MahÃÂvëra (or Mahaviracharya, "Mahavira the Teacher") was a 9th-century Indian Jain mathematician possibly born in Mysore, in India. He authored Gaá¹Âita-sÃÂra-saá¹ graha (Ganita Sara Sangraha) or the Compendium on the gist of Mathematics in 850 CE. He was patronised by the Rashtrakuta emperor 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. MahÃÂvëra's eminence spread throughout southern 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.
He discovered algebraic identities like a<sup>3</sup> = a (a + b) (a − b) + b<sup>2</sup> (a − b) + b<sup>3</sup>. He also found out the formula for <sup>n</sup>C<sub>r</sub> as <br/>[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. Arithmetic operations utilized in his works like Gaá¹Âita-sÃÂra-saá¹ graha(Ganita Sara Sangraha) uses decimal place-value system and include the use of zero. However, he erroneously states that a number divided by zero remains unchanged.
MahÃÂvëra's Gaá¹Âita-sÃÂra-saá¹ graha 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 à Âulba Sà «tras' giving an approximation of equivalent to .
In the Gaá¹Âita-sÃÂra-saá¹ graha (GSS), the second section of the chapter on arithmetic is named kalÃÂ-savará¹Âa-vyavahÃÂra (lit. "the operation of the reduction of fractions"). In this, the bhÃÂgajÃÂti section (verses 55âÂÂ98) gives rules for the following:
Some further rules were given in the Gaá¹Âita-kaumudi of NÃÂrÃÂyaá¹Âa in the 14th century.