MODULE 1 - ARYABHATTA

ARYABHATTA (476 - 550)

Born	476 AD (1707-04-15)
Died	550 AD
Residence	Kusumapura or Pataliputra (Patna)
Nationality	Indian
Fields	Mathematics, Astronomy
Institutions	Nalanda University

Aryabhatta was the first of great Hindu mathematician. He is also known as Aryabhatta I. He lived at Kusumapura or Pataliputra in ancient Magadhar or modern Patna.  He was born in 476 AD.

 At the age of 23 years Aryabhatt wrote two books on astronomy (1) Aryabhatiya (2) Arya-siddhanta. The aryabhatt deals with both mathematics and astronomy. It contains 121 stanzas in all. Aryabhatt is divided into 4 chapters called Pada (section)

v  Pada 1 – GitikaPada – 13 stanzas of basis definition of important astronomical parameters and tables.

v  Pada 2 – Ganita Pada – 33 stanzas deals with mathematics. The topics are geometrical figures with their properties and mensurations, series, linear and quadratic equations, methods for extracting the square roots, the cube roots etc.

v  Pada 3 – Kalakriya Pada – 25 stanzas deals with the true position of sun, moon and planets.

v  Pada 4 – Gola Pada – 50 stanzas deals with the motion of sun, moon and planets on the celestial sphere.

ARYABHATTA CONTRIBUTIONS TO MATHEMATICS

NUMBER NOTATION

Ø  Numerical values: he made a notation system in which digits are denoted with the help of alphabet numerals e.g., 1 = ka, 2 = Kha, etc.

Aryabhatta assigned numerical values to the 33 consonants of the Indian alphabet to represent 1,2,3…25,30,40,50,60,70,80,90,100.

Ø  Notation system: He invented a notation system consisting of alphabet numerals Digits were denoted by alphabet numerals. In this system devanagiri script contain varga letters (consonants) and avarga letters (vowels).1-25 are denoted by 1^st 25 varga letters.

Ø  Place-value: Aryabhatta was familiar with the place-value system.

Ø  He knew numeral symbols and the sign for zero

Ø  Square root & cube root: His calculations on square root and cube root would not have been possible without the knowledge of place values system and zero. He has given methods of extracting square root cube root along with their explanation.

Ø  Interest: He formulated for the first time in India the formula for interest, time and other related ones, in the problems of interest.

ALGEBRA

Ø  Integer solutions: Aryabhatta was the first one to explore integer solutions to the equations of the form by =ax+c and by =ax-c, where a,b,c are integers. He used kuttuka method to solve problems.

Ø  Indeterminate equations: He gave general solutions to linear indeterminate equations ax+by+c= 0 by the method of continued fraction.

Ø  Identities: He had dealt with identities like (a+b)²=a²+2ab+b²and ab={(a+b)²-(a²-b²)}/2

Ø  He has given the following formula in aryabhatia

1²+2²+3²+---------+n²=n(n+1)(2n+1)/6

1³+2³+3³+---------+n³ = (1+2+3+------------+)²= {n²(n+1)²}/4

Ø  Algebraic quantities: He has given the method of addition, subtraction, multiplication of simple and compound algebraic quantities

Ø  Arithmetic series: He was given a formula for summing up of the arithmetic series after the P^th term The rule is S= n[a+{(n-1)/2+p} d]

S=(a+1) n/2

GEOMETRY

Ø  Discover the P  Value : The credit for discovering the exact values P may be ascribed to the celebrated mathematician Aryabhatta.

Rule: Add 4 to 100, multiply by 8, add 62000. 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 P =62832/20000=3.1416. Which is an accurate value of P. Aryabhatta discovered this value independently and also realized that P is an irrational number

Ø  Pythagorean Theorem: The Pythagorean theorem is stated as follows in his work “the square of the Bhuja (base) plus the square of the koti (perpendicular) is the square of the Karna”

(Buja and koti are the sides of a right-angled triangle. The Karna is the hypotenuse)

Ø  Circle Theorem: He has postulated a theorem relating to circle as follows “In a circle the product of two Saras is the square of the half chord of the two arcs” i.e. a*b=c² where c is half the chord and the saras or arrows are the segments of a diameter which bisect any chord.

Ø  Formula: Aryabhatta gives formulae for the areas of a triangle, square, rectangle, rhombus, circle etc.

TRIGONOMETRY

Ø  Sine Table: Aryabhatta gave a table of sines for calculating the approximate values at intervals of 90/24 = 3 45’. This was done using the formula for

sin (n+1)x -  sin nx in terms of sin nx and sin (n-1) x.

Ø  Versine: He introduced the versine (versin = 1-cosine) into trigonometry.

ASTRONOMY

Ø  Earth: Aryabhatta gave the circumference of the earth as 4 967 yojanas and its diameter as 1 5811/24 yojanas. Since1 yojana =5miles this gives the circumference as 24,835 miles, which is an excellent approximation to the currently accepted value of 24,902 miles.

Ø  He believes that the orbits of the planets are ellipses. He correctly explains the caused of eclipses of the Sun and the Moon.

Ø  Length of year: 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 and 6 hours.

Aryabhatta was one of those ancient scholars of India who is hardly surpassed by any one else of his time in his treatise on mathematics and astronomy. In appreciation of his great contributions to mathematics and astronomy, the government of India named the first satellite sent into space on 19-4-1975 as aryabhatta, after him.