### Mahaviracharya

**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. There are several points worth noting about the work of Mahavira. 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^{2}.^{[3]} He also found out the formula for ^{r}C_{n} 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 $1\; +\; \backslash tfrac13\; +\; \backslash tfrac1\{3\backslash cdot4\}\; -\; \backslash tfrac1\{3\backslash cdot4\backslash cdot34\}$.^{[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].

- $1\; =\; \backslash frac1\{1\; \backslash cdot\; 2\}\; +\; \backslash frac1\{3\}\; +\; \backslash frac1\{3^2\}\; +\; \backslash dots\; +\; \backslash frac1\{3^\{n-2\}\}\; +\; \backslash frac1\{\backslash frac23\; \backslash cdot\; 3^\{n-1\}\}$

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

- $1\; =\; \backslash frac1\{2\backslash cdot\; 3\; \backslash cdot\; 1/2\}\; +\; \backslash frac1\{3\; \backslash cdot\; 4\; \backslash cdot\; 1/2\}\; +\; \backslash dots\; +\; \backslash frac1\{(2n-1)\; \backslash cdot\; 2n\; \backslash cdot\; 1/2\}\; +\; \backslash frac1\{2n\; \backslash cdot\; 1/2\}$

- To express a unit fraction $1/q$ as the sum of
*n*other fractions with given numerators $a\_1,\; a\_2,\; \backslash dots,\; a\_n$ (GSS*kalāsavarṇa*78, examples in 79):

- $\backslash frac1q\; =\; \backslash frac\{a\_1\}\{q(q+a\_1)\}\; +\; \backslash frac\{a\_2\}\{(q+a\_1)(q+a\_1+a\_2)\}\; +\; \backslash dots\; +\; \backslash frac\{a\_\{n-1\}\}\{q+a\_1+\backslash dots+a\_\{n-2\})(q+a\_1+\backslash dots+a\_\{n-1\})\}\; +\; \backslash frac\{a\_n\}\{a\_n(q+a\_1+\backslash dots+a\_\{n-1\})\}$

- 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 $\backslash tfrac\{q+i\}\{p\}$ is an integer*r*, then write- $\backslash frac\{p\}\{q\}\; =\; \backslash frac\{1\}\{r\}\; +\; \backslash frac\{i\}\{r\; \backslash cdot\; q\}$

- 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]}

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

- 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]}

- $\backslash frac\{p\}\{q\}\; =\; \backslash frac\{a\}\{\backslash frac\{ai+b\}\{p\}\backslash cdot\backslash frac\{q\}\{i\}\}\; +\; \backslash frac\{b\}\{\backslash frac\{ai+b\}\{p\}\; \backslash cdot\; \backslash frac\{q\}\{i\}\; \backslash cdot\{i\}\}$ 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]}

## Notes

## References

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