# Bhāskara I

Bhāskara (c. 600 – c. 680) (commonly called Bhaskara I to avoid confusion with the 12th-century mathematician Bhāskara II) was a 7th-century mathematician, who was the first to write numbers in the Hindu decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's work.[1] This commentary, Āryabhaṭīyabhāṣya, written in 629 CE, is among the oldest known prose works in Sanskrit on mathematics and astronomy. He also wrote two astronomical works in the line of Aryabhata's school, the Mahābhāskarīya and the Laghubhāskarīya.[2]

Bhāskara I
Bornc. 600 CE
NationalityIndian
OccupationMathematician; scientist
Known forBhaskara I's sine approximation formula

On 7 June 1979 the Indian Space Research Organisation launched Bhaskara I honouring the mathematician.[3]

## Biography

Little is known about Bhāskara's life. He was probably a Brahmin astronomer.[4] He was born in India in 7th century.

His astronomical education was given by his father. Bhaskara is considered the most important scholar of Aryabhata's astronomical school. He and Brahmagupta are two of 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 had been known to Indian astronomers approximately 500 years prior to this work. However, these numbers, prior to Bhaskara, were written not 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.

His system is truly positional, since the same words representing, can also be used to represent the values 40 or 400.[5] 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 . Contrary to his word 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.

## Further contributions

Bhaskara wrote three astronomical contributions. In 629 he annotated 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 sin x, that is

${\displaystyle \sin x\approx {\frac {16x(\pi -x)}{5\pi ^{2}-4x(\pi -x)}},\qquad (0\leq x\leq \pi )}$

which he assigns to Aryabhata. It reveals a relative error of less than 1.9% (the greatest deviation ${\displaystyle {\frac {16}{5\pi }}-1\approx 1.859\%}$  at ${\displaystyle x=0}$ ). Moreover, relations between sine and cosine, as well as between the sine of an angle >90° >180° or >270° to the sine of an angle <90° are given. Parts of Mahabhaskariya were later translated into Arabic.

Bhaskara already dealt with the assertion that if p is a prime number, then 1 + (p–1)! is divisible by p.[dubious ][citation needed] 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 ${\displaystyle 8x^{2}+1=y^{2}}$ . 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).

## References

2. ^ Keller (2006, p. xiii)
3. ^ Bhaskara NASA 16 September 2017
4. ^ Keller (2006, p. xiii) cites [K S Shukla 1976; p. xxv-xxx], and Pingree, Census of the Exact Sciences in Sanskrit, volume 4, p. 297.
5. ^ B. van der Waerden: Erwachende Wissenschaft. Ägyptische, babylonische und griechische Mathematik. Birkäuser-Verlag Basel Stuttgart 1966 p. 90

## Sources

(From Keller (2006))

• M. C. Apaṭe. The Laghubhāskarīya, with the commentary of Parameśvara. Anandāśrama, Sanskrit series no. 128, Poona, 1946.
• v.harish Mahābhāskarīya of Bhāskarācārya with the Bhāṣya of Govindasvāmin and Supercommentary Siddhāntadīpikā of Parameśvara. Madras Govt. Oriental series, no. cxxx, 1957.
• K. S. Shukla. Mahābhāskarīya, Edited and Translated into English, with Explanatory and Critical Notes, and Comments, etc. Department of mathematics, Lucknow University, 1960.
• K. S. Shukla. Laghubhāskarīya, Edited and Translated into English, with Explanatory and Critical Notes, and Comments, etc., Department of mathematics and astronomy, Lucknow University, 2012.
• K. S. Shukla. Āryabhaṭīya of Āryabhaṭa, with the commentary of Bhāskara I and Someśvara. Indian National Science Academy (INSA), New- Delhi, 1999.