Mahāvīra (mathematician) - Revision history

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	{{Use Indian English\|date=December 2020}}		{{Use Indian English\|date=December 2020}}
			{{Infobox religious biography
			\| name = Mahāvīrā (Mahāvīrāchārya)
			\| religion = [[Jainism]]
			\| sect = [[Digambara]]
			\| era = 9th century CE
			\| birth_place = Karnataka, [[Rashtrakuta]] Kingdom
			\| works = "Gaṇita Sāra Saṅgraha"
			\| occupation = Mathematician, Philosopher
			\| dynasty = [[Rashtrakuta]]
			\| image = Mahāvīrā (Mathematician) 25-07-26 22-58-50-463~2.png
			\| caption = Jain Matheamatician Mahāvīrā (Mahāvīrāchārya)
			}}
			'''Mahāvīra''' (or '''Mahaviracharya''', "Mahavira the Teacher") was a 9th-century Indian [[Jain]] [[mathematician]] possibly born in [[Mysore]], in [[India]].{{sfn\|Pingree\|1970}}{{sfn\|O'Connor\|Robertson\|2000}}{{sfn\|Tabak\|2009\|p=42}} He authored ''[[Gaṇita-sāra-saṅgraha]]'' (''Ganita Sara Sangraha'') or the Compendium on the gist of Mathematics in 850 CE.{{sfn\|Puttaswamy\|2012\|p=231}} He was patronised by the [[Rashtrakuta]] emperor [[Amoghavarsha]].{{sfn\|Puttaswamy\|2012\|p=231}} He separated [[Jyotisha\|astrology]] from mathematics. It is the earliest Indian text entirely devoted to mathematics.<ref>The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the ... by Clifford A. Pickover: page 88</ref> 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.<ref>Algebra: Sets, Symbols, and the Language of Thought by John Tabak: p.43</ref> 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.<ref>Geometry in Ancient and Medieval India by T. A. Sarasvati Amma: page 122</ref> Mahāvīra's eminence spread throughout southern India and his books proved inspirational to other mathematicians in [[Southern India]].{{sfn\|Hayashi\|2013}} It was translated into the [[Telugu language]] by [[Pavuluri Mallana]] as ''Saara Sangraha Ganitamu''.<ref>Census of the Exact Sciences in Sanskrit by David Pingree: page 388</ref>

	'''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]].{{sfn\|Pingree\|1970}}{{sfn\|O'Connor\|Robertson\|2000}}{{sfn\|Tabak\|2009\|p=42}} He authored ''[[Gaṇitasārasan̄graha]]'' (''Ganita Sara Sangraha'') or the Compendium on the gist of Mathematics in 850 AD.{{sfn\|Puttaswamy\|2012\|p=231}} He was patronised by the [[Rashtrakuta]] king [[Amoghavarsha]].{{sfn\|Puttaswamy\|2012\|p=231}} He separated [[astrology]] from mathematics. It is the earliest Indian text entirely devoted to mathematics.<ref>The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the ... by Clifford A. Pickover: page 88</ref> 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.<ref>Algebra: Sets, Symbols, and the Language of Thought by John Tabak: p.43</ref> 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.<ref>Geometry in Ancient and Medieval India by T. A. Sarasvati Amma: page 122</ref> Mahāvīra's eminence spread throughout South India and his books proved inspirational to other mathematicians in [[Southern India]].{{sfn\|Hayashi\|2013}} It was translated into the [[Telugu language]] by [[Pavuluri Mallana]] as ''Saara Sangraha Ganitamu''.<ref>Census of the Exact Sciences in Sanskrit by David Pingree: page 388</ref>		He discovered algebraic identities like ''a''<sup>3</sup> = ''a'' (''a'' + ''b'') (''a'' − ''b'') + ''b''<sup>2</sup> (''a'' − ''b'') + ''b''<sup>3</sup>.{{sfn\|Tabak\|2009\|p=42}} 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].{{sfn\|Tabak\|2009\|p=43}} 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.{{sfn\|Krebs\|2004\|p=132}} He asserted that the [[square root]] of a [[negative number]] does not exist.{{sfn\|Selin\|2008\|p=1268}} Arithmetic operations utilized in his works like Gaṇita-sāra-saṅgraha(Ganita Sara Sangraha) uses [[Positional notation\|decimal place-value system]] and include the use of [[0\|zero]]. However, he erroneously states that a number divided by zero remains unchanged.<ref>{{Cite book \|last= \|first= \|url=http://archive.org/details/in.ernet.dli.2015.502083 \|title=A Concise History of Science in India (Eds.) D. M. Bose, S. N. Sen and B.V. Subbarayappa. \|date=1971-10-15 \|publisher=Indian National Science Academy \|pages=167 \|language=English}}</ref>

	He discovered algebraic identities like ''a''<sup>3</sup> = ''a'' (''a'' + ''b'') (''a'' − ''b'') + ''b''<sup>2</sup> (''a'' − ''b'') + ''b''<sup>3</sup>.{{sfn\|Tabak\|2009\|p=42}} 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].{{sfn\|Tabak\|2009\|p=43}} 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.{{sfn\|Krebs\|2004\|p=132}} He asserted that the [[square root]] of a [[negative number]] does not exist.{{sfn\|Selin\|2008\|p=1268}}

	==Rules for decomposing fractions==		==Rules for decomposing fractions==
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	* To express 1 as the sum of ''n'' unit fractions (GSS ''kalāsavarṇa'' 75, examples in 76):<ref name=k497/>		* To express 1 as the sum of ''n'' unit fractions (GSS ''kalāsavarṇa'' 75, examples in 76):<ref name=k497/>
	{{~~quote~~\|rūpāṃśakarāśīnāṃ rūpādyās triguṇitā harāḥ kramaśaḥ /<br/>		{{blockquote\|rūpāṃśakarāśīnāṃ rūpādyās triguṇitā harāḥ kramaśaḥ /<br/>
	dvidvitryaṃśābhyastāv ādimacaramau phale rūpe //}}		dvidvitryaṃśābhyastāv ādimacaramau phale rūpe //}}
	{{~~quote~~\|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].}}		{{blockquote\|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].}}
	:: <math> 1 = \frac1{1 \cdot 2} + \frac1{3} + \frac1{3^2} + \dots + \frac1{3^{n-2}} + \frac1{\frac23 \cdot 3^{n-1}} </math>		:: <math> 1 = \frac1{1 \cdot 2} + \frac1{3} + \frac1{3^2} + \dots + \frac1{3^{n-2}} + \frac1{\frac23 \cdot 3^{n-1}} </math>

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	==References==		==References==
			{{sfn whitelist \|CITEREFPingree1970}}
	*Bibhutibhusan Datta and Avadhesh Narayan Singh (1962). ''[[History of Hindu Mathematics: A Source Book]]''.		*Bibhutibhusan Datta and Avadhesh Narayan Singh (1962). ''[[History of Hindu Mathematics: A Source Book]]''.
	*{{DSB\|first=David\|last=Pingree\|authorlink=David Pingree\|title=Mahāvīra\|date=1970}} (Available, along with many other entries from other encyclopaedias for other Mahāvīra-s, [http://www.encyclopedia.com/doc/1G2-2830902766.html online].)		*{{DSB\|first=David\|last=Pingree\|authorlink=David Pingree\|title=Mahāvīra\|date=1970}} (Available, along with many other entries from other encyclopaedias for other Mahāvīra-s, [http://www.encyclopedia.com/doc/1G2-2830902766.html online].)
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	*{{MacTutor Biography\|id=Mahavira\|title=Mahavira\|date=2000}}		*{{MacTutor Biography\|id=Mahavira\|title=Mahavira\|date=2000}}
	*{{citation\|last=Tabak\|first=John\|title=Algebra: Sets, Symbols, and the Language of Thought\|url=https://books.google.com/books?id=h-zRieb7VbwC&pg=PA42\|year=2009\|publisher=Infobase Publishing\|isbn=978-0-8160-6875-3}}		*{{citation\|last=Tabak\|first=John\|title=Algebra: Sets, Symbols, and the Language of Thought\|url=https://books.google.com/books?id=h-zRieb7VbwC&pg=PA42\|year=2009\|publisher=Infobase Publishing\|isbn=978-0-8160-6875-3}}
	*{{citation\|last=Krebs\|first=Robert E.\|title=Groundbreaking Scientific Experiments, Inventions, and Discoveries of the Middle Ages and the Renaissance\|url=https://books.google.~~combooks~~?id=MTXdplfiz-cC&pg=PA131\|year=2004\|publisher=Greenwood Publishing Group\|isbn=978-0-313-32433-8}}		*{{citation\|last=Krebs\|first=Robert E.\|title=Groundbreaking Scientific Experiments, Inventions, and Discoveries of the Middle Ages and the Renaissance\|url=https://books.google.com/books?id=MTXdplfiz-cC&pg=PA131\|year=2004\|publisher=Greenwood Publishing Group\|isbn=978-0-313-32433-8}}
	*{{citation\|last=Puttaswamy\|first=T.K\|title=Mathematical Achievements of Pre-modern Indian Mathematicians\|url=https://books.google.com/books?id=DAPLaxw-53IC&pg=PA231\|year=2012\|publisher=Newnes\|isbn=978-0-12-397938-4}}		*{{citation\|last=Puttaswamy\|first=T.K\|title=Mathematical Achievements of Pre-modern Indian Mathematicians\|url=https://books.google.com/books?id=DAPLaxw-53IC&pg=PA231\|year=2012\|publisher=Newnes\|isbn=978-0-12-397938-4}}
	* {{citation \| last=Kusuba\|first=Takanori \| contribution=Indian Rules for the Decomposition of Fractions \| year=2004 \| title=Studies in the History of the Exact Sciences in Honour of David Pingree \| publisher=[[Brill Publishers\|Brill]] \| isbn=9004132023 \| issn=0169-8729 \| editor1=Charles Burnett \| editor2=Jan P. Hogendijk \| editor3=[[Kim Plofker]] \|display-editors = 3 \| editor4=Michio Yano}}		* {{citation \| last=Kusuba\|first=Takanori \| contribution=Indian Rules for the Decomposition of Fractions \| year=2004 \| title=Studies in the History of the Exact Sciences in Honour of David Pingree \| publisher=[[Brill Publishers\|Brill]] \| isbn=9004132023 \| issn=0169-8729 \| editor1=Charles Burnett \| editor2=Jan P. Hogendijk \| editor3=[[Kim Plofker]] \|display-editors = 3 \| editor4=Michio Yano}}
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	[[Category:Scholars from Karnataka]]		[[Category:Scholars from Karnataka]]
	[[Category:Acharyas]]		[[Category:Acharyas]]
			[[Category:Rashtrakuta people]]

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{{short description|9th-century Indian mathematician}}
{{Use dmy dates|date=December 2015}}
{{Use Indian English|date=December 2020}}

'''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]].{{sfn|Pingree|1970}}{{sfn|O'Connor|Robertson|2000}}{{sfn|Tabak|2009|p=42}} He authored ''[[Gaṇitasārasan̄graha]]'' (''Ganita Sara Sangraha'') or the Compendium on the gist of Mathematics in 850 AD.{{sfn|Puttaswamy|2012|p=231}} He was patronised by the [[Rashtrakuta]] king [[Amoghavarsha]].{{sfn|Puttaswamy|2012|p=231}} He separated [[astrology]] from mathematics. It is the earliest Indian text entirely devoted to mathematics.<ref>The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the ... by Clifford A. Pickover: page 88</ref> 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.<ref>Algebra: Sets, Symbols, and the Language of Thought by John Tabak: p.43</ref> 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.<ref>Geometry in Ancient and Medieval India by T. A. Sarasvati Amma: page 122</ref> Mahāvīra's eminence spread throughout South India and his books proved inspirational to other mathematicians in [[Southern India]].{{sfn|Hayashi|2013}} It was translated into the [[Telugu language]] by [[Pavuluri Mallana]] as ''Saara Sangraha Ganitamu''.<ref>Census of the Exact Sciences in Sanskrit by David Pingree: page 388</ref>

He discovered algebraic identities like ''a''3 = ''a'' (''a'' + ''b'') (''a'' − ''b'') + ''b''2 (''a'' − ''b'') + ''b''3.{{sfn|Tabak|2009|p=42}} He also found out the formula for ''n''C''r'' as [''n'' (''n'' − 1) (''n'' − 2) ... (''n'' − ''r'' + 1)] / [''r'' (''r'' − 1) (''r'' − 2) ... 2 * 1].{{sfn|Tabak|2009|p=43}} 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.{{sfn|Krebs|2004|p=132}} He asserted that the [[square root]] of a [[negative number]] does not exist.{{sfn|Selin|2008|p=1268}}

==Rules for decomposing fractions==
Mahāvīra's ''Gaṇita-sāra-saṅgraha'' gave systematic rules for expressing a fraction as the [[Egyptian fraction|sum of unit fractions]].<ref name=k497>{{Harvnb|Kusuba|2004|pp=497–516}}</ref> This follows the use of unit fractions in [[Indian mathematics]] in the Vedic period, and the [[Shulba Sutras|Śulba Sūtras]]' giving an approximation of {{radic|2}} equivalent to <math>1 + \tfrac13 + \tfrac1{3\cdot4} - \tfrac1{3\cdot4\cdot34}</math>.<ref name=k497/>

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:<ref name=k497/>

* To express 1 as the sum of ''n'' unit fractions (GSS ''kalāsavarṇa'' 75, examples in 76):<ref name=k497/>
{{quote|rūpāṃśakarāśīnāṃ rūpādyās triguṇitā harāḥ kramaśaḥ / 
dvidvitryaṃśābhyastāv ādimacaramau phale rūpe //}}
{{quote|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].}}
:: <math> 1 = \frac1{1 \cdot 2} + \frac1{3} + \frac1{3^2} + \dots + \frac1{3^{n-2}} + \frac1{\frac23 \cdot 3^{n-1}} </math>

* To express 1 as the sum of an odd number of unit fractions (GSS ''kalāsavarṇa'' 77):<ref name=k497/>
:: <math>1 = \frac1{2\cdot 3 \cdot 1/2} + \frac1{3 \cdot 4 \cdot 1/2} + \dots + \frac1{(2n-1) \cdot 2n \cdot 1/2} + \frac1{2n \cdot 1/2} </math>

* To express a unit fraction <math>1/q</math> as the sum of ''n'' other fractions with given numerators <math>a_1, a_2, \dots, a_n</math> (GSS ''kalāsavarṇa'' 78, examples in 79):
:: <math>\frac1q = \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})}</math>

* To express any fraction <math>p/q</math> as a sum of unit fractions (GSS ''kalāsavarṇa'' 80, examples in 81):<ref name=k497/>
: Choose an integer ''i'' such that <math>\tfrac{q+i}{p}</math> is an integer ''r'', then write
:: <math> \frac{p}{q} = \frac{1}{r} + \frac{i}{r \cdot q} </math>
: 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):<ref name=k497/>
:: <math>\frac1{n} = \frac1{p\cdot n} + \frac1{\frac{p\cdot n}{n-1}}</math> where <math>p</math> is to be chosen such that <math>\frac{p\cdot n}{n-1}</math> is an integer (for which <math>p</math> must be a multiple of <math>n-1</math>).
:: <math>\frac1{a\cdot b} = \frac1{a(a+b)} + \frac1{b(a+b)}</math>

* To express a fraction <math>p/q</math> as the sum of two other fractions with given numerators <math>a</math> and <math>b</math> (GSS ''kalāsavarṇa'' 87, example in 88):<ref name=k497/>
:: <math>\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}}</math> where <math>i</math> is to be chosen such that <math>p</math> divides <math>ai + b</math>

Some further rules were given in the ''Gaṇita-kaumudi'' of [[Narayana Pandit|Nārāyaṇa]] in the 14th century.<ref name=k497/>

==See also==
* [[List of Indian mathematicians]]

==Notes==
{{reflist|2}}

==References==
*Bibhutibhusan Datta and Avadhesh Narayan Singh (1962). ''[[History of Hindu Mathematics: A Source Book]]''.
*{{DSB|first=David|last=Pingree|authorlink=David Pingree|title=Mahāvīra|date=1970}} (Available, along with many other entries from other encyclopaedias for other Mahāvīra-s, [http://www.encyclopedia.com/doc/1G2-2830902766.html online].)
*{{citation|last=Selin|first=Helaine|title=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures|url=https://books.google.com/books?id=kt9DIY1g9HYC&pg=PA1268|year=2008|publisher=Springer|bibcode=2008ehst.book.....S|isbn=978-1-4020-4559-2}}
*{{citation|first=Takao|last=Hayashi|author-link=Takao Hayashi|title=Mahavira|encyclopedia=Encyclopædia Britannica|url=http://www.britannica.com/EBchecked/topic/853508/Mahavira|year=2013}}
*{{MacTutor Biography|id=Mahavira|title=Mahavira|date=2000}}
*{{citation|last=Tabak|first=John|title=Algebra: Sets, Symbols, and the Language of Thought|url=https://books.google.com/books?id=h-zRieb7VbwC&pg=PA42|year=2009|publisher=Infobase Publishing|isbn=978-0-8160-6875-3}}
*{{citation|last=Krebs|first=Robert E.|title=Groundbreaking Scientific Experiments, Inventions, and Discoveries of the Middle Ages and the Renaissance|url=https://books.google.combooks?id=MTXdplfiz-cC&pg=PA131|year=2004|publisher=Greenwood Publishing Group|isbn=978-0-313-32433-8}}
*{{citation|last=Puttaswamy|first=T.K|title=Mathematical Achievements of Pre-modern Indian Mathematicians|url=https://books.google.com/books?id=DAPLaxw-53IC&pg=PA231|year=2012|publisher=Newnes|isbn=978-0-12-397938-4}}
* {{citation | last=Kusuba|first=Takanori | contribution=Indian Rules for the Decomposition of Fractions | year=2004 | title=Studies in the History of the Exact Sciences in Honour of David Pingree | publisher=[[Brill Publishers|Brill]] | isbn=9004132023 | issn=0169-8729 | editor1=Charles Burnett | editor2=Jan P. Hogendijk | editor3=[[Kim Plofker]] |display-editors = 3 | editor4=Michio Yano}}


{{Indian mathematics}}

{{Authority control}}

{{DEFAULTSORT:Mahavira}}
[[Category:9th-century Indian mathematicians]]
[[Category:9th-century Indian Jains]]
[[Category:Scholars from Karnataka]]
[[Category:Acharyas]]