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Duration : c. 600 - c. 680

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Bhaskara I was an Indian mathematician of the 7th century, who probably lived between c.600- c.680. He was most likely the first to use a circle for the zero in the Hindu-Arabic decimal system, and while commenting on Aryabhata's work, he evaluated an extraordinary rational approximation of the sine function. There is very little information about Bhaskara's life. He is said to be born near Saurashtra in Gujarat and died in Ashmaka. He was educated by his father in astronomy. He is considered to be a follower of Aryabhata I and one of the most renowned scholars of Aryabhata's astronomical school. Bhaskara I wrote two treatises, the Mahabhaskariya and the Laghubhaskariya. He also wrote commentaries on the work of Aryabhata I entitled Aryabhatiyabhasya. The Mahabhaskariya comprises of eight chapters dealing with mathematical astronomy. The book deals with topics such as: the longitudes of the planets; association of the planets with each other and also with the bright stars; the lunar crescent; solar and lunar eclipses; and rising and setting of the planets. Bhaskara I suggested a formula which was astonishingly accurate value of Sine. The formula is: sin x = 16x (p - x)/[5p2 - 4x (p - x)]

Bhaskara I wrote the Aryabhatiyabhasya in 629,, which is a commentary on the Aryabhatiya written by Aryabhata I. Bhaskara I commented only on the 33 verses of Aryabhatiya which is about mathematical astronomy and discusses the problems of the first degree of indeterminate equations and trigonometric formula. While discussing about Aryabhatiya he discussed about cyclic quadrilateral. He was the first mathematician to discuss about quadrilaterals whose four sides are not equal with none of the opposite sides parallel.

For many centuries, the approximate value of p was considered v10. But Bhaskara I did not accept this value and believed that p had an irrational value which later proved to be true. Some of the contributions of Bhaskara I to mathematics are: numbers and symbolism, the categorization of mathematics, the names and solution of the first degree equations, quadratic equations, cubic equations and equations which have more than one unknown value, symbolic algebra, the algorithm method to solve linear indeterminate equations which was later suggested by Euclid, and formulated certain tables for solving equations that occurred in astronomy.

Biography
We know little about Bhāskara's life. Presumably he was born in Kerala. His astronomical education was given by his father. Bhaskara is considered the most important scholar of Aryabhata's astronomical school. He and Brahmagupta are the most renowned Indian mathematicians who made considerable contributions to the study of fractions.

Representation of numbers
Bhaskara's probably most important mathematical contribution concerns the representation of numbers in a positional system. The first positional representations were known to Indian astronomers about 500. However, the numbers were not written in figures, but in words or allegories, and were organized in verses. For instance, the number 1 was given as moon, since it exists only once; the number 2 was represented by wings, twins, or eyes, since they always occur in pairs; the number 5 was given by the (5) senses. Similar to our current decimal system, these words were aligned such that each number assigns the factor of the power of ten corresponding to its position, only in reverse order: the higher powers were right from the lower ones. For example,

1052 = wings senses void moon.
Why did the Indian scientists use words instead of the already known Brahmi numerals? The texts were written in Sanskrit, the "language of the gods", which played a similar role as Latin in Europe, the spoken languages were quite different dialects. Presumably, the Brahmi numerals which were used in every-day life were regarded as too vulgar for the gods (Ifrah 2000, p. 431).

About 510, Aryabhata used a different method ("Aryabhata cipher") assigning syllables to the numbers. His number system has the basis 100, and not 10 (Ifrah 2000, p. 449). In his commentary to Aryabhata's Aryabhatiya in 629, Bhaskara modified this system to a true positional system with the base 10, containing a zero. He used properly defined words for the numbers, began with the ones, then writes the tens, etc. For instance, he wrote the number 4,320,000 as

viyat ambara akasha sunya yama rama veda
sky atmosphere ether void primordial couple (Yama & Yami) Rama Veda
0 0 0 0 2 3 4

His system is truly positional, since the same words representing, e.g. the number 4 (like veda), can also be used to represent the values 40 or 400 (van der Waerden 1966, p. 90). Quite remarkably, he often explains a number given in this system, using the formula ankair api ("in figures this reads"), by repeating it written with the first nine Brahmi numerals, using a small circle for the zero (Ifrah 2000, p. 415). Contrary to his word number system, however, the figures are written in descending valuedness from left to right, exactly as we do it today. Therefore, at least since 629 the decimal system is definitely known to the Indian scientists. Presumably, Bhaskara did not invent it, but he was the first having no compunctions to use the Brahmi numerals in a scientific contribution in Sanskrit.

The first, however, to compute with the zero as a number and to know negative numbers, was Bhaskara's contemporary Brahmagupta.

[edit] Further contributions
Bhaskara wrote three astronomical contributions. In 629 he commented the Aryabhatiya, written in verses, about mathematical astronomy. The comments referred exactly to the 33 verses dealing with mathematics. There he considered variable equations and trigonometric formulae.

His work Mahabhaskariya divides into eight chapters about mathematical astronomy. In chapter 7, he gives a remarkable approximation formula for sinx, that is

which he assigns to Aryabhata. It reveals a relative error of less than 1.9% (the greatest deviation at x = 0). Moreover, relations between sine and cosine, as well as between the sine of an angle , or to the sine of an angle are given. Parts of Mahabhaskariya were later translated into Arabic.

Bhaskara already dealt with the assertion: If p is a prime number, then 1 + (p − 1)! is divisible by p. It was proved later by Al-Haitham, also mentioned by Fibonacci, and is now known as Wilson's theorem.

Moreover, Bhaskara stated theorems about the solutions of today so called Pell equations. For instance, he posed the problem: "Tell me, O mathematician, what is that square which multiplied by 8 becomes - together with unity - a square?" In modern notation, he asked for the solutions of the Pell equation 8x2 + 1 = y2. It has the simple solution x = 1, y = 3, or shortly (x,y) = (1,3), from which further solutions can be constructed, e.g., (x,y) = (6,17).

[edit] References
H.-W. Alten, A. Djafari Naini, M. Folkerts, H. Schlosser, K.-H. Schlote, H. Wußing: 4000 Jahre Algebra. Springer-Verlag Berlin Heidelberg 2003 [ISBN 3-540-43554-9], §3.2.1
S. Gottwald, H.-J. Ilgauds, K.-H. Schlote (Hrsg.): Lexikon bedeutender Mathematiker. Verlag Harri Thun, Frankfurt a. M. 1990 [ISBN 3-8171-1164-9]
G. Ifrah: The Universal History of Numbers. John Wiley & Sons, New York 2000 [ISBN 0-471-39340-1]
B. van der Waerden: Erwachende Wissenschaft. Ägyptische, babylonische und griechische Mathematik. Birkäuser-Verlag Basel Stuttgart 1966

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