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Contribution of aryabhatta in mathematics

Number notation

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.

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 1st 25 varga letters.

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

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.

Algebra

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.

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

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

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

Geometry

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.

Trigonometry

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.

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

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.

Contribution of Varaha mihira in mathematics

Varahamihira (505 – 587) was an Indian astrologer whose main work was a treatise on mathematical astronomy which summarised earlier astronomical treatises. He discovered a version of Pascal’s triangle and worked on magic squares. He was aware of gravity over a millennium before Isaac Newton.

Varahamihira worked as one of the Navaratnas for Chandragupta Vikramaditya. His book Pancasiddhantika (or Pancha-Siddhantika, The Five Astronomical Canons) dated 575 AD gives us information about older Indian texts which are now lost. The work is a treatise on mathematical astronomy and it summarises five earlier astronomical treatises, namely the Surya, Romaka, Paulisa, Vasistha and Paitamaha siddhantas.

Varahamihira is said to have origins from Eastern Iran from a sect known as Maga Brahmins.(Quote: Ramesh Chitor). In more ways than one, the Surya Siddhanta or Treatise on Sun hints that Mihira was from Iran as Iran was the only South Asian country following the practice of SUN worship. Varaha was a name coined by Vikramaditya- king of Ujjain. Mihira(meaning “friend” in Persian)accurately predicted death of Vikramidtya’s son during the 18th year. The entire army, intelligence and the king could not save this fatal incident. This will remain as the greatest astrological prediction ever made by Mihira. VarahaMihira’s painting can be found in the Indian Parliament alongside Aryabhatta.

Some important trigonometric results attributed to Varahamihira