# Mahāvīra (mathematician)

Mahāvīra (or Mahaviracharya, "Mahavira the Teacher") was a 9th-century Jain mathematician possibly born in Mysore, in India.[1][2][3] He authored Gaṇita-sāra-saṅgraha (Ganita Sara Sangraha) or the Compendium on the gist of Mathematics in 850 AD.[4] 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 throughout South India and his books proved inspirational to other mathematicians in Southern India.[8] It was translated into the Telugu language by Pavuluri Mallana as Saara Sangraha Ganitamu.[9]

He discovered algebraic identities like a3 = a (a + b) (ab) + b2 (ab) + b3.[3] He also found out the formula for nCr as
[n (n − 1) (n − 2) ... (nr + 1)] / [r (r − 1) (r − 2) ... 2 * 1].[10] He devised a formula which approximated the 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 does 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 ${\displaystyle 1+{\tfrac {1}{3}}+{\tfrac {1}{3\cdot 4}}-{\tfrac {1}{3\cdot 4\cdot 34}}}$ .[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].

${\displaystyle 1={\frac {1}{1\cdot 2}}+{\frac {1}{3}}+{\frac {1}{3^{2}}}+\dots +{\frac {1}{3^{n-2}}}+{\frac {1}{{\frac {2}{3}}\cdot 3^{n-1}}}}$
• To express 1 as the sum of an odd number of unit fractions (GSS kalāsavarṇa 77):[13]
${\displaystyle 1={\frac {1}{2\cdot 3\cdot 1/2}}+{\frac {1}{3\cdot 4\cdot 1/2}}+\dots +{\frac {1}{(2n-1)\cdot 2n\cdot 1/2}}+{\frac {1}{2n\cdot 1/2}}}$
• To express a unit fraction ${\displaystyle 1/q}$  as the sum of n other fractions with given numerators ${\displaystyle a_{1},a_{2},\dots ,a_{n}}$  (GSS kalāsavarṇa 78, examples in 79):
${\displaystyle {\frac {1}{q}}={\frac {a_{1}}{q(q+a_{1})}}+{\frac {a_{2}}{(q+a_{1})(q+a_{1}+a_{2})}}+\dots +{\frac {a_{n-1}}{(q+a_{1}+\dots +a_{n-2})(q+a_{1}+\dots +a_{n-1})}}+{\frac {a_{n}}{a_{n}(q+a_{1}+\dots +a_{n-1})}}}$
• To express any fraction ${\displaystyle p/q}$  as a sum of unit fractions (GSS kalāsavarṇa 80, examples in 81):[13]
Choose an integer i such that ${\displaystyle {\tfrac {q+i}{p}}}$  is an integer r, then write
${\displaystyle {\frac {p}{q}}={\frac {1}{r}}+{\frac {i}{r\cdot q}}}$
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]
${\displaystyle {\frac {1}{n}}={\frac {1}{p\cdot n}}+{\frac {1}{\frac {p\cdot n}{n-1}}}}$  where ${\displaystyle p}$  is to be chosen such that ${\displaystyle {\frac {p\cdot n}{n-1}}}$  is an integer (for which ${\displaystyle p}$  must be a multiple of ${\displaystyle n-1}$ ).
${\displaystyle {\frac {1}{a\cdot b}}={\frac {1}{a(a+b)}}+{\frac {1}{b(a+b)}}}$
• To express a fraction ${\displaystyle p/q}$  as the sum of two other fractions with given numerators ${\displaystyle a}$  and ${\displaystyle b}$  (GSS kalāsavarṇa 87, example in 88):[13]
${\displaystyle {\frac {p}{q}}={\frac {a}{{\frac {ai+b}{p}}\cdot {\frac {q}{i}}}}+{\frac {b}{{\frac {ai+b}{p}}\cdot {\frac {q}{i}}\cdot {i}}}}$  where ${\displaystyle i}$  is to be chosen such that ${\displaystyle p}$  divides ${\displaystyle ai+b}$

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

## Notes

1. ^ Pingree 1970.
2. ^
3. ^ a b Tabak 2009, p. 42.
4. ^ a b 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