Mahāvīra (or Mahaviracharya, "Mahavira the Teacher") was a 9th-century Jainmathematician from Bihar, India.^{[1]}^{[2]}^{[3]} He was the author of Gaṇitasārasan̄graha (or Ganita Sara Samgraha, c. 850), which revised the Brāhmasphuṭasiddhānta.^{[1]} He was patronised by the Rashtrakuta king Amoghavarsha.^{[4]} He separated astrology from mathematics. It is the earliest Indian text entirely devoted to mathematics.^{[5]} He expounded on the same subjects on which Aryabhata and Brahmagupta contended, but he expressed them more clearly. His work is a highly syncopated approach to algebra and the emphasis in much of his text is on developing the techniques necessary to solve algebraic problems.^{[6]} He is highly respected among Indian mathematicians, because of his establishment of terminology for concepts such as equilateral, and isosceles triangle; rhombus; circle and semicircle.^{[7]} Mahāvīra's eminence spread in all South India and his books proved inspirational to other mathematicians in Southern India.^{[8]} It was translated into Telugu language by Pavuluri Mallana as Saar Sangraha Ganitam.^{[9]}
He discovered algebraic identities like a^{3}=a(a+b)(a-b) +b^{2}(a-b) + b^{3}.^{[3]} He also found out the formula for ^{n}C_{r} as [n(n-1)(n-2)...(n-r+1)]/r(r-1)(r-2)...2*1.^{[10]} He devised formula which approximated area and perimeters of ellipses and found methods to calculate the square of a number and cube roots of a number.^{[11]} He asserted that the square root of a negative number did not exist.^{[12]}
Mahāvīra's Gaṇita-sāra-saṅgraha gave systematic rules for expressing a fraction as the sum of unit fractions.^{[13]} This follows the use of unit fractions in Indian mathematics in the Vedic period, and the Śulba Sūtras' giving an approximation of √2 equivalent to $1+{\tfrac {1}{3}}+{\tfrac {1}{3\cdot 4}}-{\tfrac {1}{3\cdot 4\cdot 34}}$.^{[13]}
In the Gaṇita-sāra-saṅgraha (GSS), the second section of the chapter on arithmetic is named kalā-savarṇa-vyavahāra (lit. "the operation of the reduction of fractions"). In this, the bhāgajāti section (verses 55–98) gives rules for the following:^{[13]}
To express 1 as the sum of n unit fractions (GSS kalāsavarṇa 75, examples in 76):^{[13]}
When the result is one, the denominators of the quantities having one as numerators are [the numbers] beginning with one and multiplied by three, in order. The first and the last are multiplied by two and two-thirds [respectively].
To express a unit fraction $1/q$ as the sum of n other fractions with given numerators $a_{1},a_{2},\dots ,a_{n}$ (GSS kalāsavarṇa 78, examples in 79):
and repeat the process for the second term, recursively. (Note that if i is always chosen to be the smallest such integer, this is identical to the greedy algorithm for Egyptian fractions.)
To express a unit fraction as the sum of two other unit fractions (GSS kalāsavarṇa 85, example in 86):^{[13]}
${\frac {1}{n}}={\frac {1}{p\cdot n}}+{\frac {1}{\frac {p\cdot n}{n-1}}}$ where $p$ is to be chosen such that ${\frac {p\cdot n}{n-1}}$ is an integer (for which $p$ must be a multiple of $n-1$).
To express a fraction $p/q$ as the sum of two other fractions with given numerators $a$ and $b$ (GSS kalāsavarṇa 87, example in 88):^{[13]}
${\frac {p}{q}}={\frac {a}{{\frac {ai+b}{p}}\cdot {\frac {q}{i}}}}+{\frac {b}{{\frac {ai+b}{p}}\cdot {\frac {q}{i}}\cdot {i}}}$ where $i$ is to be chosen such that $p$ divides $ai+b$
Some further rules were given in the Gaṇita-kaumudi of Nārāyaṇa in the 14th century.^{[13]}
Kusuba, Takanori (2004), "Indian Rules for the Decomposition of Fractions", in Charles Burnett; Jan P. Hogendijk; Kim Plofker; et al., Studies in the History of the Exact Sciences in Honour of David Pingree, Brill, ISBN9004132023, ISSN0169-8729