Mahāvīra 
Born 
India 
Occupation 
Mathematician 
Mahāvīra (or Mahaviracharya, "Mahavira the Teacher") was a 9thcentury Jain mathematician from Karnataka, India. He was the author of Gaṇitasārasan̄graha (or Ganita Sara Samgraha, c. 850), which revised the Brāhmasphuṭasiddhānta. He was patronised by the Rashtrakuta king Amoghavarsha. 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. 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}. 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]. He devised a formula which approximated the area and perimeters of ellipses and found methods to calculate the square of a number and cube roots of a number. He asserted that the square root of a negative number did not exist.
Rules for decomposing fractions
Mahāvīra's Gaṇitasārasaṅgraha 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 Gaṇitasārasaṅgraha (GSS), the second section of the chapter on arithmetic is named kalāsavarṇavyavahāra (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]}
rūpāṃśakarāśīnāṃ rūpādyās triguṇitā harāḥ kramaśaḥ /
dvidvitryaṃśābhyastāv ādimacaramau phale rūpe //
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 twothirds [respectively].

 $1={\frac {1}{1\cdot 2}}+{\frac {1}{3}}+{\frac {1}{3^{2}}}+\dots +{\frac {1}{3^{n2}}}+{\frac {1}{{\frac {2}{3}}\cdot 3^{n1}}}$
 To express 1 as the sum of an odd number of unit fractions (GSS kalāsavarṇa 77):^{[13]}

 $1={\frac {1}{2\cdot 3\cdot 1/2}}+{\frac {1}{3\cdot 4\cdot 1/2}}+\dots +{\frac {1}{(2n1)\cdot 2n\cdot 1/2}}+{\frac {1}{2n\cdot 1/2}}$
 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):

 ${\frac {1}{q}}={\frac {a_{1}}{q(q+a_{1})}}+{\frac {a_{2}}{(q+a_{1})(q+a_{1}+a_{2})}}+\dots +{\frac {a_{n1}}{(q+a_{1}+\dots +a_{n2})(q+a_{1}+\dots +a_{n1})}}+{\frac {a_{n}}{a_{n}(q+a_{1}+\dots +a_{n1})}}$
 To express any fraction $p/q$ as a sum of unit fractions (GSS kalāsavarṇa 80, examples in 81):^{[13]}
 Choose an integer i such that ${\tfrac {q+i}{p}}$ is an integer r, then write
 ${\frac {p}{q}}={\frac {1}{r}}+{\frac {i}{r\cdot q}}$
 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]}

 ${\frac {1}{n}}={\frac {1}{p\cdot n}}+{\frac {1}{\frac {p\cdot n}{n1}}}$ where $p$ is to be chosen such that ${\frac {p\cdot n}{n1}}$ is an integer (for which $p$ must be a multiple of $n1$).
 ${\frac {1}{a\cdot b}}={\frac {1}{a(a+b)}}+{\frac {1}{b(a+b)}}$
 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 Gaṇitakaumudi of Nārāyaṇa in the 14th century.^{[13]}
See also
Notes
 ^ The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the ... by Clifford A. Pickover: page 88
 ^ Algebra: Sets, Symbols, and the Language of Thought by John Tabak: p.43
 ^ Geometry in Ancient and Medieval India by T. A. Sarasvati Amma: page 122
 ^ Census of the Exact Sciences in Sanskrit by David Pingree: page 388
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} ^{i} Kusuba 2004, pp. 497–516
References
 Bibhutibhusan Datta and Avadhesh Narayan Singh (1962). History of Hindu Mathematics: A Source Book.
 Pingree, David (1970). "Mahāvīra". Dictionary of Scientific Biography. New York: Charles Scribner's Sons. ISBN 9780684101149.
 Selin, Helaine (2008), Encyclopaedia of the History of Science, Technology, and Medicine in NonWestern Cultures, Springer, ISBN 9781402045592
 Hayashi, Takao (2013), "Mahavira", Encyclopædia Britannica
 O'Connor, John J.; Robertson, Edmund F. (2000), "Mahavira", MacTutor History of Mathematics archive, University of St Andrews .
 Tabak, John (2009), Algebra: Sets, Symbols, and the Language of Thought, Infobase Publishing, ISBN 9780816068753
 Krebs, Robert E. (2004), Groundbreaking Scientific Experiments, Inventions, and Discoveries of the Middle Ages and the Renaissance, Greenwood Publishing Group, ISBN 9780313324338
 Puttaswamy, T.K (2012), Mathematical Achievements of Premodern Indian Mathematicians, Newnes, ISBN 9780123979384
 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, ISBN 9004132023, ISSN 01698729

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