Mahāvīra (mathematician)

Mahāvīra
Born	India
Occupation	Mathematician
Religion	Jainism

Mahāvīra (or Mahaviracharya, "Mahavira the Teacher") was a 9th-century Jain mathematician from Mysore, 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 ⁿC_r as [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 the square root of a negative number did not exist.^[12]

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 $1 + \tfrac13 + \tfrac1{3\cdot4} - \tfrac1{3\cdot4\cdot34}$.^[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]

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].

$1 = \frac1{1 \cdot 2} + \frac1{3} + \frac1{3^2} + \dots + \frac1{3^{n-2}} + \frac1{\frac23 \cdot 3^{n-1}}$

• To express 1 as the sum of an odd number of unit fractions (GSS kalāsavarṇa 77):^[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 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):

$\frac1q = \frac{a_1}{q(q+a_1)} + \frac{a_2}{(q+a_1)(q+a_1+a_2)} + \dots + \frac{a_{n-1}}{q+a_1+\dots+a_{n-2})(q+a_1+\dots+a_{n-1})} + \frac{a_n}{a_n(q+a_1+\dots+a_{n-1})}$

• To express any fraction $p/q$ as a sum of unit fractions (GSS kalāsavarṇa 80, examples in 81):^[13]

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):^[13]

$\frac1{n} = \frac1{p\cdot n} + \frac1{\frac{p\cdot n}{n-1}}$ 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 $n-1$).

$\frac1{a\cdot b} = \frac1{a(a+b)} + \frac1{b(a+b)}$

• 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):^[13]

$\frac{p}{q} = \frac{a}{\frac{ai+b}{p}\cdot\frac{q}{i}} + \frac{b}{\frac{ai+b}{p} \cdot \frac{q}{i} \cdot{i}}$ 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.^[13]

See also

List of Indian mathematicians

Notes

1. Pingree 1970.
2. O'Connor & Robertson 2000.
3. 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. 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.
• 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