Mahāvīrā (Mahāvīrāchārya)
Jain Matheamatician Mahāvīrā (Mahāvīrāchārya)
Personal life
Born	Karnataka,Rashtrakuta Kingdom
Era	9th century CE
Notable work(s)	"Gaṇita Sāra Saṅgraha"
Occupation	Mathematician, Philosopher
Religious life
Religion	Jainism
Sect	Digambara
Dynasty	Rashtrakuta

Mahāvīra (orMahaviracharya, "Mahavira the Teacher") was a 9th-century IndianJainmathematician possibly born inMysore, inIndia.^[1][2][3] He authoredGaṇita-sāra-saṅgraha (Ganita Sara Sangraha) or the Compendium on the gist of Mathematics in 850 CE.^[4] He was patronised by theRashtrakuta emperorAmoghavarsha.^[4] He separatedastrology from mathematics. It is the earliest Indian text entirely devoted to mathematics.^[5] He expounded on the same subjects on whichAryabhata andBrahmagupta 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 ofterminology 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 inSouthern India.^[8] It was translated into theTelugu language byPavuluri Mallana asSaara Sangraha Ganitamu.^[9]

He discovered algebraic identities likea³ =a (a +b) (a −b) +b² (a −b) +b³.^[3] He also found out the formula forⁿC_r as
[n (n − 1) (n − 2) ... (n −r + 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 thesquare root of anegative number does not exist.^[12] Arithmetic operations utilized in his works like Gaṇita-sāra-saṅgraha(Ganita Sara Sangraha) usesdecimal place-value system and include the use ofzero. However, he erroneously states that a number divided by zero remains unchanged.^[13]

Mahāvīra'sGaṇita-sāra-saṅgraha gave systematic rules for expressing a fraction as thesum of unit fractions.^[14] This follows the use of unit fractions inIndian mathematics in the Vedic period, and theŚulba Sūtras' giving an approximation of√2 equivalent to${\displaystyle 1+\frac{1}{3}+\frac{1}{3\cdot 4}-\frac{1}{3\cdot 4\cdot 34}}$.^[14]

In theGaṇita-sāra-saṅgraha (GSS), the second section of the chapter on arithmetic is namedkalā-savarṇa-vyavahāra (lit. "the operation of the reduction of fractions"). In this, thebhāgajāti section (verses 55–98) gives rules for the following:^[14]

• To express 1 as the sum ofn unit fractions (GSSkalāsavarṇa 75, examples in 76):^[14]

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].

${\displaystyle 1=\frac{1}{1\cdot 2}+\frac{1}{3}+\frac{1}{3^2}+\cdots +\frac{1}{3^{n-2}}+\frac{1}{\frac{2}{3}\cdot 3^{n-1}}}$

• To express 1 as the sum of an odd number of unit fractions (GSSkalāsavarṇa 77):^[14]

${\displaystyle 1=\frac{1}{2\cdot 3\cdot 1/2}+\frac{1}{3\cdot 4\cdot 1/2}+\cdots +\frac{1}{(2n-1)\cdot 2n\cdot 1/2}+\frac{1}{2n\cdot 1/2}}$

• To express a unit fraction${\displaystyle 1/q}$ as the sum ofn other fractions with given numerators${\displaystyle a_1,a_2,\dots ,a_n}$ (GSSkalāsavarṇa 78, examples in 79):

${\displaystyle \frac{1}{q}=\frac{a_1}{q(q+a_1)}+\frac{a_2}{(q+a_1)(q+a_1+a_2)}+\cdots +\frac{a_{n-1}}{(q+a_1+\cdots +a_{n-2})(q+a_1+\cdots +a_{n-1})}+\frac{a_n}{a_n(q+a_1+\cdots +a_{n-1})}}$

• To express any fraction${\displaystyle p/q}$ as a sum of unit fractions (GSSkalāsavarṇa 80, examples in 81):^[14]

Choose an integeri such that${\displaystyle \frac{q+i}{p}}$ is an integerr, 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 ifi is always chosen to be thesmallest such integer, this is identical to thegreedy algorithm for Egyptian fractions.)

• To express a unit fraction as the sum of two other unit fractions (GSSkalāsavarṇa 85, example in 86):^[14]

${\displaystyle \frac{1}{n}=\frac{1}{p\cdot n}+\frac{1}{\frac{p\cdot n}{n-1}}}$ 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${\displaystyle b}$ (GSSkalāsavarṇa 87, example in 88):^[14]

${\displaystyle \frac{p}{q}=\frac{a}{\frac{ai+b}{p}\cdot \frac{q}{i}}+\frac{b}{\frac{ai+b}{p}\cdot \frac{q}{i}\cdot i}}$ where${\displaystyle i}$ is to be chosen such that${\displaystyle p}$ divides${\displaystyle ai+b}$

Some further rules were given in theGaṇita-kaumudi ofNārāyaṇa in the 14th century.^[14]

• List of Indian mathematicians

• Bibhutibhusan Datta and Avadhesh Narayan Singh (1962).History of Hindu Mathematics: A Source Book.
• Pingree, David (1970). "Mahāvīra".Dictionary of Scientific Biography. New York: Charles Scribner's Sons.ISBN 978-0-684-10114-9. (Available, along with many other entries from other encyclopaedias for other Mahāvīra-s,online.)
• Selin, Helaine (2008),Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer,Bibcode:2008ehst.book.....S,ISBN 978-1-4020-4559-2
• Hayashi, Takao (2013),"Mahavira",Encyclopædia Britannica
• O'Connor, John J.;Robertson, Edmund F. (2000),"Mahavira",MacTutor History of Mathematics Archive,University of St Andrews
• Tabak, John (2009),Algebra: Sets, Symbols, and the Language of Thought, Infobase Publishing,ISBN 978-0-8160-6875-3
• Krebs, Robert E. (2004),Groundbreaking Scientific Experiments, Inventions, and Discoveries of the Middle Ages and the Renaissance, Greenwood Publishing Group,ISBN 978-0-313-32433-8
• Puttaswamy, T.K (2012),Mathematical Achievements of Pre-modern Indian Mathematicians, Newnes,ISBN 978-0-12-397938-4
• Kusuba, Takanori (2004), "Indian Rules for the Decomposition of Fractions", in Charles Burnett; Jan P. Hogendijk;Kim Plofker; et al. (eds.),Studies in the History of the Exact Sciences in Honour of David Pingree,Brill,ISBN 9004132023,ISSN 0169-8729

• 9th-century Indian mathematicians
• 9th-century Indian Jains
• Scholars from Karnataka
• Acharyas
• Rashtrakuta people

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