Indian mathematics emerged in the Indian subcontinent until the end of the 18th century. 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.

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

thumb|Cubical weights standardised in the Indus Valley civilisation

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 integer 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

Samhitas and Brahmanas

The texts of the Vedic Period provide evidence for the use of large numbers. By the time of the Yajurveda| (1200–900 BCE), numbers as high as were being included in the texts.

Śulba Sūtras

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.</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. 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: , , , , and , 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."

::<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... This expression is similar in structure to the expression found on a Mesopotamian tablet from the Old Babylonian period (1900–1600 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&nbsp;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:

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.

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&nbsp;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.

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:

  • <math>\sin^2(x) + \cos^2(x) = 1</math>
  • <math>\sin(x)=\cos\left(\frac{\pi}{2}-x\right)</math>
  • <math>\frac{1-\cos(2x)}{2}=\sin^2(x)</math>

Seventh and eighth centuries

thumb|right|200px|[[Brahmagupta's theorem states that AF = FD.]]

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 ' (literally "mathematics of algorithms") and ' (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 Brahmasphutasiddhanta| (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:

Chapter 18 contained 103 Sanskrit verses which began with rules for arithmetical operations involving zero and negative numbers

This is equivalent to:

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

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

:<math>\ x^2-Ny^2=1, </math>

where <math>N</math> is a nonsquare integer. He did this by discovering the following identity:

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

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

could be approximately expressed as

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

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

  • Computed π, correct to five decimal places.
  • Calculated the solar year to 9 decimal places.

Trigonometry:

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

Medieval and early modern mathematics (1300–1800)

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

thumb|Chain of teachers of [[Kerala school of astronomy and mathematics]]

thumb|220x220px|Pages from the [[Yuktibhāṣā|Yuktibhasa c.1530]]

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,

  • 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. The Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as:

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 Yuktibhāṣās 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:

: <math>

\begin{align}

& x + y = a,\ x - y = b,\ xy = c, x^2 + y^2 = d, \\[8pt]

& 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. 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:

Translated into the modern mathematical language of recurrence sequences:

: for ,

with initial values

:.

The first few terms are 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88,... .

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":

<blockquote>[Their work] takes on board some of the objections raised about the classical Eurocentric trajectory. The awareness [of Indian and Arabic mathematics] is all too likely to be tempered with dismissive rejections of their importance compared to Greek mathematics. The contributions from other 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"</blockquote>

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. 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.