In the Brahmasphutasiddhanta (Opening of the Universe, also seen as Brahmasiddhanta) of 628, Brahmagupta introduces zero and negative numbers. It is not know whether he pioneered the concepts or whether they were known in Indian mathematics before that. This is the first major advance on Greek mathematics.

He called positive numbers fortunes and negative numbers debts, defined them and zero, and introduced rules of arithmetic for them all, correct apart from an attempt to define division by zero: he set 0/0 = 0.

Brahmagupta was head of the leading astronomical observatory of India, at Ujjain, which might have been his birthplace (in 598).

The Brahmasphutasiddhanta is in 25 chapters. The first ten discuss "mean longitudes of the planets; true longitudes of the planets; the three problems of diurnal rotation; lunar eclipses; solar eclipses; risings and settings; the moon's crescent; the moon's shadow; conjunctions of the planets with each other; and conjunctions of the planets with the fixed stars." The remaining 15 are additions and summaries plus more on gnomons and other instruments.

At the age of 67, in 665, he wrote the Khandakhadyaka, a work in eight chapters covering similar astronomical topics.

Brahmagupta (588-670), son of Jisnugupta

Brahmagupta also introduced the formula which bears his name:

Brahmagupta's formula

Given a cyclic quadrilateral (one which can be inscribed in a circle) with side lengths a,b,c,d, the area of the quadrilateral is ((s-a)(s-b)(s-c)(s-d))1/2, where s = (a+b+c+d)/2

The proof of this rather elegant statement is quite hideous and involves lots of trigonometry and messy algebra. It is quite probable that Brahmagupta did not prove it, but rather took it as a generalisation of Heron's formula. It is quite easy to see that Heron's formula follows, simply by considering the case where d=0.

Other notable achievements of Brahmagupta are rules for the summing of various series (first n squares, first n cubes). He developed a method of long multiplication very similar to that in common use today and found quite accurate methods for approximating square roots and sines.

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