BRAHMA GUPTA - GREAT INDIAN MATHEMATICIAN

Brahma Gupta was a great Indian mathematician of 7th century C E. He was born in a village called Billamalla in Rajasthan in the year 598 CE. He moved to Ujjain in central India which had a famous school of mathematics. In that school, his predecessors were the famous astronomers Varahamihira and Aryabhata.

In those days, astronomers did lot of mathematical work too and the distinction between astronomy and mathematics was not clear cut. Brahma Gupta was also known as a great astronomer and became head of the Ujjain observatory. 

1  Number Theory

Perhaps his greatest contribution was in number theory. He developed the use of zero with precise equations: if n is a number ,then 0 + n = 0, 0 x 0 =0 and so on.

He also enunciated that negative numbers could be used in what we now call "algebra". He found out that while taking the square root we get two roots--one positive and one negative. Thus square root of 9 is either +3 or -3.

2  Brahma Gupta Formula

For a cyclic quadrilateral --- that is a quadrilateral inscribed in a circle ---- the area A is given by :

A = square root( s-a)(s-b)(s-c)(s-d)

where a, b,c,and d are sides of the quadrilateral and s is the semi-perimeter: s= (a+b+c+d)/2

[This can be extended to non-cyclic quadrilateral also;] This reminds us of the formula for area of a triangle given by Heron [of Alexandria].

Note that BG's formula reduces to Heron's formula when d goes to zero:

Heron's formula:   Area = sqrt( s (s-a)(s-ab)(s-a))

It is a moot point whether BG wanted to extend Heron's formula or derived this independently. It is quite possible that he was aware of Heron's formula.

3  Approximation to Pi

Almost all astronomers and mathematicians have been fascinated by the irrational number pi and had approximated it in several ways. They needed the value of pi for many computations. 

Egyptians used the ratio of 256/81 = 3.1605 as pi for all calculations. 256/81 = (4x4x4x4)/(3x3x3x3)

Early Greeks used pi = 3 or  following Archimedes work, pi= 3 + 10/71 or simply, pi = 3 +1/7 or pi = 22/7 --  a ratio   often used by school students even today.

Brahma Gupta approximated pi to square root of 10 which is 3.16 . This is close to 3.14159 and was perhaps convenient to use in astronomical calculations.

{Bhaskara II used the ratio of 355/113 for pi,yielding 3.14159.]

Brahma Gupta's work was translated by Arab mathematicians into Arabic and became part of Arab math in their schools,especially the one that developed in Baghdad. 

The book "Sindhind" contained his works on number theory for Arab mathematicians. The noted    mathematician al-Khwarizmi wrote his book of Algebra in 830 CE, including BG's works. 

By 12th century, the work of Brahma Gupta was widely known in Europe.It was at this time Bhaskara wrote his further work    on Pell's equation--around 1150 CE.

It should be noted that BG's works, like works of other Indian mathematicians, were written in verse form in Sanskrit language .

Like other Indian astronomer-mathematicians,much of his work might have been motivated by astronomical problems.

BG wrote two books, the second one at the age of 69.