# 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 a3=a(a+b)(a-b) +b2(a-b) + b3.[3] He also found out the formula for nCr 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 $1 + \tfrac13 + \tfrac1{3\cdot4} - \tfrac1{3\cdot4\cdot34}$.[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].

$1 = \frac1{1 \cdot 2} + \frac1{3} + \frac1{3^2} + \dots + \frac1{3^{n-2}} + \frac1{\frac23 \cdot 3^{n-1}}$
• To express 1 as the sum of an odd number of unit fractions (GSS kalāsavarṇa 77):[13]
$1 = \frac1{2\cdot 3 \cdot 1/2} + \frac1{3 \cdot 4 \cdot 1/2} + \dots + \frac1{(2n-1) \cdot 2n \cdot 1/2} + \frac1{2n \cdot 1/2}$
• 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):
$\frac1q = \frac{a_1}{q(q+a_1)} + \frac{a_2}{(q+a_1)(q+a_1+a_2)} + \dots + \frac{a_{n-1}}{q+a_1+\dots+a_{n-2})(q+a_1+\dots+a_{n-1})} + \frac{a_n}{a_n(q+a_1+\dots+a_{n-1})}$
• 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]
$\frac1{n} = \frac1{p\cdot n} + \frac1{\frac{p\cdot n}{n-1}}$ where $p$ is to be chosen such that $\frac{p\cdot n}{n-1}$ is an integer (for which $p$ must be a multiple of $n-1$).
$\frac1{a\cdot b} = \frac1{a(a+b)} + \frac1{b(a+b)}$
• 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]
$\frac{p}{q} = \frac{a}{\frac{ai+b}{p}\cdot\frac{q}{i}} + \frac{b}{\frac{ai+b}{p} \cdot \frac{q}{i} \cdot{i}}$ where $i$ is to be chosen such that $p$ divides $ai + b$

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

## Notes

1. ^ a b
2. ^
3. ^ a b Tabak 2009, p. 42.
4. ^ Puttaswamy 2012, p. 231.
5. ^ The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the ... by Clifford A. Pickover: page 88
6. ^ Algebra: Sets, Symbols, and the Language of Thought by John Tabak: p.43
7. ^ Geometry in Ancient and Medieval India by T. A. Sarasvati Amma: page 122
8. ^
9. ^ Census of the Exact Sciences in Sanskrit by David Pingree: page 388
10. ^ Tabak 2009, p. 43.
11. ^ Krebs 2004, p. 132.
12. ^ Selin 2008, p. 1268.
13. Kusuba 2004, pp. 497–516