Bhaskara II Biography
Bhaskara II was the head of the astronomical observatory at Ujjain, the chief mathematical center of ancient India. It goes to the credit of Varahamihira and Brahmagupta, the leading mathematicians who worked there and built up this school of mathematical astronomy. He wrote six books and the seventh book, which is attributed to him, is considered to be a forgery. The subjects of his six works are arithmetic, algebra, trigonometry, calculus, geometry, and astronomy. The six works are: Lilavati on mathematics; Bijaganita on algebra; the Siddhantasiromani which is divided into two parts: mathematical astronomy and sphere; the Vasanabhasya of Mitaksara which is Bhaskaracharya’s views on the Siddhantasiromani ; the Karanakutuhala or Brahmatulya in which he simplified the concepts of Siddhantasiromani ; and the Vivarana which comments on the Shishyadhividdhidatantra of Lalla. From the mathematical point of view the first three of these works are the most interesting.
Bhaskara II wrote Siddhanta Shiromani at the age of 36 in 1150 AD. This colossal work is divided into four parts Lilawati, Beejaganit, Ganitadhyaya and Goladhyaya and consists of about 1450 verses. Each part of the book consists of huge number of verses and can be considered as a separate book: Lilawati has 278, Beejaganit has 213, Ganitadhyaya has 451 and Goladhyaya has 501 verses. He formulated simple ways of calculations from Arithmetic to Astronomy in this book. He wrote Lilawati is an excellent lucid and poetic language. It has been translated in various languages throughout the world.
Few important contributions of BhaskarII to mathematics are as follows:
Terms for numbers
In English, the multiples of 1000 are termed as thousand, million, billion, trillion, quadrillion etc. These terms were named recently in English, but Bhaskaracharya gave the terms for numbers in multiples of ten which are as follows: eka(1), dasha(10), shata(100), sahastra(1000), ayuta(10,000), laksha(100,000), prayuta (1,000,000=million), koti(107), arbuda(108), abja(109=billion), kharva (1010), nikharva (1011), mahapadma (1012=trillion), shanku(1013), jaladhi(1014), antya(1015=quadrillion), Madhya (1016) and parardha(1017).
Kuttak according to modern mathematics is ‘indeterminate equation of first order’. In the western world, the method of solving such equations was called as ‘pulverizer’. Bhaskara suggested a generalized solution to get multiple answers for these equations.
Chakrawaal, according to western mathematics is the “indeterminate equation of second order”. It is also called Pell’s equation. The equation was first solved by an ancient Indian mathematician, Brahmagupta (628 AD) and given in his Brahmasphutasiddhanta. Bhaskara II converted the method and gave a generalized solution.
Simple mathematical methods
Bhaskara II suggested simple methods to calculate the squares, square roots, cube, and cube roots of big numbers. The Pythagoras theorem was proved by him in only two lines. Bhaskara’s ‘Khandameru’is the famous Pascal Triangle. Pascal, the European mathematician was born 500 years after Bhaskara. In Lilawati, he solved several problems on permutations and combinations and called the method as ‘ankapaash’. He even gave an approximate value of PI as 22/7, which is 3.1416. He was even familiar with the concept of infinity and called it as ‘khahar rashi’, which means ‘anant’.
He died in 1185 in Ujjain, India