The corpus, every claim citable.

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

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

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

Brahmagupta's bhāvanā lemma (Brahmasphutasiddhanta XVIII.64-65, 628 CE; rendered §76-77 in Colebrooke 1817): if (x₁, y₁) solves x²N + k₁ = y² and (x₂, y₂) solves x²N + k₂ = y², then their cross-product (x₁y₂ ± x₂y₁, y₁y₂ ± Nx₁x₂) solves x²N + (k₁k₂) = y². The mathematical foundation Bhāskara II's chakravala (1150 CE) stands on. Lagrange's 1768 European solution to "Pell's equation" rediscovers the same composition + iteration structure.

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