Brahmagupta – The Genius of Mathematics

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Born

Brahmagupta 123
Brahmagupta

Brahmagupta, son of an astrologer named Jisnugupta, was born in 598 A.D perhaps in bhinmal (Old name was Bhillamala). However, there is no conclusive evidence available relating to the birth place of Brahmagupta. Now Bhinmal is a town in the Jalore District of RajasthanIndia but at that time it was the apparent capital of the Gurjaradesa, the second largest kingdom of Western India, comprising the southern Rajasthan and northern Gujarat in modern-day India. It was also a center of learning for mathematics and astronomy. He was a Shaivite by religion. Shaivism is one of the major branches of Hinduism and it is devoted to worship of the god Shiva.

Education & Career

Brahmagupta became an astronomer of the Brahmapaksha school (one of the four major schools of Indian astronomy during that period). He studied the five traditional siddhanthas on Indian astronomy as well as the work of other astronomers including Aryabhata I, Latadeva, PradyumnaVarahamihira, Simha, Srisena, Vijayanandin and Vishnuchandra.

At the age of 30, he composed Brahma-sphuta-siddhanta (Correctly Established Doctrine of Brahma) in 628 AD having 25 chapters while living in Bhinmal.  The first ten chapters cover the topics of mean longitudestrue longitudes of the planets, the three problems of diurnal rotation, lunar eclipses & solar eclipses, risings and settings; the moon’s crescent; the moon’s shadow, conjunctions of the planets with each other and conjunctions of the planets with the fixed stars.

The remaining fifteen chapters contain key chapters on mathematics including algebra, geometry, trigonometry and algorithmic. Τhe text is notable for its mathematical content as it contains rules regarding zero, negative and positive numbers, rules for computing square roots, methods of solving linear and quadratic equations, rules for summing series, Brahmagupta’s identity, Brahmagupta’s formula, Brahmagupta’s matrix and Brahmagupta’s theorem.

Later, Brahmagupta moved to Ujjain as it was also a major centre for astronomy in India. At the mature age of 67, he composed his next well known work Khanda-khadyaka. The other books he wrote “Durkeamynarda” and “Cadamakela”.

Known For

Brahmagupta was known mostly through his works, which cover mathematical and astronomical topics and significantly combine the two. He did tremendous work in several streams of  mathematics and astronomy. Here is the list of his most famous works;

Arithmetic

Indian arithmetic was well known in Medieval Europe as “Modus Indoram” meaning method of the Indians. In Brahma-sphuta-siddhanta, Multiplication was named Gomutrika. In the beginning of chapter 12 of Brahma-sphuta-siddhanta, entitled Calculation, Brahmagupta explained how to find the cube and cube-root of an integer and gave rules facilitating the computation of squares and square roots. He also gave rules for dealing with five types of combinations of fractions:

1) a/cb/c;  2) a/c × b/d;  3) a/1 + b/d;  4) a/c + b/d × a/c = a(d + b)/cd; and  5) a/c − b/d × a/c = a(d − b)/cd.

Series

Brahmagupta gave the sum of the squares of the first n natural numbers as n(n+1)(2n+1)/6  and the sum of the cubes of the first n natural numbers as (n(n+1)/2)2. However, no proofs has been given about how did he discover these formulae.

Number System

Brahmagupta is the first who gave concrete rules of using positive numbers, negative numbers, and zero. The Brahma-sphuta-siddhanta is one the earliest known text to treat zero as a number. In Chapter 18 of his Brahma-sphuta-siddhanta, he established the basic mathematical rules which are-

  • The sum of two positive quantities is positive [A + B =C].
  • The sum of two negative quantities is negative [-A + (-B) = – C].
  • A negative number minus zero is a negative number [-A – 0 = -A].
  • A positive number minus zero is a positive number [A – 0 = A].
  • Zero minus zero is a zero [0 – 0 = 0].
  • The sum of zero and zero is zero [0 + 0 = 0].
  • The sum of a positive and a negative is their difference [A + (-B) = A -B] or if they are equal, zero [A + (-A) = 0].
  • A negative number subtracted from zero is a positive number [0 – (-A) = A].
  • A positive number subtracted from zero is a negative number [0 – (A) = -A].
  • The product of zero by a negative number or positive number is zero [0 x –A = 0] or [0 x A = 0]
  • The product of zero by zero is zero [0 x 0 = 0].
  • The product of two positive numbers is a positive number [A x B = C].
  • The product of two negative numbers is a positive number [-A x -B = C].
  • The product of a negative number and a positive number is a negative number [-A x B = -C].
  • The product of a positive number and a negative number is a negative number [A x -B = -C]
  • Positive number divided by positive or negative by negative is positive [ A/B = C Or -A/-B = C ]
  • Positive divided by negative is negative. Negative divided by positive is negative [ A/-B = -C Or -A/B = -C ]
  • A positive or negative number when divided by zero is a fraction with the zero as denominator
  • Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator
  • Zero divided by zero is zero [ 0/0 = 0 ]

Here we can see that the Brahmagupta’s postulates relating to negative and positive numbers and zero are quite close to the modern understanding, except that in modern mathematics division by zero is left undefined.

Algebra

In chapter 18 of Brahma-sphuta-siddhanta, he developed the solution of the general linear equation of form bx + c = dx + e,

equivalent to x = e − c/b − dwhere c and e are constants.

He further gave two equivalent solutions to the general quadratic equation of form ax2 + bx = c, equivalent tofa99820c27af6a7bd392479da0b794f5

and,

b9ac955bac4f38b89aee0a8c78e36131

He went on to solve systems of simultaneous indeterminate equations stating that the desired variable must first be isolated, and then the equation must be divided by the desired variable’s coefficient. In particular, he recommended using “the pulverizer” to solve equations with multiple unknowns.

Brahmagupta’s formula

Picture1In Chapter 12 of Brahma-sphuta-siddhanta , he gave his remarkable and most famous result in geometry is his formula for cyclic quadrilaterals (one that can be inscribed in a circle). It gives the area K of a cyclic quadrilateral whose sides have lengths a, b, c, d as9abfb272dbcb743ffd64e7b590ab6295

where S, the semi-perimeter and is given as,4867241a727566e85882bd12b27938b6

 

Brahmagupta’s theorem

In Chapter 12 of Brahma-sphuta-siddhanta, his theorem states that if a cyclic quadrilateral is orthodiagonal (that is, has perpendicular diagonals), then 234px-Brahmaguptra's_theorem.svgthe perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side.

More specifically, let ABC and D be four points on a circle such that the lines AC and BD are perpendiculars. The intersection of AC and BD is M. Now, Drop the perpendicular from M to the line BC, calling the intersection E. Let be the intersection of the line EM and the edge AD. Then, the theorem states that F is the midpoint of AD or AF=FD.

Brahmagupta–Fibonacci identity

In algebra, the Brahmagupta–Fibonacci identity or simply Brahmagupta identity says that the product of two sums each of two squares is itself a sum of two squares. In other words, the set of all sums of two squares is closed under multiplication. Specifically:dc9f93109fa9ba772138edb6d34a531d

The identity was actually first found in Diophantus’ Arithmetica (III, 19), of the third century A.D. It was rediscovered by Brahmagupta, who generalized it to the Brahmagupta identity and used it in his study of what is now called Pell’s equation, namely X2 − NY2 = 1

Brahmagupta matrix

To solve the indeterminate equation, Brahmagupta  gave an iterative method of deriving new solutions from the known ones by his samasa-bhavana, the principle of composition. He gave the following matrix ,e9b14ec5dbb68aa1313e700564ed8510

which satisfies , 71a2be8c806b744bbd6c9f91bedda220

where B is called as the Brahmagupta matrix. To know more about Brahmagupta matrix and Brahmagupta polynomials follow the link; THE BRAHMAGUPTA POLYNOMIALS.

Astronomy 

In 7th Chapter of Brahma-sphuta-siddhanta entitled Lunar Crescent, Some of the important contributions made by Brahmagupta are: methods for calculating the position of heavenly bodies over time (ephemerides), positions of the planets, their rising and setting, planetary conjunctions, and the calculation of solar and lunar eclipses.

In his second work Khanda-khadyaka, he uses  the interpolation formula to compute values of sines.

Died

He is believed to have died in Ujjain between 665 to 670 A.D. He was one of the greatest mathematicians in Indian history and his contributions to mathematics and science have made significant. A writer of his own time, Bhaksara II, called him Ganita Chakra Chudamani”, which means “the gem in the circle of mathematicians.”

 

References
  1. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook by Victor J. Katz, Annette Imhausen

  2. A History of Mathematics  by Carl B. Boyer

  3. THE BRAHMAGUPTA POLYNOMIALS by E. R. Suryanarayan, Department of Mathematics, University of Rhode Island, Kingston, RI02881 (Submitted May 1994)