Mahāvīra (mathematician)

Mahāvīra (or Mahaviracharya, "Mahavira the Teacher") was a 9th-century Indian

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]

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 ${\displaystyle 1+{\tfrac {1}{3}}+{\tfrac {1}{3\cdot 4}}-{\tfrac {1}{3\cdot 4\cdot 34}}}$.^[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]

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].

${\displaystyle 1={\frac {1}{1\cdot 2}}+{\frac {1}{3}}+{\frac {1}{3^{2}}}+\dots +{\frac {1}{3^{n-2}}}+{\frac {1}{{\frac {2}{3}}\cdot 3^{n-1}}}}$

• To express 1 as the sum of an odd number of unit fractions (GSS kalāsavarṇa 77):^[13]

${\displaystyle 1={\frac {1}{2\cdot 3\cdot 1/2}}+{\frac {1}{3\cdot 4\cdot 1/2}}+\dots +{\frac {1}{(2n-1)\cdot 2n\cdot 1/2}}+{\frac {1}{2n\cdot 1/2}}}$

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

${\displaystyle {\frac {1}{q}}={\frac {a_{1}}{q(q+a_{1})}}+{\frac {a_{2}}{(q+a_{1})(q+a_{1}+a_{2})}}+\dots +{\frac {a_{n-1}}{(q+a_{1}+\dots +a_{n-2})(q+a_{1}+\dots +a_{n-1})}}+{\frac {a_{n}}{a_{n}(q+a_{1}+\dots +a_{n-1})}}}$

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

Choose an integer i such that ${\displaystyle {\tfrac {q+i}{p}}}$ is an integer r, then write

${\displaystyle {\frac {p}{q}}={\frac {1}{r}}+{\frac {i}{r\cdot q}}}$

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]

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

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

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

${\displaystyle {\frac {p}{q}}={\frac {a}{{\frac {ai+b}{p}}\cdot {\frac {q}{i}}}}+{\frac {b}{{\frac {ai+b}{p}}\cdot {\frac {q}{i}}\cdot {i}}}}$ where ${\displaystyle i}$ is to be chosen such that ${\displaystyle p}$ divides ${\displaystyle ai+b}$

Some further rules were given in the Gaṇita-kaumudi of

• List of Indian mathematicians

Notes

References

• Bibhutibhusan Datta and Avadhesh Narayan Singh (1962).
  History of Hindu Mathematics: A Source Book
  .
• 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,
  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;
  ISSN 0169-8729