Thursday, February 21, 2008

Ancient Indian greats - 2 Brahmagupta (Nov 07)

Brahmagupta Theorem

Brahmagupta was an Indian astronomer and mathematician. He was the head of the astronomical observatory at Ujjain. His main, but not sole, achievements in the field of mathematics were the introduction of zero and negative numbers.

Like many Indian mathematicians of this and later periods, Brahmagupta was producing work that was many centuries ahead of the equivalent work being carried out in the western world. Brahmagupta wrote two main texts, both of which deal with arithmetic and astronomy. His first work in 628 was Brahmasphuta siddhanta (The Opening of the Universe) (catchy name, ain’t it?), and in 665 he published Khandakhadyaka. Both of these texts are actually written in verse!!! It is hard to imagine a book written on advanced mathematics without mathematical variables but totally in verse!

Undoubtedly, he must have been a very influential person. The astronomical observatory at Ujjain was the best in the world at that point of time. His school, which was a rival to that of Aryabhata, was very influential in western and northern India. He made his presence at an age when Indians were brimming with mathematicians. Brahmagupta had a plethora of criticism directed towards the work of rival astronomers. In fact he wrote a whole chapter disagreeing choice of astronomical parameters and theories put forward by other Indian mathematicians.

But let’s get back to his contribution. The fact that he introduced the concept of zero and negative numbers is probably known by all. But did you know that one of his chief contributions is the solution of a certain second order indeterminate equation which is of great significance in number theory? Or that he knew that both Earth and the universe are round? Or that he knew about gravitation force? Or that he was the first to make a sine table. This "or" sequence could actually go on for hours ... oh and yeah ... he also contributed a lot in the field of progressions. He has contributed so much that it makes me wonder if he had any Grad students working under him.

Under the Gurukul system which was prevalent then, it is very much possible. But unfortunately all his books just state the results and does not give the proof. Infact it would have been impossible for him to give the proof for every theorem that he put forward. That was because books were to be written only in

To give a glimpse of how difficult it was for him, let me quote a few lines from his books.

Diminish by the middle [number] the square-root of the rupas multiplied by four times the square and increased by the square of the middle [number]; divide the remainder by twice the square. [The result is] the middle [number].

Guessed what that was all about? Stop forcing your already-exploited brain, the above verse was about the solution to a quadratic equation. Or get a load of this: The sines: The Progenitors, twins; Ursa Major, twins, the Vedas; the gods, fires, six; flavors, dice, the gods; the moon, five, the sky, the moon the moon, arrows, suns .....

That’s sine table for you. Yes, you guessed right. He used names of objects to represent the digits of place-value numerals. The only red mark on his report card is this definition of division of zero by zero: A positive divided by a positive or a negative divided by a negative is positive; a zero divided by a zero is zero; a positive divided by a negative is negative; a negative divided by a positive is negative.

Perhaps the greatest tribute to Brahmagupta's mathematical talents is that his book was used to introduce the basic ideas of algebra to Islamic mathematicians, who were later given credit for its invention. Brahmagupta died between 660 and 670. At the time of his death he was widely acknowledged as the greatest mathematician of this period of Indian history, and one of the greatest Indian mathematicians of any time.

-V.Kranti Kumar

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