ARYABHATA THE INDIAN MATHEMATICIAN

Aryabhata

Born- 476 AD in Kusumpur Patna

Died -550 AD

Contribution:- Gave exact value of Pi up to 4 decimal places, formula for Area and Volume of    

                        Sphere, Trigonometry table of Sine etc .

Aryabhata , the greatest mathematical genius of India was born in Kusumpur now in Patna in Bihar around 476 AD at the time when Patna was the capital of Gupta Empire. Patna was the major centre of learning.  The exact date and timing of his birth  is not confirmed but from the sloka of Aryabhatiya it is evident that he was born in 476 AD.  He writes that he was twenty three years of age when he wrote Aryabhatiya which he finished in 499.

“k”V~;Cnkuka “kf”V;Znk O;rhrkL=;Üp~ ;qxiknk% A

«;kf/kdk foa’kfrjCnkLrnsg ee tUeuks·rhrk %AA

Aryabhata had written at least three books on astronomy but the surviving one is the Aryabhata’s masterpiece Aryabhatiya which is a small astronomical treatise written in 118 verses giving a summary of Hindu mathematics up to that time. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry and spherical trigonometry. It also contains continued fractions, quadratic equations, sums of power series and a table of sines.

Aryabhata also invented a numeral system for representing numbers that consists of giving numerical 90, 100. The higher numbers are denoted by these consonants followed by a vowel to obtain 100, 10000,  .... In fact the system allows numbers up to 10¹⁸ to be represented with an alphabetical notation.

oxkZ{kjkf.k oxsZ·oxsZ·oxkZ{kjkf.k dkr~³~ekS ;% A

[kf}uods Lojk uo oxsZ·oxsZ uokUR;oxsZ ok AA

In the Devnagri alphabets, the twenty five varghiya vyanjannas (calssified consonants) from k to m constituting the five vargas (calsses) viz., ka- varga (d &oxZ) etc—represents the number 1 to 25 as shown in the given table---

Varga	Letters representing numbers
d oxZ	d 1	[k 2	Xk 3	?k 4	³  5
Pk oxZ	Pk  6	N 7	Tk 8	>  9	´ 10
V oxZ	V  11	B 12	M 13	< 14	.k 15
r oxZ	r  16	Fk 17	n 18	/k 19	Uk 20
i oxZ	i  21	Q 22	c 23	Hk 24	e 25

Ifrah argues that ---it is extremely likely that Aryabhata knew the sign for zero and the numerals of the place value system. This supposition is based on the following two facts: first, the invention of his alphabetical counting system would have been impossible without zero or the place-value system; secondly, he carries out calculations on square and cubic roots which are impossible if the numbers in question are not written according to the place-value system and zero.

Aryabhata gave an accurate approximation for pi

prqjf/kda ‘kre”Vxq.ka }k”kf”VLrFkk lglzk.kke~ A

v;qr};fo”dEHkL;kléks o`rifj.kkga AA

He wrote in the Aryabhatiya the following:-

Add four to one hundred, multiply by eight and then add sixty-two thousand. the result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given.

This gives pi = ⁶²⁸³²/₂₀₀₀₀ = 3.1416 which is a surprisingly accurate value. 

He gave a table of sines calculating the approximate values at intervals of 90/24 = 3 45'. In order to do this he used a formula for sin(n+1)x - sin nx in terms of sin nx and sin (n-1)x.

Aryabhata gives a systematic treatment of the position of the planets in space. He gave 62832 miles as the circumference of the earth, which is an excellent approximation. He believed that the apparent rotation of the heavens was due to the axial rotation of the Earth. He correctly explains the causes of eclipses of the Sun and the Moon. The Indian belief up to that time was that eclipses were caused by a demon called Rahu. His value for the length of the year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since the true value is less than 365 days 6 hours.

For the system of equationons of the form by = ax + c and by = ax - c, where a, b, care integers, Aryabhata used the kuttaka method. The word kuttaka means "to pulverise" and the method consisted of breaking the problem down into new problems where the coefficients became smaller and smaller with each step. 

The importance of Aryabhata in Indian mathematical world can be summed up with the following quotes of Bhaskaracharya-

 Aryabhata is the master who, after reaching the furthest shores and plumbing the inmost depths of the sea of ultimate knowledge of mathematics, kinematics and spherics, handed over the three sciences to the learned world.