Mathematics

The Tantrasaṅgraha of Nīlakaṇṭha Somayājī (Kerala, c. 1500 CE) states the alternating series π/4 = 1 − 1/3 + 1/5 − 1/7 + … as a verse rule for the circumference of a circle of given diameter, together with a rational end-correction that sharply accelerates convergence. The Kerala school's commentaries attribute the series to Mādhava (c. 1340–1425). Leibniz published the same series in Europe in 1673; Charles Whish first reported the Kerala texts to European scholarship in 1834.

On the Hindu Quadrature of the Circle, and the Infinite Series of the Proportion of the Circumference to the Diameter Exhibited in the Four Sastras · 1500

Bhāskara II's Bījagaṇita (1150 CE) defines the quotient 3/0 as khahara — "termed an infinite quantity" — and states it is unaltered by adding or subtracting finite amounts. A deliberate algebraic definition of division by zero, five centuries before Europe acquired a working infinity symbol (Wallis, 1655). His wider system still misfires — elsewhere a quantity multiplied then divided by zero recovers its finite value — but the definition itself is the first of its kind.

Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara · 628

Mahāvīra's Ganita-sara-sangraha I.52 (850 CE) gives the first explicit recognition that √(negative) is undefined on reals. The reasoning: every square is non-negative, so no real number squared yields a negative. Cardano (1545) calls these roots "fictitious"; Bombelli (1572) treats them as imaginary; Gauss (1799) gives the complex-number foundation. The impossibility acknowledgement is Mahāvīra's, ~700 years before Cardano.

The Ganita-sara-sangraha of Mahaviracarya · 850

Mahāvīra's Ganita-sara-sangraha VI.218 (850 CE) gives the general algorithmic statement of the nCr formula: write 1..n ascending and n..1 descending in two rows; the product of the top r entries divided by the product of the bottom r is nCr. Pingala (~200 BCE) had the binomial-prosody special case; Pascal's Traité (1654 CE) gives the European systematic form. Mahāvīra's algorithm is the general procedural statement, 800 years before Pascal.

The Ganita-sara-sangraha of Mahaviracarya · 850

Aryabhatiya II.32-33 (499 CE) gives the kuttaka algorithm for solving the linear indeterminate equation ax + by = c in integers, via reciprocal (Euclidean) division of a and b, then working the quotient chain backwards. Same algorithm later named "Chinese Remainder Theorem" via Qin Jiushao (1247 CE) and powers modern RSA key recovery (1977). Aryabhata's motivation was astronomical: computing when planets would all return to a given longitude.

The Aryabhatiya of Aryabhata · 499