## Frequent Links

# Mahāvīra (mathematician)

**Mahāvīra** (or **Mahaviracharya**, "Mahavira the Teacher") was a 9th-century Jain mathematician from Mysore, 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]}

## Rules for decomposing fractions

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 <math>1 + \tfrac13 + \tfrac1{3\cdot4} - \tfrac1{3\cdot4\cdot34}</math>.^{[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]}

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 two-thirds [respectively].

- <math> 1 = \frac1{1 \cdot 2} + \frac1{3} + \frac1{3^2} + \dots + \frac1{3^{n-2}} + \frac1{\frac23 \cdot 3^{n-1}} </math>

- To express 1 as the sum of an odd number of unit fractions (GSS
*kalāsavarṇa*77):^{[13]}

- <math>1 = \frac1{2\cdot 3 \cdot 1/2} + \frac1{3 \cdot 4 \cdot 1/2} + \dots + \frac1{(2n-1) \cdot 2n \cdot 1/2} + \frac1{2n \cdot 1/2} </math>

- To express a unit fraction <math>1/q</math> as the sum of
*n*other fractions with given numerators <math>a_1, a_2, \dots, a_n</math> (GSS*kalāsavarṇa*78, examples in 79):

- <math>\frac1q = \frac{a_1}{q(q+a_1)} + \frac{a_2}{(q+a_1)(q+a_1+a_2)} + \dots + \frac{a_{n-1}}{q+a_1+\dots+a_{n-2})(q+a_1+\dots+a_{n-1})} + \frac{a_n}{a_n(q+a_1+\dots+a_{n-1})}</math>

- To express any fraction <math>p/q</math> as a sum of unit fractions (GSS
*kalāsavarṇa*80, examples in 81):^{[13]}

- Choose an integer
*i*such that <math>\tfrac{q+i}{p}</math> is an integer*r*, then write- <math> \frac{p}{q} = \frac{1}{r} + \frac{i}{r \cdot q} </math>

- 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]}

- <math>\frac1{n} = \frac1{p\cdot n} + \frac1{\frac{p\cdot n}{n-1}}</math> where <math>p</math> is to be chosen such that <math>\frac{p\cdot n}{n-1}</math> is an integer (for which <math>p</math> must be a multiple of <math>n-1</math>).
- <math>\frac1{a\cdot b} = \frac1{a(a+b)} + \frac1{b(a+b)}</math>

- To express a fraction <math>p/q</math> as the sum of two other fractions with given numerators <math>a</math> and <math>b</math> (GSS
*kalāsavarṇa*87, example in 88):^{[13]}

- <math>\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}}</math> where <math>i</math> is to be chosen such that <math>p</math> divides <math>ai + b</math>

Some further rules were given in the *Gaṇita-kaumudi* of Nārāyaṇa in the 14th century.^{[13]}

## Notes

- ^
^{a}^{b}Pingree 1970. **^**O'Connor & Robertson 2000.- ^
^{a}^{b}Tabak 2009, p. 42. **^**Puttaswamy 2012, p. 231.**^**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**^**Hayashi 2013.**^**Census of the Exact Sciences in Sanskrit by David Pingree: page 388**^**Tabak 2009, p. 43.**^**Krebs 2004, p. 132.**^**Selin 2008, p. 1268.- ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}^{h}^{i}Kusuba 2004, pp. 497–516

## See also

## References

- Bibhutibhusan Datta and Avadhesh Narayan Singh (1962).
*History of Hindu mathematics: a source book*. - Template:DSB
- Selin, Helaine (2008),
*Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures*, Springer, ISBN 978-1-4020-4559-2 - 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 978-0-8160-6875-3 - Krebs, Robert E. (2004),
*Groundbreaking Scientific Experiments, Inventions, and Discoveries of the Middle Ages and the Renaissance*, Greenwood Publishing Group, ISBN 978-0-313-32433-8 - Puttaswamy, T.K (2012),
*Mathematical Achievements of Pre-modern Indian Mathematicians*, Newnes, ISBN 978-0-12-397938-4 - 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 0169-8729

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