Bhāskara I

Bhaskara also Bhaskara I, (c. 600 in Saurashtra, Gujarat; ? † around 680 in Ashmaka ) was an Indian mathematician and astronomer.


About Bhaskaras life almost nothing is known. He received his astronomical training from his father. Bhaskara is the preeminent representative of the founded by Aryabhata astronomical school.

Representation of numbers

Bhaskaras perhaps the most important mathematical achievement relates to the representation of numbers in place value systems. The first weight representations are already known Indian astronomers to 500. The numbers are written but not in numbers, but in word numbers or symbols and held in verses. Thus the moon is specified, for example, the number 1, because it exists only once; the number 2 are wings, twins or eyes, as they always occur as a pair; for the number 5 are the (five) senses ( cf. today's mnemonics ). These words are similar to our current decimal system lined up, only began with the units and ended with the highest magnitude, contrary to the original Brahmi notation. For example,

Why the Indian scientists used number words and not at that time also known Brahmi numbers? The texts were, were spoken in Sanskrit, the written " language of the gods ", which is now played in India a role similar to Latin in Europe has quite different dialects. Presumably, the Brahmi numerals used in everyday use were initially considered too vulgar for the gods ( Ifrah 2000, p 431).

Aryabhata made ​​use later to 510 but a different method ( " Aryabhata Code " ), by describing the numbers of syllables. His system is also based on the base 100, and not 10 ( Ifrah 2000, p 449 ). Bhaskara modified in his commentary on Aryabhatiya from the year 629 Aryabhatas syllabic digit case to a place value system to the base 10, which contains a zero. However, it uses a fixed number words, begins with the units, then the tens, etc., for example, he writes the number 4320000 as

His system is genuine positionally, as the same words, for example, a 4 are (as here veda ), can also have the value 40 or 400 ( van der Waerden 1966, p 90). Most remarkable, however, is that he often ankair after such a number word with the words api ( " in digits is this " ) is the same number with the first nine Brahmi numerals and a small round circle for the zero writes ( Ifrah 2000, p 415 ). However, contrary to the numeral notation it writes the digits in descending value from left to right, just as we do today. Thus, our present decimal system is known to the Indian scholars later than 629. Bhaskara probably not invent it, but had to use the first no concerns, the digits in a scientific facility in Sanskrit.

The first one, however, who reckoned as a number with zero and negative numbers knew, was Bhaskaras contemporary of Brahmagupta.

Other Production

Bhaskara wrote three astronomical work. In the year 629 he commented, written in verse form Aryabhatiya work on mathematical astronomy, precisely those 33 verses that dealt with mathematics. He considered it an indeterminate equations of the first degree and trigonometric formulas.

His work Mahabhaskariya is divided into eight chapters on mathematical astronomy. In Chapter 7, he gives a remarkably accurate approximation formula for, namely

Which he attributes to Aryabhata. It gives a relative error of less than 1.9 % ( the largest deviation is obtained for ). At the boundary points and the approximation is exact (ie returns 0 or 1 ). Furthermore, relations between sine and cosine, and between the sine of an angle, or by the sine of an angle are listed. Parts of the Mahabhaskariya have been translated into Arabic later.

Bhaskara already dealt with the statement: If a prime number, it is divisible by. It was proved later by Al- Haitham first mentioned by Fibonacci and is now known as a set of Wilson.

Furthermore, Bhaskara formulated sentences about the solutions of today so called Pell equation. So he set the task: "Tell me, O mathematician, what is the square which multiplied by 8 together with the unit gives a square? " In today's terms the Pell equation is needed for the solution. It has a simple solution, or separately, making it possible to construct other solutions, such as.