Mahāvīra (mathematician)

















Mahāvīra
Born
India
Occupation Mathematician

Mahāvīra (or Mahaviracharya, "Mahavira the Teacher") was a 9th-century Jain mathematician possibly born in or close to the present day city of Mysore, in southern India.[1][2][3] He was the author of Gaṇitasārasan̄graha (or Ganita Sara Samgraha, c. 850), which revised the Brāhmasphuṭasiddhānta.[1] He was patronised by the Rashtrakuta king 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 in all South India and his books proved inspirational to other mathematicians in Southern India.[8] It was translated into Telugu language by Pavuluri Mallana as Saar Sangraha Ganitam.[9]


He discovered algebraic identities like a3 = a (a + b) (a − b) + b2 (a − b) + b3.[3] He also found out the formula for nCr 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 did not exist.[12]




Contents






  • 1 Rules for decomposing fractions


  • 2 See also


  • 3 Notes


  • 4 References





Rules for decomposing fractions


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 1+13+13⋅4−13⋅4⋅34{displaystyle 1+{tfrac {1}{3}}+{tfrac {1}{3cdot 4}}-{tfrac {1}{3cdot 4cdot 34}}}1+{tfrac 13}+{tfrac 1{3cdot 4}}-{tfrac 1{3cdot 4cdot 34}}.[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]

.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 40px}.mw-parser-output .templatequote .templatequotecite{line-height:1.5em;text-align:left;padding-left:1.6em;margin-top:0}

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=11⋅2+13+132+⋯+13n−2+123⋅3n−1{displaystyle 1={frac {1}{1cdot 2}}+{frac {1}{3}}+{frac {1}{3^{2}}}+dots +{frac {1}{3^{n-2}}}+{frac {1}{{frac {2}{3}}cdot 3^{n-1}}}}{displaystyle 1={frac {1}{1cdot 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):[13]

1=12⋅3⋅1/2+13⋅4⋅1/2+⋯+1(2n−1)⋅2n⋅1/2+12n⋅1/2{displaystyle 1={frac {1}{2cdot 3cdot 1/2}}+{frac {1}{3cdot 4cdot 1/2}}+dots +{frac {1}{(2n-1)cdot 2ncdot 1/2}}+{frac {1}{2ncdot 1/2}}}{displaystyle 1={frac {1}{2cdot 3cdot 1/2}}+{frac {1}{3cdot 4cdot 1/2}}+dots +{frac {1}{(2n-1)cdot 2ncdot 1/2}}+{frac {1}{2ncdot 1/2}}}

  • To express a unit fraction 1/q{displaystyle 1/q}1/q as the sum of n other fractions with given numerators a1,a2,…,an{displaystyle a_{1},a_{2},dots ,a_{n}}a_{1},a_{2},dots ,a_{n} (GSS kalāsavarṇa 78, examples in 79):

1q=a1q(q+a1)+a2(q+a1)(q+a1+a2)+⋯+an−1(q+a1+⋯+an−2)(q+a1+⋯+an−1)+anan(q+a1+⋯+an−1){displaystyle {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})}}}{displaystyle {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{displaystyle p/q}p/q as a sum of unit fractions (GSS kalāsavarṇa 80, examples in 81):[13]


Choose an integer i such that q+ip{displaystyle {tfrac {q+i}{p}}}{displaystyle {tfrac {q+i}{p}}} is an integer r, then write
pq=1r+ir⋅q{displaystyle {frac {p}{q}}={frac {1}{r}}+{frac {i}{rcdot q}}}{displaystyle {frac {p}{q}}={frac {1}{r}}+{frac {i}{rcdot 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):[13]



1n=1p⋅n+1p⋅nn−1{displaystyle {frac {1}{n}}={frac {1}{pcdot n}}+{frac {1}{frac {pcdot n}{n-1}}}}{displaystyle {frac {1}{n}}={frac {1}{pcdot n}}+{frac {1}{frac {pcdot n}{n-1}}}} where p{displaystyle p}p is to be chosen such that p⋅nn−1{displaystyle {frac {pcdot n}{n-1}}}{displaystyle {frac {pcdot n}{n-1}}} is an integer (for which p{displaystyle p}p must be a multiple of n−1{displaystyle n-1}n-1).

1a⋅b=1a(a+b)+1b(a+b){displaystyle {frac {1}{acdot b}}={frac {1}{a(a+b)}}+{frac {1}{b(a+b)}}}{displaystyle {frac {1}{acdot b}}={frac {1}{a(a+b)}}+{frac {1}{b(a+b)}}}


  • To express a fraction p/q{displaystyle p/q}p/q as the sum of two other fractions with given numerators a{displaystyle a}a and b{displaystyle b}b (GSS kalāsavarṇa 87, example in 88):[13]


pq=aai+bp⋅qi+bai+bp⋅qi⋅i{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}}}}{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 i{displaystyle i}i is to be chosen such that p{displaystyle p}p divides ai+b{displaystyle ai+b}{displaystyle ai+b}

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



See also


  • List of Indian mathematicians


Notes





  1. ^ ab Pingree 1970.


  2. ^ O'Connor & Robertson 2000.


  3. ^ ab Tabak 2009, p. 42.


  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


  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. ^ abcdefghi 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..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


  • Selin, Helaine (2008), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer, 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., Studies in the History of the Exact Sciences in Honour of David Pingree, Brill, ISBN 9004132023, ISSN 0169-8729










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