Brahmagupta (c. 598 - c. 668) was an Indian mathematician and astronomer whose work marks a decisive stage in the history of number. [1] In his treatise Brahmasphutasiddhanta, composed in 628, he gave the earliest known systematic rules for calculating with zero and with negative numbers, along with methods for solving quadratic and indeterminate equations. [2] He directed the astronomical observatory at Ujjain, then the leading centre of Indian mathematical astronomy.
Life
Brahmagupta was born around 598, according to his own statement in Bhillamala, identified with modern Bhinmal in Rajasthan, then the capital of the Gurjara kingdom. [1] Little is recorded of his personal life. He worked in the tradition of Aryabhata and other earlier Indian astronomers, and became head of the astronomical observatory at Ujjain, the foremost such institution in India. He is thought to have died around 668.
Brahmasphutasiddhanta
In 628, at about the age of thirty, Brahmagupta completed the Brahmasphutasiddhanta, or "Correctly Established Doctrine of Brahma", a work of twenty-five chapters written in Sanskrit verse. [2] Most of it concerns astronomy, but two chapters are devoted to mathematics and constitute its most influential portion.
The treatise contains the first known statement of zero as a number in its own right rather than merely a placeholder, together with rules for its arithmetic: a number subtracted from itself gives zero, and zero added to a number leaves it unchanged. [3] Brahmagupta also formulated the rules of sign for positive and negative quantities, which he described as fortunes and debts, establishing that the product of two negatives is positive. His treatment of division by zero was incomplete and was corrected only much later.
Algebra and geometry
Brahmagupta gave a general solution to the quadratic equation, admitting negative roots, and worked extensively on linear and quadratic indeterminate equations, including the equation later known in Europe as Pell's equation. [2] His method of composition for such problems anticipated results obtained in Europe more than a thousand years afterwards.
In geometry he is known for Brahmagupta's formula, which gives the area of a cyclic quadrilateral in terms of its sides, generalising Heron's formula for the triangle, and for a theorem on the diagonals of such quadrilaterals. [1] He also produced accurate approximations for pi and interpolation methods for computing sines.
Astronomy
The astronomical chapters treat planetary longitudes, diurnal rotation, eclipses of the sun and moon, risings and settings, lunar crescents and planetary conjunctions. [3] Brahmagupta rejected the notion that the earth rotates, disagreeing with Aryabhata, but gave a reasoned estimate of the earth's circumference and explained eclipses by the shadow mechanism rather than by mythological cause. He composed a second astronomical work, the Khandakhadyaka, in 665.
Transmission and legacy
In the eighth century the Brahmasphutasiddhanta was carried to Baghdad and translated into Arabic under the title Sindhind, becoming a principal channel by which Indian arithmetic, including the decimal place-value system and zero, entered the Islamic world and from there medieval Europe. [2] Brahmagupta's rules for zero and negative numbers are among the foundational statements in the history of algebra. [1]
