Brahmagupta

Brahmagupta

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  BRAHMAGUPTABRAHMAGUPTA
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   Brahmagupta isone 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.
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   He gaveHegave 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
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   He explainedthe 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.”
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   He gavethe 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.
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   He solvedthe 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.
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   His workson 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
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   In thefield 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.
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   He alsogave, 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.
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