index of Ancient Indian mathematics, MacTutor History of Mathematics Archive, St Andrews University, 2004.
Indian Mathematics: Redressing the balance, Student Projects in the History of Mathematics. Ian Pearce. MacTutor History of Mathematics Archive, St Andrews University, 2002.
Indian Mathematics on In Our Time at the BBC
InSIGHT 2009, a workshop on traditional Indian sciences for school children conducted by the Computer Science department of Anna University, Chennai, India.
Mathematics in ancient India by R. Sridharan
Combinatorial methods in ancient India
Mathematics before S. Ramanujan

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