**Indian mathematics** emerged in the Indian subcontinent^{[1]} from 1200 BCE ^{[2]} until the end of the 18th century. In the classical period of Indian mathematics (400 AD to 1200 AD), important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today^{[3]} and the binary number system^{[4]} were first recorded in Indian mathematics.^{[5]} Indian mathematicians made early contributions to the study of the concept of zero as a number,^{[6]} negative numbers,^{[7]} arithmetic, and algebra.^{[8]} In addition, trigonometry^{[9]}
was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there.^{[10]} These mathematical concepts were transmitted to the Middle East, China, and Europe^{[8]} 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 as important as the ideas involved.^{[1]}^{[11]} 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.^{[12]}^{[13]}

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 remarkable work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series (apart from geometric series).^{[14]} However, they did not formulate a systematic theory of differentiation and integration, nor is there any *direct* evidence of their results being transmitted outside Kerala.^{[15]}^{[16]}^{[17]}^{[18]}

Some of the areas of mathematics studied in ancient and medieval India include the following:

- Arithmetic: Decimal system, Negative numbers (see Brahmagupta), Zero (see Hindu-Arabic numeral system), Binary numeral system, the modern positional notation numeral system, Floating point numbers (see Kerala school of astronomy and mathematics), Number theory, Infinity (see Yajur Veda), Transfinite numbers, Irrational numbers (see Shulba Sutras)
- Geometry: Square roots (see Bakhshali approximation), Cube roots (see Mahavira), Pythagorean triples (see Sulba Sutras; Baudhayana and Apastamba state the Pythagorean theorem without proof), Transformation (see Panini), Pascal's triangle (see Pingala)
- Algebra: Quadratic equations (see Sulba Sutras, Aryabhata, and Brahmagupta), Cubic equations and Quartic equations (biquadratic equations) (see Mahavira and Bhāskara II)
- Mathematical logic: Formal grammars, formal language theory, the Panini–Backus form (see Panini), Recursion (see Panini)
- General mathematics: Fibonacci numbers (see Pingala), Earliest forms of Morse code (see Pingala), Logarithms, indices (see Jaina mathematics), Algorithms, Algorism (see Aryabhata and Brahmagupta)
- Trigonometry: Trigonometric functions (see Surya Siddhanta and Aryabhata), Trigonometric series (see Madhava and Kerala school)

Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley Civilization have uncovered evidence of the use of "practical mathematics". The people of the IVC manufactured bricks whose dimensions were in the proportion 4:2:1, considered favorable for the stability of a brick structure. They used a standardized system of weights based on the ratios: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, with the unit weight equaling approximately 28 grams (and approximately equal to the English ounce or Greek uncia). They mass produced weights in regular geometrical shapes, which included hexahedra, barrels, cones, and cylinders, thereby demonstrating knowledge of basic geometry.^{[19]}

The inhabitants of Indus civilization also tried to standardize 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.^{[20]}^{[21]}