Brahma Gupta- a great Indian mathematician

Dr N K Srinivasan

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. But

,in this article, we focus on his mathematical

contributions.

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 x0 =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

neagtive. Thus square root of 9 is either +3 or -3.

He is well known in the western world for two things with

quadrilateral: 'Brahma Gupta Formula' and 'Brahma Gupta

Theorem.'

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.

Brahma Gupta Theorem

For a cyclic quadrilateral with diagonals perpendicular to

each other, the altitude from a side bisects the opposite

side.[See other websites for this theorem...I am unable to

reproduce the figures due to copyright issues.]

3 Pell's equations

Brahma Gupta's greatest contribution was in solving Pell's

equation of the form:

x2 -N y2 = 1

[The equation was named after John Pell by Euler much later.

Earlier work of this nature is the familiar Diophantine equations,

due to Diophantus. ]

Here N is a non square integer, say 2 or 3 or 7 and

we seek solutions where x and y are integers. {Such

problems are called 'integer solutions for non-linear

equations'.]

What is the motivation for B G to explore this? --Most

probably to find square roots of 2,3,5,7 and so on. The

square root of 2 has fascinated mathematicians since

Pythagoras, who was puzzled by its irrational root.

Let N=2 .Then x/y is a good approximation for square root

of 2.

How to solve this equation? We shall not give the complete

method but indicate the solution of a similar equation

,the first example given by BG in his book: "Brahma Sphuta

Siddhanta".

Taking N = 92, the general equation becomes :

x2 - 92 y2 = 8

A solution is : x=10 y= 1

This is written as a triple: {10,1,8} for convenience as

solution set for N = 92.

BG gave a method of 'composing' other solutions or triples

. Using this method, with the same triple of {10,1,8}, we

get another solution set: {192,20,64}

Going further, we can get {1151,120,1} which is another

solution set for the equation: x2 - 92 y2 = 1.

Note that x/y = 1151/120 = 9.592 which is a good approximation for

sqrt(920.[ sqrt(92) = 9.59166]

Returning to the question of finding the square root of 2 by this

method---via Pell's equation-- we find the triple:{17,12,1} for N

= 2.

17/12=1.4166 which is a good approximation for sqrt(2) = 1.4142.

[The error is 2 parts in 1400 or 1/7 %!]

Another solution is x= 577, y= 408 x/y = 1.4142156.

{ Archimedes found sqrt(3) by a similar method:x=1351, y= 780, x/y

= 1.73205 sqrt(3) = 1.7320508

Several centuries later, around 1150 CE, Bhaskara II extended

this method using a cyclic [chakravala] process for solving Pell's

equation.

Brahma Gupta's work has been a significant contribution to number

theory and extended into many fields.

{For this work of Brahma Gupta, refer: John Stillwell,"

Mathematics and its history " Springer verlag, 2002.}

4 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.]

5 Interpolation Method

Brahma Gupta is well known for his improvement of sine

tables.The Indian astronomers were pioneers in developing

the sine tables for trig work ,mainly for use in

astronomy. BG introduced the second order interpolation .

This is similar to second order [quadratic] approximation

in Newton's finite difference interpolation formula or

Stirling's interpolation formula.

Consider an equation y= f(x).

The values of y are known for evenly spaced x values ,such

as x1, x2, x3...and so on.

Let d= x2-x1 = x3-x2=.....

Then first order difference formula for any x value

between x2 and x3 :

f(x) = f(x2) + [f(x3)-f(x2)/d](x-x2)

For second order interpolation we take the following

difference : [ f(x3)-f(x2) + (f(x2) - f(x1)}/2 ]

.[(x-x2)/d]2 .

BG used the interpolation method to get better sine values

for his tables.

It should be noted that the works of Newton ,Gregory and

Wallis were done several centuries later.It is

acknowledged now that Brahma Gupta is the inventor of

second order interpolation method.

[Bhaskara I belonged to 7th century [600-680 CE] and gave

the formula for sin x as an rational number:

sin x = [16x(pi-x)] / [ 5 pi .pi - 4x(pi-x)]

with an error of less than 1.9%. This formula was widely

used by astronomers and astrologers at that time in

India.]

BG also extended this for interpolation in cases where the

x values are not evenly placed for the function y= f(x).

6 Sum of squares and cubes of integers

BG gave the formula for the sum of squares of integers:

s= [n(n+1)(2n+1)/6]

and for sum of cubes of integers

s' = [n(n+1)/2]2

[These formulae were , perhaps,known to Greeks.]

7 Brahma Gupta and other Indian mathematicians were using

Pythagorian theorem . BG did much work on right triangles

which was attributed later to Fibonacci of 13th century

and Francoise Vieta of 16th Century.

8 Finding square roots--

Brahma Gupta is credited with developing an algorithm

similar to Newton-Raphson iteration method. I am unable to

get the details so far.

8 Quadratic Equation

Brahma Gupta is the one who gave the solutions of

quadratic equation as we use today. This became easy for

him because he had already a good procedure for finding

square roots and also could handle negative numbers.Did he

know how to use imaginary quantity and complex roots?--

perhaps not.

After Brahma Gupta

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.

The Chinese also translated his work in the seventh

century.Colebrooke translated BG's work into English in

1817.

Brahma Gupta died in the year 668 CE.

Summary

To sum up, we can state that the contributions of

Brahma Gupta may help us to revise our opinions about

European math of later centuries , derived as much from

Indian contributions as from Greek or Chinese sources.

It is also significant to note that there was a healthy

exchange of mathematical knowledge between India, Arab

nations [especially Baghdad school]and China around 7th

century. There was much travel due to trade routes , both

sea and land, connecting India, China and Arab nations, in

5th century onwards.

Further much credit goes to Arab mathematicians for

introducing the Indian math to Middle East and Europe,

especially to Spain and Italy at that time.Indian math

books were translated into Arabic and later into Latin in

Spain.

During Renaissance time, it was Italy and the Vatican

which sent Christian missionaries to India; some of them

took interest in mathematical and astronomical studies of

India and translated the "siddhanta" texts.

References

1 John Stillwell--- Mathematics and its history

--Springer,3rd Edition 2008.

2 Victor Katz -- A history of mathematics--Pearson-addison

-- 2004

3 Victor Katz -- The mathematics of Egypt,----,a Source

Book,..Princeton U Press, 2007

4 Numerous web pages in the Internet.