Place Value system and zero
The number place-value system, first seen in the 3rd century Bakhshali Manuscript was clearly in place in his work. Whether Aryabhata knew about zero or not remains in doubt; he certainly did not use the symbol, but the French mathematician Georges Ifrah argues that knowledge of zero was implicit in Aryabhata's place-value system as a place holder for the powers of ten with null coefficients
However, Aryabhata did not use the brahmi numerals; continuing the Sanskritic tradition from Vedic times, he used letters of the alphabet to denote numbers, expressing quantities (such as the table of sines) in a mnemonic form.
Pi as Irrational
Aryabhata worked on the approximation for Pi (?), and may have realized that ? is irrational. In the second part of the Aryabhatiyam (ga?itap?da 10), he writes:
"Add four to 100,
multiply by eight and then add 62,000. By this rule the circumference of a
circle of diameter 20,000 can be approached."
In other words, ?= ~ 62832/20000 = 3.1416, correct to five digits. The commentator Nilakantha Somayaji, (Kerala School, 15th c.) interprets the word ?sanna (approaching), appearing just before the last word, as saying that not only that is this an approximation, but that the value is incommensurable (or irrational). If this is correct, it is quite a sophisticated insight, for the irrationality of pi was proved in Europe only in 1761 by Lambert.
After Aryabhatiya was translated into Arabic (ca. 820 AD) this approximation was mentioned in Al-Khwarizmi's book on algebra.
Menstruation and trigonometry
Ganitapada 6, Aryabhata gives the area of triangle as
translates to: for a triangle, the result of a perpendicular with the half-side
is the area.
A problem of great interest to Indian mathematicians since ancient times has been to find integer solutions to equations that have the form ax + b = cy, a topic that has come to be known as diophantine equations. Here is an example from Bhaskara's commentary on
Find the number which gives 5 as the remainder when divided by 8; 4 as the remainder when divided by 9; and 1 as the remainder when divided by 7.i.e. find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general, diophantine equations can be notoriously difficult. Such equations were considered extensively in the ancient Vedic text Sulba Sutras, the more ancient parts of which may date back to 800 BCE. Aryabhata's method of solving such problems, called the ku??aka method. Kuttaka means pulverizing, that is breaking into small pieces, and the method involved a recursive algorithm for writing the original factors in terms of smaller numbers. Today this algorithm, as elaborated by Bhaskara in AD 621, is the standard method for solving first order Diophantine equations, and it is often referred to as the Aryabhata algorithm. The diophantine equations are of interest in cryptology, and the RSA Conference, 2006, focused on the kuttaka method and earlier work in the Sulvasutras.