The corpus, every claim citable.

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

Bhaskara II's Bijaganita (1150 CE) gives a complete cyclic algorithm (chakravala) for solving Nx² + 1 = y², the equation Europeans would later call "Pell's equation" after a 17th-c. misattribution. Applied to N=61, the algorithm yields x=226,153,980 and y=1,766,319,049 — the smallest positive-integer solution. Fermat proposed exactly this case to Frenicle and Wallis in 1657 as a challenge problem; Lagrange was the first European to find a general method in 1768, six centuries after Bhaskara.

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

Aryabhata gives π ≈ 62832/20000 = 3.1416 in Aryabhatiya II.10 (the Ganitapada). Crucially, the Sanskrit word for "approximately" he uses is *āsanna* — "near, approaching but not reaching." This is the earliest explicit acknowledgement in any tradition that π is an irrational constant that can only be approximated, predating Lambert's 1761 formal proof by ~1,262 years.

The Aryabhatiya of Aryabhata · 499

The decimal place-value system originated in India and spread in documented stages: epigraphic evidence (including a 605 CE Cambodian inscription) shows coverage "roughly of the size of Europe" by the end of the 6th century; Arab mathematicians adopted it in the 8th century; common European use came only around the 16th — popular almanacs of 1557–96 still print Roman numerals. Datta & Singh (1938) document each stage from primary evidence.

History of Hindu Mathematics — A Source Book · 1938

The Yajurveda Saṁhitā (Vājasaneyī xvii.2, c. 1200–800 BCE) lists thirteen decimal denominations — eka (1) through parārdha (10¹²) — each ten times the preceding; the same list recurs in the Taittirīya Saṁhitā. Datta & Singh (1938) contrast this with Greek terminology, which stopped at the myriad (10⁴), and Roman, at mille (10³). Named decuple ranks are a documented Vedic-era feature of Sanskrit, many centuries before written place-value numerals.

History of Hindu Mathematics — A Source Book · 1938

Pingala's Chandahsutra (~200 BCE) gives a four-aphorism recursive algorithm for counting metrical arrangements of n syllables. The rules ("halve; subtract one when odd; multiply by two; square when halved") implement exponentiation-by-squaring — the same recurrence modern computers use to compute 2ⁿ in O(log n) steps. Halayudha's 10th-century commentary makes the recursion explicit. The algorithm predates Leibniz's binary arithmetic (1703) by ~1,900 years.

History of Hindu Mathematics — A Source Book · 1938

Brahmasphutasiddhanta XVIII.22 (628 CE) gives the four sign-multiplication rules: (−)·(+) = (−), (−)·(−) = (+), (+)·(+) = (+), with the parallel zero rules. The hardest case — negative times negative — was hedged in European mathematics for centuries: Cardano (1545) calls negative roots "fictitious"; Wallis (1685) is the first to argue geometrically for the rule; Hankel (1867) gives the first formal construction. Brahmagupta states all four rules without apology in 628 CE.

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

Brahmasphutasiddhanta XII.21 (628 CE) gives the exact-area formula for any cyclic quadrilateral with sides a, b, c, d: K = √[(s−a)(s−b) (s−c)(s−d)], where s = (a+b+c+d)/2. First known generalization of Heron's triangle area formula to four sides. Rediscovered in Europe by Carl Strehlke in 1842 — 1,214 years later. Bhaskara II preserved the rule in Lilavati §167 (1150 CE), source of the verbatim quote.

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

Brahmasphutasiddhanta XVIII.19 (628 CE) gives explicit rules for arithmetic with zero as a number: addition, subtraction, multiplication, square root. Brahmagupta also writes a rule for division by zero — getting it wrong (treats x/0 as finite) but pioneering the question itself. Bhaskara II refined the rule ~500 years later (1150 CE) treating 1/0 as khahara — closer to modern infinity-as-limit.

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

The Lalitavistara, a canonical Buddhist life of the Buddha, stages a counting contest: the examiner Arjuna asks the Bodhisattva to count beyond a koti (10⁷), and he recites a centesimal ladder — each name a hundred times the last — through ayuta, niyuta, kaṅkara and onward to tallakṣaṇa = 10⁵³, with further series beyond. High-magnitude number-naming was so culturally prized that a religious biography made it proof of the hero's perfection. Datta & Singh (1938) translate the dialogue.

History of Hindu Mathematics — A Source Book · 1938

The Vedāṅga Jyotiṣa — the calendar manual among the six Vedāṅgas, late Vedic period — declares gaṇita (mathematics/computation) the highest of the auxiliary sciences: "As the crests on the heads of peacocks, as the gems on the hoods of snakes, so is gaṇita at the top of the sciences known as the Vedāṅga" (trans. Datta & Singh 1938). It is the earliest known text to rank mathematics supreme among the sciences — a cultural charter the Sanskrit mathematical tradition cited for centuries.

History of Hindu Mathematics — A Source Book · 1938

Mahāvīra's Gaṇita-sāra-saṅgraha (850 CE) poses the hundred-birds problem: pigeons at 5 for 3 paṇas, sārasas at 7 for 5, swans at 9 for 7, peacocks at 3 for 9 — buy 100 birds for 100 paṇas: two equations, four unknowns, integer solutions required. The same problem-type appears in Zhang Qiujian's Chinese classic (c. 475 CE, priority) and Alcuin's Latin puzzles (c. 800) — a marker problem for the circulation of mathematics across Eurasia.

History of Hindu Mathematics — A Source Book · 1938

Bhāskara II's Bījagaṇita §139 (1150 CE, Colebrooke trans.) poses (x/8)² + 12 = x and derives both roots, x = 48 and x = 16, noting both satisfy the conditions. Companion problems show the other case: where a root is negative or otherwise inconsistent with the problem, it is declared incongruous and dropped — "people do not approve a negative absolute number." Two-root awareness plus root-validity screening, standard curriculum in 1150; Descartes was still calling negative roots "false" in 1637.

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

The Sadratnamālā of Śaṅkara Varman (Kerala, 1819) gives the circumference of a circle of diameter 10¹⁷ parts as 314,159,265,358,979,324 — π correct to seventeen figures — encoded in one verse via the kaṭapayādi consonant-to-digit cipher and computed with the Kerala school's series methods. Whish reported it to the Royal Asiatic Society in 1834. Europe held longer digit records by then; the claim is the encoding and the unbroken lineage, not the record.

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

Classical Sanskrit mathematics and astronomy used bhūta-saṁkhyā ("object numerals"): numbers written as words — candra (moon) = 1, netra (eyes) = 2, agni (fires) = 3, sāgara (oceans) = 4 — arranged by place value. Datta & Singh (1938) document the system's rationale: scientific works were metrical, and word numerals with many synonyms per digit let any number be versified. One number could be written hundreds of ways; the convention remains in use for numbers in Sanskrit verse.

History of Hindu Mathematics — A Source Book · 1938

The Līlāvatī of Bhāskara II (1150 CE) is an arithmetic and geometry textbook composed in Sanskrit verse, with word problems addressed to a woman — by tradition Bhāskara's daughter Līlāvatī. Colebrooke §54: a swarm of bees splits into fifths and thirds among named flowers, one bee hovers between a jasmine and a pandanus; find the swarm. The book stayed the subcontinent's standard mathematics text for roughly 700 years and was translated into Persian at Akbar's court (Fyzī, 1587).

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