MAHVRA (MATHEMATICIAN)HELP IMPROVE THIS ARTICLESourced from World Heritage Encyclopedia licensed underCC BY-SA 3.0Help to improve this article, make contributions at theCitational Source1158573MAHVRA (MATHEMATICIAN)Mahvra(orMahaviracharya, "Mahavira the Teacher") was a 9th-centuryJainmathematicianfromMysore,India.[1][2][3]He was the author ofGaitasrasangraha(orGanita Sara Samgraha, c. 850), which revised theBrhmasphuasiddhnta.[1]He was patronised by theRashtrakutakingAmoghavarsha.[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. There are several points worth noting about the work of Mahavira. 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]Mahvra's eminence spread in all South India and his books proved inspirational to other mathematicians inSouthern India.[8]It was translated intoTelugu languagebyPavuluri MallanaasSaar Sangraha Ganitam.[9]He discovered algebraic identities like a3=a(a+b)(a-b) +b2(a-b) + b2.[3]He also found out the formula forrCnas [n(n-1)(n-2)...(n-r+1)]/r(r-1)(r-2)...2*1.[10]He devised formula which approximated 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 rootof anegative numberdid not exist.[12]RULES FOR DECOMPOSING FRACTIONSMahvra'sGaita-sra-sagrahagave systematic rules for expressing a fraction as thesum of unit fractions.[13]This follows the use of unit fractions inIndian mathematicsin the Vedic period, and theulba Stras' giving an approximation of 2 equivalent to1 + \tfrac13 + \tfrac1{3\cdot4} - \tfrac1{3\cdot4\cdot34}.[13]In theGaita-sra-sagraha(GSS), the second section of the chapter on arithmetic is namedkal-savara-vyavahra(lit. "the operation of the reduction of fractions"). In this, thebhgajtisection (verses 5598) gives rules for the following:[13] To express 1 as the sum ofnunit fractions (GSSkalsavara75, examples in 76):[13]rpakarn rpdys triguit har kramaa /

dvidvitryabhyastv dimacaramau phale rpe //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 = \frac1{1 \cdot 2} + \frac1{3} + \frac1{3^2} + \dots + \frac1{3^{n-2}} + \frac1{\frac23 \cdot 3^{n-1}} To express 1 as the sum of an odd number of unit fractions (GSSkalsavara77):[13]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} To express a unit fraction1/qas the sum ofnother fractions with given numeratorsa_1, a_2, \dots, a_n(GSSkalsavara78, examples in 79):\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})} To express any fractionp/qas a sum of unit fractions (GSSkalsavara80, examples in 81):[13]Choose an integerisuch that\tfrac{q+i}{p}is an integerr, then write\frac{p}{q} = \frac{1}{r} + \frac{i}{r \cdot q}and repeat the process for the second term, recursively. (Note that ifiis always chosen to be thesmallestsuch integer, this is identical to thegreedy algorithm for Egyptian fractions.) To express a unit fraction as the sum of two other unit fractions (GSSkalsavara85, example in 86):[13]\frac1{n} = \frac1{p\cdot n} + \frac1{\frac{p\cdot n}{n-1}}wherepis to be chosen such that\frac{p\cdot n}{n-1}is an integer (for whichpmust be a multiple ofn-1).\frac1{a\cdot b} = \frac1{a(a+b)} + \frac1{b(a+b)} To express a fractionp/qas the sum of two other fractions with given numeratorsaandb(GSSkalsavara87, example in 88):[13]\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}}whereiis to be chosen such thatpdividesai + bSome further rules were given in theGaita-kaumudiofNryaain the 14th century.[13]Mahvra, a 9th-century Indian mathematician, was the first to state that square roots of negative numbers don't exist.[15]

NewMahvra(orMahaviracharya, "Mahavira the Teacher") was a 9th-centuryJainmathematicianfromMysore,India.[1][2][3]He was the author ofGaitasrasangraha(orGanita Sara Samgraha, c. 850), which revised theBrhmasphuasiddhnta.[1]He was patronised by theRashtrakutakingAmoghavarsha.[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]Mahvra's eminence spread in all South India and his books proved inspirational to other mathematicians inSouthern India.[8]It was translated intoTelugu languagebyPavuluri MallanaasSaar 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 fornCras [n(n-1)(n-2)...(n-r+1)]/r(r-1)(r-2)...2*1.[10]He devised formula which approximated 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 rootof anegative numberdid not exist.[12]Contents[hide] 1Rules for decomposing fractions 2Notes 3See also 4ReferencesRules for decomposing fractions[edit]Mahvra'sGaita-sra-sagrahagave systematic rules for expressing a fraction as thesum of unit fractions.[13]This follows the use of unit fractions inIndian mathematicsin the Vedic period, and theulba Stras' giving an approximation of 2 equivalent to.[13]In theGaita-sra-sagraha(GSS), the second section of the chapter on arithmetic is namedkal-savara-vyavahra(lit. "the operation of the reduction of fractions"). In this, thebhgajtisection (verses 5598) gives rules for the following:[13] To express 1 as the sum ofnunit fractions (GSSkalsavara75, examples in 76):[13]rpakarn rpdys triguit har kramaa /dvidvitryabhyastv dimacaramau phale rpe //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].

To express 1 as the sum of an odd number of unit fractions (GSSkalsavara77):[13]

To express a unit fractionas the sum ofnother fractions with given numerators(GSSkalsavara78, examples in 79):

To express any fractionas a sum of unit fractions (GSSkalsavara80, examples in 81):[13]Choose an integerisuch thatis an integerr, then write

and repeat the process for the second term, recursively. (Note that ifiis always chosen to be thesmallestsuch integer, this is identical to thegreedy algorithm for Egyptian fractions.) To express a unit fraction as the sum of two other unit fractions (GSSkalsavara85, example in 86):[13]whereis to be chosen such thatis an integer (for whichmust be a multiple of).

To express a fractionas the sum of two other fractions with given numeratorsand(GSSkalsavara87, example in 88):[13]whereis to be chosen such thatdividesSome further rules were given in theGaita-kaumudiofNryaain the 14th century.[13]Mahavira(orMahaviracharyameaning Mahavira the Teacher) was of the Jaina religion and was familiar with Jaina mathematics. He worked in Mysore in southern Indian where he was a member of a school of mathematics. If he was not born in Mysore then it is very likely that he was born close to this town in the same region of India. We have essentially no other biographical details although we can gain just a little of his personality from the acknowledgement he gives in the introduction to his only known work, see below. However Jain in [10] mentions six other works which he credits to Mahavira and he emphasises the need for further research into identifying the complete list of his works.The only known book by Mahavira isGanita Sara Samgraha, dated 850 AD, which was designed as an updating ofBrahmagupta's book. Filliozat writes [6]:-This book deals with the teaching ofBrahmaguptabut contains both simplifications and additional information. ... Although like all Indian versified texts, it is extremely condensed, this work, from a pedagogical point of view, has a significant advantage over earlier texts.It consisted of nine chapters and included all mathematical knowledge of mid-ninth century India. It provides us with the bulk of knowledge which we have of Jaina mathematics and it can be seen as in some sense providing an account of the work of those who developed this mathematics. There were many Indian mathematicians before the time of Mahavira but, perhaps surprisingly, their work on mathematics is always contained in texts which discuss other topics such as astronomy. TheGanita Sara Samgrahaby Mahavira is the earliest Indian text which we possess which is devoted entirely to mathematics.In the introduction to the work Mahavira paid tribute to the mathematicians whose work formed the basis of his book. These mathematicians includedAryabhata I,Bhaskara I, andBrahmagupta. Mahavira writes:-With the help of the accomplished holy sages, who are worthy to be worshipped by the lords of the world ... I glean from the great ocean of the knowledge of numbers a little of its essence, in the manner in which gems are picked from the sea, gold from the stony rock and the pearl from the oyster shell; and I give out according to the power of my intelligence, the Sara Samgraha, a small work on arithmetic, which is however not small in importance.The nine chapters of theGanita Sara Samgrahaare:1. Terminology2. Arithmetical operations3. Operations involving fractions4. Miscellaneous operations5. Operations involving the rule of three6. Mixed operations7. Operations relating to the calculations of areas8. Operations relating to excavations9. Operations relating to shadowsThroughout the work a place-value system with nine numerals is used or sometimes Sanskrit numeral symbols are used. Of interest in Chapter 1 regarding the development of a place-value number system is Mahavira's description of the number 12345654321 which he obtains after a calculation. He describes the number as:-... beginning with one which then grows until it reaches six, then decreases in reverse order.Notice that this wording makes sense to us using a place-value system but would not make sense in other systems. It is a clear indication that Mahavira is at home with the place-value number system.Among topics Mahavira discussed in his treatise was operations with fractions including methods to decompose integers and fractions into unit fractions. For example2/17 = 1/12 + 1/51 + 1/68.He examined methods of squaring numbers which, although a special case of multiplying two numbers, can be computed using special methods. He also discussed integer solutions of first degree indeterminate equation by a method called kuttaka. The kuttaka (or the "pulveriser") method is based on the use of the Euclidean algorithm but the method of solution also resembles the continued fraction process ofEulergiven in 1764. The work kuttaka, which occurs in many of the treatises of Indian mathematicians of the classical period, has taken on the more general meaning of "algebra".An example of a problem given in theGanita Sara Samgrahawhich leads to indeterminate linear equations is the following:Three merchants find a purse lying in the road. One merchant says "If I keep the purse, I shall have twice as much money as the two of you together". "Give me the purse and I shall have three times as much" said the second merchant. The third merchant said "I shall be much better off than either of you if I keep the purse, I shall have five times as much as the two of you together". How much money is in the purse? How much money does each merchant have?If the first merchant hasx, the secondy, the thirdzandpis the amount in the purse thenp+x= 2(y+z),p+y= 3(x+z),p+z= 5(x+y).There is no unique solution but the smallest solution in positive integers isp= 15,x= 1,y= 3,z= 5. Any solution in positive integers is a multiple of this solution as Mahavira claims.Mahavira gave special rules for the use of permutations and combinations which was a topic of special interest in Jaina mathematics. He also described a process for calculating the volume of a sphere and one for calculating the cube root of a number. He looked at some geometrical results including right-angled triangles with rational sides, see for example [4].Mahavira also attempts to solve certain mathematical problems which had not been studied by other Indian mathematicians. For example, he gave an approximate formula for the area and the perimeter of an ellipse. In [8] Hayashi writes:-The formulas for a conch-like figure have so far been found only in the works of Mahavira and Narayana.It is reasonable to ask what a "conch-like figure" is. It is two unequal semicircles (with diametersABandBC) stuck together along their diameters. Although it might be reasonable to suppose that the perimeter might be obtained by considering the semicircles, Hayashi claims that the formulae obtained:-... were most probably obtained not from the two semicircles AB and BC.