# 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 a3=a(a+b)(a-b) +b2(a-b) + b3.Template:Sfn He also found out the formula for nCr 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 ${\displaystyle 1+{\tfrac {1}{3}}+{\tfrac {1}{3\cdot 4}}-{\tfrac {1}{3\cdot 4\cdot 34}}}$.[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].

${\displaystyle 1={\frac {1}{1\cdot 2}}+{\frac {1}{3}}+{\frac {1}{3^{2}}}+\dots +{\frac {1}{3^{n-2}}}+{\frac {1}{{\frac {2}{3}}\cdot 3^{n-1}}}}$
• To express 1 as the sum of an odd number of unit fractions (GSS kalāsavarṇa 77):[5]
${\displaystyle 1={\frac {1}{2\cdot 3\cdot 1/2}}+{\frac {1}{3\cdot 4\cdot 1/2}}+\dots +{\frac {1}{(2n-1)\cdot 2n\cdot 1/2}}+{\frac {1}{2n\cdot 1/2}}}$
${\displaystyle {\frac {1}{q}}={\frac {a_{1}}{q(q+a_{1})}}+{\frac {a_{2}}{(q+a_{1})(q+a_{1}+a_{2})}}+\dots +{\frac {a_{n-1}}{q+a_{1}+\dots +a_{n-2})(q+a_{1}+\dots +a_{n-1})}}+{\frac {a_{n}}{a_{n}(q+a_{1}+\dots +a_{n-1})}}}$
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):[5]
${\displaystyle {\frac {1}{n}}={\frac {1}{p\cdot n}}+{\frac {1}{\frac {p\cdot n}{n-1}}}}$ 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 ${\displaystyle n-1}$).
${\displaystyle {\frac {1}{a\cdot b}}={\frac {1}{a(a+b)}}+{\frac {1}{b(a+b)}}}$
${\displaystyle {\frac {p}{q}}={\frac {a}{{\frac {ai+b}{p}}\cdot {\frac {q}{i}}}}+{\frac {b}{{\frac {ai+b}{p}}\cdot {\frac {q}{i}}\cdot {i}}}}$ 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.[5]

## Notes

1. The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the ... by Clifford A. Pickover: page 88
2. Algebra: Sets, Symbols, and the Language of Thought by John Tabak: p.43
3. Geometry in Ancient and Medieval India by T. A. Sarasvati Amma: page 122
4. Census of the Exact Sciences in Sanskrit by David Pingree: page 388
5. Template:Harvnb

## References

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