Mahāvīra (mathematician)

He discovered algebraic identities like a³ = a (a + b) (a − b) + b² (a − b) + b³. 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]. 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. He asserted that the square root of a negative number does not exist. Arithmetic operations utilized in his works like Gaṇita-sāra-saṅgraha(Ganita Sara Sangraha) uses decimal place-value system and include the use of zero. However, he erroneously states that a number divided by zero remains unchanged.

Rules for decomposing fractions

Mahāvīra's Gaṇita-sāra-saṅgraha gave systematic rules for expressing a fraction as the sum of unit fractions. 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 ${\displaystyle 1+\frac{1}{3}+\frac{1}{3\cdot 4}-\frac{1}{3\cdot 4\cdot 34}}$.

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:

• To express 1 as the sum of n unit fractions (GSS kalāsavarṇa 75, examples in 76):

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].

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

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

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

• 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):

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

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

Choose an integer i such that ${\displaystyle \frac{q+i}{p}}$ 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):

${\displaystyle \frac{1}{n}=\frac{1}{p\cdot n}+\frac{1}{\frac{p\cdot n}{n-1}}}$ 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):

${\displaystyle \frac{p}{q}=\frac{a}{\frac{ai+b}{p}\cdot \frac{q}{i}}+\frac{b}{\frac{ai+b}{p}\cdot \frac{q}{i}\cdot i}}$ 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 Nārāyaṇa in the 14th century.