**Bhaskara II** or **Bhaskaracharya** was an Indian mathematician and astronomer who extended Brahmagupta's work on number systems.

- Since he is known in India as Bhaskaracharya we will refer to him throughout this article by that name.
- Bhaskaracharya became head of the astronomical observatory at Ujjain, the leading mathematical centre in India at that time.
- In many ways Bhaskaracharya represents the peak of mathematical knowledge in the 12th century.
- Six works by Bhaskaracharya are known but a seventh work, which is claimed to be by him, is thought by many historians to be a late forgery.
- The six works are: "Lilavati" (The Beautiful) which is on mathematics; "Bijaganita" (Seed Counting or Root Extraction) which is on algebra; the "Siddhantasiromani" which is in two parts, the first on mathematical astronomy with the second part on the sphere; the "Vasanabhasya" of Mitaksara which is Bhaskaracharya's own commentary on the "Siddhantasiromani" ; the "Karanakutuhala" (Calculation of Astronomical Wonders) or "Brahmatulya" which is a simplified version of the Siddhantasiromani ; and the "Vivarana" which is a commentary on the Shishyadhividdhidatantra of Lalla.
- Given that he was building on the knowledge and understanding of Brahmagupta it is not surprising that Bhaskaracharya understood about zero and negative numbers.
- There is also a theory that Lilavati was Bhaskaracharya's wife.
- In dealing with numbers Bhaskaracharya, like Brahmagupta before him, handled efficiently arithmetic involving negative numbers.
- Although this claim is not without foundation, perhaps it is seeing ideas beyond what Bhaskaracharya intended.
- Bhaskaracharya gave two methods of multiplication in his Lilavati.
- As well as the rule of three, Bhaskaracharya discusses examples to illustrate rules of compound proportions, such as the rule of five (Pancarasika), the rule of seven (Saptarasika), the rule of nine (Navarasika), etc.
- In the final chapter on combinations Bhaskaracharya considers the following problem.
- At first sight we might be tempted to believe that Bhaskaracharya has it correct, but of course he does not.
- The problem leads to a quadratic equation and Bhaskaracharya says that the two solutions, namely 16 and 48, are equally admissible.
- Of course such problems do not have a unique solution as Bhaskaracharya is fully aware.
- In particular Bhaskaracharya seems more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation.
- Bhaskaracharya rightly achieved an outstanding reputation for his remarkable contribution.
- In 1207 an educational institution was set up to study Bhaskaracharya's works.

Born 1114, Vijayapura, India. Died 1185, Ujjain, India.

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Algebra, Analysis, Ancient Babylonian, Ancient Indian, Astronomy, Geometry, Origin India, Number Theory, Special Numbers And Numerals

**Oâ€™Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive