Mathematician BRAHMAGUPTA

1. BRAHMAGUPTABRAHMAGUPTA

2.  Brahmagupta is one of the mostBrahmagupta is one of the most
distinguished Mathematician anddistinguished Mathematician and
Astronomer in the 7th century. He was theAstronomer in the 7th century. He was the
son of Vishnu Gupta and was born inson of Vishnu Gupta and was born in
Punjab. He lived in Ujjain and worked inPunjab. He lived in Ujjain and worked in
great astrological laboratory at Ujjain. Hegreat astrological laboratory at Ujjain. He
wrote his first book ‘wrote his first book ‘Brahm-sp-huta-Brahm-sp-huta-
sidhantasidhanta’ or ‘’ or ‘BrahmasidhantaBrahmasidhanta’ at this’ at this
place at the age ofplace at the age of 3030. It consists of. It consists of 2121
chapter and contains great knowledge onchapter and contains great knowledge on
arithmetic, geometry, algebra andarithmetic, geometry, algebra and
astronomy.astronomy.

3.  He gaveHe gave 22/722/7 as value ofas value of ππ and suggestedand suggested 33
as a practical value. In chapter an arithmetic,as a practical value. In chapter an arithmetic,
he has given a detailed account of progression,he has given a detailed account of progression,
areas of triangles and quadrilaterals, volumesareas of triangles and quadrilaterals, volumes
of trenches and slopes and amount of grains, inof trenches and slopes and amount of grains, in
heaps etc. He also invented four differentheaps etc. He also invented four different
methods of multiplication, namelymethods of multiplication, namely
1.1. Gan MutrikaGan Mutrika
2.2. KhandaKhanda
3.3. BhedaBheda
4.4. IstaIsta

4.  He explained the method of inversion forHe explained the method of inversion for
the first time in the following way:the first time in the following way:
““Beginning from the end, make theBeginning from the end, make the
multiplier divisor, the divisor multiplier,multiplier divisor, the divisor multiplier,
make addition subtraction and subtractionmake addition subtraction and subtraction
addition, make square, square-root andaddition, make square, square-root and
square-root. This gives the requiredsquare-root. This gives the required
quantity.”quantity.”

5.  He gave the method of squaring, cubing,He gave the method of squaring, cubing,
extracting square root as well as cubeextracting square root as well as cube
root. Also he gave the exact concept ofroot. Also he gave the exact concept of
zerozero. He defined it as. He defined it as a-aa-a==0.0. He gaveHe gave
the following rules to deal withthe following rules to deal with negativenegative
numbersnumbers,,
1.1. Negative multiplied or divided byNegative multiplied or divided by
negative becomes positive.negative becomes positive.
2.2. Negative subtracted from zero isNegative subtracted from zero is
also positive.also positive.

6.  He solved the equationHe solved the equation xx22
-10x-10x==-9-9 by aby a
rule which is equivalent to the quadraticrule which is equivalent to the quadratic
formula.formula. He multiplied the constantHe multiplied the constant
term by the coefficient of xterm by the coefficient of x22
, added, added
the square of half the coefficient ofthe square of half the coefficient of
x and found the square root of thisx and found the square root of this
sum. He then subtracted half thesum. He then subtracted half the
coefficient of x and divided it by thecoefficient of x and divided it by the
coefficient of xcoefficient of x22
. The quotient gave. The quotient gave
the solution of the equation.the solution of the equation.

7.  His works on arithmetic includesHis works on arithmetic includes
integer, fractions, progressions, barter,integer, fractions, progressions, barter,
simple interest, the mensuration of planesimple interest, the mensuration of plane
figures and problems on volumes.figures and problems on volumes.
 He found the formula for addition ofHe found the formula for addition of
geometrical progression,geometrical progression,
a+ar+ara+ar+ar22
+…….n terms+…….n terms == a(ra(rn-1n-1
)/r-1)/r-1

8.  In the field of geometry, he elaboratedIn the field of geometry, he elaborated
upon the properties of right angledupon the properties of right angled
triangles and for the first time gave thetriangles and for the first time gave the
solution of a right angled triangle by givingsolution of a right angled triangle by giving
the following value of its sides;the following value of its sides;
aa==2mn; b2mn; b==mm22
-n-n22
; c; c==mm22
+n+n22
andand
aa==mm1/21/2
;;
hh==(m/n-n)/2; c(m/n-n)/2; c==(m/n+n)/2 where(m/n+n)/2 where
m and n are two in equal integers.m and n are two in equal integers.

9.  He also gave, for the first time, suggestions for theHe also gave, for the first time, suggestions for the
construction of a cyclic quadrilateral having its sides asconstruction of a cyclic quadrilateral having its sides as
rational numbers. Two of the following formulae given byrational numbers. Two of the following formulae given by
him are in use even at present,him are in use even at present,
1.1. AreaArea of a cyclic quadrilateral havingof a cyclic quadrilateral having a, b, ca, b, c andand dd as itsas its
2.2. sidessides is equal tois equal to √√(s-a)(s-b)(s-c)(s-d)(s-a)(s-b)(s-c)(s-d) wherewhere
a+b+c+da+b+c+d==2s, s2s, s isis perimeter of the quadrilateralperimeter of the quadrilateral ..
3.3. Length of one of the diagonalsLength of one of the diagonals of the cyclicof the cyclic
quadrilateral is equal toquadrilateral is equal to (bc+ad/ab+cd)(ac+bd)(bc+ad/ab+cd)(ac+bd)
4.4. Length of the other diagonalLength of the other diagonal is equal tois equal to
(ab+cd/bc+ad)(ac+bd)(ab+cd/bc+ad)(ac+bd)

Brahmagupta was the first Indian writer, whoBrahmagupta was the first Indian writer, who
applied algebra to astronomy. He was a greatapplied algebra to astronomy. He was a great
mathematician, an astronomer and a poet.mathematician, an astronomer and a poet.

10. THANKYOUTHANKYOU