महावीराचार्य Mahaviracharya - The First Mathematician in the world to generalize Permutations and Combinations formula

महावीराचार्य

Mahaviracharya

Mahaviracharya was a great Bhartiya Jain Mathematician of the Ninth-Century. He was born in kundalpur , in India. He was patronised by the  राष्ट्रकूट Rashtrakuta king  अमोघवर्ष प्रथमAmoghavarsha The First. He did very remarkable work on  क्रमचय-संचय combinatorics and was the first in the world to present the generalized formula to calculate the number of क्रमचयों permutations and संचयों combinations. He authored गणितसारसंग्रहाGaṇitasārasan̄graha (Ganita Sara Sangraha) in 850AD in which many topics of algebra  बीजगणित  and ज्यामिति  geometry are discussed.    पावुलूरि मल्लन Pavuluri Mallan translated this treatise into Telugu under the name 'सारसंग्रह गणितम्' 'Sarasanghara Gnatham'. He separated astrology from mathematics. It is the earliest Indian text entirely devoted to mathematics. 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. 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. His eminence spread throughout South India and his books proved inspirational to other mathematicians in Southern India.

He discovered Important formulas and Identities

a^³ = a (a + b) (a − b) + b^² (a − b) + b^³

ⁿC =  [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.

He named Big Numbers 

Numbers	Names	Numbers	Names	Numbers	Names	Numbers	Names
10^1	दशं	10^2	शतं	10^3	सहस्रं	10^4	दशसहस्रं
10^5	लक्षं	10^6	दशलक्षं	10^7	कोटि	10^8	दशकोटि
10^9	शतकोटि	10^10	अर्बुदं	10^11	न्यर्बुदं	10^12	खर्‌व्वं
10^13	महाखर्‌व्वं	10^14	पद्मं	10^15	महापद्मं	10^16	क्षोणि
10^17	महाक्षोणि	10^18	शंखं	10^19	महाशंखं	10^20	क्षिति
10^21	महाक्षिति	10^22	क्षोभं	10^23	महाक्षोभं

[Names are , obviously , in Sanskrit.]

He gave Rules for Decomposing Fractions

His Gaṇita-sāra-saṅgraha गणितसारसंग्रहा 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 .

The following rule is given to express 1 as the sum of unit fractions.

रूपांशकराशीनां रूपाद्यास्त्रिगुणिता हराः क्रमशः।

द्विद्वित्र्यंशाभ्यस्ताव आदिमचरमौ फले रूपे ॥ (गतिणसारसंग्रह कलासवर्ण ७५)

Translation to Hindi :  जब फल  १ हो तो १ अंश वाले भिन्न, जिनके हर १ से शुरू होकर क्रमशः ३ से गुणित होते जायेंगे। प्रथम और अन्तिम को (क्रमशः) २ तथा २/३ से गुणा किया जायेगा।

Translation to English : When the result is 1, then fractions with 1 degree, whose denominators start at 1 and will be multiplied by 3 respectively. The first and last will be multiplied by (respectively) 2 and 2/3.    

• To express 1 as the sum of an odd number of unit fractions (Gaṇita-sāra-saṅgraha  गणितसारसंग्रहा kalāsavarṇa कलासवर्ण: 77):

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

• To express a unit fraction ${\displaystyle 1/q}$ as the sum of n other fractions with given numerators ${\displaystyle a_{1},a_{2},\dots ,a_{n}}$ (Gaṇita-sāra-saṅgraha  गणितसारसंग्रहा kalāsavarṇa कलासवर्ण: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 (Gaṇita-sāra-saṅgraha  गणितसारसंग्रहा kalāsavarṇa कलासवर्ण:80, examples in 81):

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.

• To express a unit fraction as the sum of two other unit fractions (Gaṇita-sāra-saṅgraha  गणितसारसंग्रहा kalāsavarṇa कलासवर्ण: 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 ${\displaystyle b}$ (Gaṇita-sāra-saṅgraha  गणितसारसंग्रहा kalāsavarṇa  कलासवर्ण: , 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 ${\displaystyle ai+b}$

He Gave formula for high order equations

Mahavir presented the solution of the following types of equations of n degree and higher order. The name of the second chapter of  गणितसारसंग्रह Ganitsaarsangraha is कलासवर्ण: Kala-Savarna-Varana (the operation of the reduction of fractions). 

 ${\displaystyle \ ax^{n}=q}$ and 

He gave Formula of cyclic quadrilateral

Aditya and (before him) Brahmagupta had highlighted the merits of cyclic quadrilaterals. After this, Mahavira gave equations to find the length of the sides and diagonals of cyclic quadrilaterals.

If a, b, c, d are the sides of a cyclic quadrilateral and the lengths of its diagonals are x and y, then

$$
{\displaystyle \ x={\sqrt {{\frac {ad+bc}{ab+cd}}(ac+bd)}}}

and

${\displaystyle y={\sqrt {{\frac {ab+cd}{ad+bc}}(ac+bd)}}}$

Hence , ${\displaystyle \ xy=ac+bd}$