# Brahmagupta

The era of Greek mathematics had extended over roughly 900 years. These so called ‘Greek’ mathematicians did not all live in Greece and were not all Greek. Many of them lived in Alexandria in northern Egypt. But they were all influenced by Greek philosophy and were part of a line that stretched from early mathematicians, such as Thales of Miletus (around 585 BC), to the last of the great mathematicians, Pappus, who lived at the end of the third century and the beginning of the fourth century AD.

After Pappus there were others of lesser importance, but the Greek era really came to an end in 641 AD when the Library at Alexandria was burnt down by the Arabs. After that there were no other mathematicians of note in the western world for nearly 1,000 years – the so called ‘Dark Ages’. Instead the centre of the mathematical world shifted east to India and Arab countries.

During this period there were several prominent Hindu mathematicians, including two named Aryabhata, Brahmagupta, Mahavira and Bhaskara. Of these, Brahmagupta is considered the most influential.

Brahmagupta was born in 598 AD in Bhinmal city in the state of Rajasthan in northwest India, but worked most of his life at the astronomical centre of Ujjain in central India. Like most of the mathematicians of that period, Brahmagupta’s discoveries were related to astronomy. Brahmagupta’s famous work on mathematics and astronomy, written in 628 AD, is named *Brahmasphutasiddhanta* *,* which means ‘The system of the god of creation in astronomy’. He also wrote three other texts of note: the *Cadamekela* , the *Khandakhadyaka* and the *Durkeamynarda* . These works were written in verse form and contained no proofs.

The *Brahmasphutasiddhanta* contains a collection of rules for arithmetic, including rules we take for granted today such as:

- a positive number multiplied by a positive number is positive.
- a positive number multiplied by a negative number is negative.
- a negative number multiplied by a positive number is negative.
- a negative number multiplied by a negative number is positive.

It also contains some incorrect statements, such as zero divided by zero is zero.

The work also contains some algebra and geometry. The arithmetic and algebra are entirely rhetorical i.e. solutions to problems are given using only words, with no abbreviations or symbols. He works out the rules for arithmetic sequences, and shows how to solve quadratic equations with real roots. He solves simultaneous equations, gives the formulae for the sum of the squares and cubes of the first *n* integers and provides an algorithm for computing square roots.

In geometry, Brahmagupta stated the Pythagorean Theorem for a right angled triangle:

He gave the formulae for the area of a triangle and the area of a cyclic quadrilateral in terms of the sides:

He also contributed to trigonometry, giving a table of values of the sine function.

Much of the *Brahmasphutasiddhanta* is about astronomy and deals with eclipses of the sun and the moon, planetary conjunctions and the positions of the planets.

Brahmagupta died in 670 AD. His work was also an influence on the great Arabian mathematician Mohammed ibn Musa al-Khowarizmi.

**A contribution of Brahmagupta to mathematics**

But t could be replaced by any other number, so Brahmagupta has given a **general** solution to the problem.

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