# Mahāvīra (mathematician)

Mahāvīra | |
---|---|

Born | India |

Occupation | Mathematician |

Religion | Jainism |

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

## 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 .^{[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].

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

- To express a unit fraction as the sum of
*n*other fractions with given numerators (GSS*kalāsavarṇa*78, examples in 79):

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

- Choose an integer
*i*such that is an integer*r*, then write - 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]}

- where is to be chosen such that is an integer (for which must be a multiple of ).

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

- where is to be chosen such that divides

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

## See also

## Notes

- 1 2 Pingree 1970.
- ↑ O'Connor & Robertson 2000.
- 1 2 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.
- 1 2 3 4 5 6 7 8 9 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 978-0-684-10114-9. - 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