Mahāvīra (mathematician)

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) (a − b) + b² (a − b) + b³.^[3] He also found out the formula for ⁿC_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 the square root of a negative number does not exist.^[12] Arithmetic operations utilized in his works like Gaṇita-sāra-saṅgraha(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.^[13]

Rules for decomposing fractions

Mahāvīra's Gaṇita-sāra-saṅgraha gave systematic rules for expressing a fraction as the sum of unit fractions.^[14] 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 $1+{\tfrac {1}{3}}+{\tfrac {1}{3\cdot 4}}-{\tfrac {1}{3\cdot 4\cdot 34}}$ .^[14]

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:^[14]

To express 1 as the sum of n unit fractions (GSS kalāsavarṇa 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].

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):^[14]

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}}

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):

{\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})}}

To express any fraction $p/q$ as a sum of unit fractions (GSS kalāsavarṇa 80, examples in 81):^[14]

Choose an integer i such that

{\tfrac {q+i}{p}}

is an integer r, then write

{\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):^[14]

{\frac {1}{n}}={\frac {1}{p\cdot n}}+{\frac {1}{\frac {p\cdot n}{n-1}}}

where

p

is to be chosen such that

{\frac {p\cdot n}{n-1}}

is an integer (for which

p

must be a multiple of

n-1

).

{\frac {1}{a\cdot b}}={\frac {1}{a(a+b)}}+{\frac {1}{b(a+b)}}

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):^[14]

{\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

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.^[14]

Notes

^ Pingree 1970.
^ O'Connor & Robertson 2000.
^ ^a ^b Tabak 2009, p. 42.
^ ^a ^b Puttaswamy 2012, p. 231.
^ The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the ... by Clifford A. Pickover: page 88
^ Algebra: Sets, Symbols, and the Language of Thought by John Tabak: p.43
^ Geometry in Ancient and Medieval India by T. A. Sarasvati Amma: page 122
^ Hayashi 2013.
^ Census of the Exact Sciences in Sanskrit by David Pingree: page 388
^ Tabak 2009, p. 43.
^ Krebs 2004, p. 132.
^ Selin 2008, p. 1268.
^ A Concise History of Science in India (Eds.) D. M. Bose, S. N. Sen and B.V. Subbarayappa. Indian National Science Academy. 15 October 1971. p. 167.{{cite book}}: CS1 maint: date and year (link)
^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ Kusuba 2004, pp. 497–516

References

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

[FOOTNOTEPingree1970-1] Pingree 1970.

[FOOTNOTEO'ConnorRobertson2000-2] O'Connor & Robertson 2000.

[FOOTNOTETabak200942-3] Tabak 2009, p. 42.

[FOOTNOTEPuttaswamy2012231-4] 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

[FOOTNOTEHayashi2013-8] Hayashi 2013.

[9] Census of the Exact Sciences in Sanskrit by David Pingree: page 388

[FOOTNOTETabak200943-10] Tabak 2009, p. 43.

[FOOTNOTEKrebs2004132-11] Krebs 2004, p. 132.

[FOOTNOTESelin20081268-12] Selin 2008, p. 1268.

[13] A Concise History of Science in India (Eds.) D. M. Bose, S. N. Sen and B.V. Subbarayappa. Indian National Science Academy. 15 October 1971. p. 167.{{cite book}}: CS1 maint: date and year (link)

[k497-14] ^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ Kusuba 2004, pp. 497–516

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Mahāvīra (mathematician)

Rules for decomposing fractions

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References