from the lower ones. Bhāskara's numeral system was truly positional, in contrast to word representations, where the same word could represent multiple values (such as 40 or 400). He often explained a number given in his numeral system by stating ankair api ("in figures this reads"), and then repeating it written with the first nine Brahmi numerals, using a small circle for the zero. Contrary to the word system, however, his numerals were written in descending values from left to right, exactly as we do it today. Therefore, since at least 629, the decimal system was definitely known to the Indian scientists. Presumably, Bhāskara did not invent it, but he was the first to openly use the Brahmi numerals in a scientific contribution in Sanskrit. Further contributions Mathematics Bhāskara I wrote three astronomical contributions. In 629, he annotated the Āryabhaṭīya, an astronomical treatise by Aryabhata written in verses. Bhāskara's comments referred exactly to the 33 verses dealing with mathematics, in which he considered variable equations and trigonometric formulae. In general, he emphasized proving mathematical rules instead of simply relying on tradition or expediency. His work Mahābhāskarīya is divided into eight chapters about mathematical astronomy. In chapter 7, he gives a remarkable approximation formula for sin x: which he assigns to Aryabhata. It reveals a relative error of less than 1.9% (the greatest deviation at ). Additionally, he gives relations between sine and cosine, as well as relations between the sine of an angle less than 90° and the sines of angles 90°–180°, 180°–270°, and greater than 270°. Bhāskara already dealt with the assertion that if is a prime number, then is divisible by . This was later proved by Al-Haitham, mentioned by Fibonacci, and is now known as Wilson's theorem. Moreover, Bhāskara stated theorems about the solutions to equations now known as Pell's 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 . This equation has the simple solution x = 1, y = 3, or shortly (x,y) = (1,3), from which further solutions can be constructed, such as (x,y) = (6,17). Bhāskara clearly believed that was irrational. In support of Aryabhata's approximation of , he criticized its approximation to , a practice common among Jain mathematicians. He was the