Mahāvīra (mathematician)

Mahāvīra (or Mahaviracharya, "Mahavira the Teacher") was a 9th-century Jain mathematician possibly born in Mysore, in India. He authored '' Gaṇitasārasan̄graha'' (''Ganita Sara Sangraha'') or the Compendium on the gist of Mathematics in 850 AD. He was patronised by the Rashtrakuta king Amoghavarsha. He separated astrology from mathematics. It is the earliest Indian text entirely devoted to mathematics. 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. 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. Mahāvīra's eminence spread throughout South India and his books proved inspirational to other mathematicians in Southern India. It was translated into the Telugu language by Pavuluri Mallana as ''Saara Sangraha Ganitamu''.Census of the Exact Sciences in Sanskrit by David Pingree: page 388

He discovered algebraic identities like ''a''³ = ''a'' (''a'' + ''b'') (''a'' − ''b'') + ''b''² (''a'' − ''b'') + ''b''³. 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 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. He asserted that the square root of a negative number does not exist.

Rules for decomposing fractions

Mahāvīra's ''Gaṇita-sāra-saṅgraha'' gave systematic rules for expressing a fraction as the sum of unit fractions. This follows the use of unit fractions in Indian mathematics in the Vedic period, and the Śulba Sūtras' giving an approximation of equivalent to $1 + \tfrac13 + \tfrac1 - \tfrac1$.

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:

* To express 1 as the sum of ''n'' unit fractions (GSS ''kalāsavarṇa'' 75, examples in 76):

:: $1 = \frac1 + \frac1 + \frac1 + \dots + \frac1 + \frac1$

* To express 1 as the sum of an odd number of unit fractions (GSS ''kalāsavarṇa'' 77):
:: $1 = \frac1 + \frac1 + \dots + \frac1 + \frac1$

* 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):
:: $\frac1q = \frac + \frac + \dots + \frac + \frac$

* To express any fraction $p/q$ as a sum of unit fractions (GSS ''kalāsavarṇa'' 80, examples in 81):
: Choose an integer ''i'' such that $\tfrac$ is an integer ''r'', then write
:: $\frac = \frac + \frac$
: 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):
:: $\frac1 = \frac1 + \frac1$ where $p$ is to be chosen such that $\frac$ is an integer (for which $p$ must be a multiple of $n-1$).
:: $\frac1 = \frac1 + \frac1$

* 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):
:: $\frac = \frac + \frac$ 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.

See also

* List of Indian mathematicians

Notes

References

*Bibhutibhusan Datta and Avadhesh Narayan Singh (1962). '' History of Hindu Mathematics: A Source Book''.
* (Available, along with many other entries from other encyclopaedias for other Mahāvīra-s
online
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{{DEFAULTSORT:Mahavira
9th-century Indian mathematicians
9th-century Indian Jains
Scholars from Karnataka
Acharyas