# Mahāvīra (mathematician)

**Mahāvīra** (or **Mahaviracharya**, "Mahavira the Teacher") was a 9th-century Jain mathematician from Mysore, India.Template:SfnTemplate:SfnTemplate:Sfn He was the author of *Gaṇitasārasan̄graha* (or *Ganita Sara Samgraha*, c. 850), which revised the Brāhmasphuṭasiddhānta.Template:Sfn He was patronised by the Rashtrakuta king Amoghavarsha.Template:Sfn He separated astrology from mathematics. It is the earliest Indian text entirely devoted to mathematics.^{[1]} 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.^{[2]} 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.^{[3]} Mahāvīra's eminence spread in all South India and his books proved inspirational to other mathematicians in Southern India.Template:Sfn It was translated into Telugu language by Pavuluri Mallana as *Saar Sangraha Ganitam*.^{[4]}

He discovered algebraic identities like a^{3}=a(a+b)(a-b) +b^{2}(a-b) + b^{3}.Template:Sfn 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.Template:Sfn 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.Template:Sfn He asserted that the square root of a negative number did not exist.Template:Sfn

## Rules for decomposing fractions

Mahāvīra's *Gaṇita-sāra-saṅgraha* gave systematic rules for expressing a fraction as the sum of unit fractions.^{[5]} 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 .^{[5]}

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:^{[5]}

- To express 1 as the sum of
*n*unit fractions (GSS*kalāsavarṇa*75, examples in 76):^{[5]}

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):^{[5]}

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

- 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):^{[5]}

- To express a fraction as the sum of two other fractions with given numerators and (GSS
*kalāsavarṇa*87, example in 88):^{[5]}

Some further rules were given in the *Gaṇita-kaumudi* of Nārāyaṇa in the 14th century.^{[5]}

## 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
- ↑
^{5.0}^{5.1}^{5.2}^{5.3}^{5.4}^{5.5}^{5.6}^{5.7}^{5.8}Template:Harvnb

## See also

## References

- Bibhutibhusan Datta and Avadhesh Narayan Singh (1962).
*History of Hindu mathematics: a source book*. - Template:DSB
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