Mahāvīra (mathematician)

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Mahāvīra (or Mahaviracharya, "Mahavira the Teacher") was a 9th-century Indian 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 CE.[4] He was patronised by the Rashtrakuta emperor 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 southern 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[edit]

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 .[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].

  • To express 1 as the sum of an odd number of unit fractions (GSS kalāsavarṇa 77):[13]
  • To express a unit fraction as the sum of n other fractions with given numerators (GSS kalāsavarṇa 78, examples in 79):
  • To express any fraction as a sum of unit fractions (GSS kalāsavarṇa 80, examples in 81):[13]
Choose an integer i such that is an integer r, then write
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]
where is to be chosen such that is an integer (for which must be a multiple of ).
  • To express a fraction as the sum of two other fractions with given numerators and (GSS kalāsavarṇa 87, example in 88):[13]
where is to be chosen such that divides

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

See also[edit]


  1. ^ Pingree 1970.
  2. ^ O'Connor & Robertson 2000.
  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. ^ Hayashi 2013.
  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. ^ a b c d e f g h i Kusuba 2004, pp. 497–516