Born

476 AD

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) Aryasiddhanta. 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).125 are denoted by 1^{st} 25 varga letters.
Ø Placevalue:
Aryabhatta was familiar with the placevalue 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 =axc,
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}=a^{2}+2ab+b^{2}and
ab={(a+b)^{2}(a^{2}b^{2})}/2
Ø He
has given the following formula in aryabhatia
1^{2}+2^{2}+3^{2}++n^{2}=n(n+1)(2n+1)/6
1^{3}+2^{3}+3^{3}++n^{3}
= (1+2+3++)^{2}= {n^{2}(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+{(n1)/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 rightangled 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=c^{2 }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 (n1) x.
Ø Versine:
He introduced the versine (versin = 1cosine) 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 1941975 as aryabhatta,
after him.
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Please give the method given by aryabhat for extracting sq root and cube root
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