Zero: India's contribution to the world of mathematics.
We always knew we were good for nothing.
(From a "Tantra" T-shirt)
The symbol and concept of zero are important for a number of different reasons:
- The concept of the empty set or the vacuum (as in classical physics).
This is clearly a difficult concept especially because one can construct "dialogues" in natural language which go like this:
A: Is nothing the absense of something? B: But then is there something like nothing? C: If it is something then it can't be nothing. D: Then there is no thing like nothing.
and so on ... as Shakespeare said "Much Ado about Nothing".
- The concept of the additive identity.
The notion of multiplicative identity is quite easily come by; 1xn = n and so on. Since one sees that addition and multiplication have similarities it is natural to postulate an additive identity 0 so that 0+n = n.
This is easy until one realises that this leads to rules like 0xn = n and n^0 = 1. Moreover certain multiplicative operations like "division by" and "taking powers of" cannot be applied to zero in a meaningful way. These difficulties make the algebraic notion of 0 a difficult concept.
- The use of 0 in symbolic place notation.
It is easy to "see" 143 as 3 sticks, 4 big sticks and 1 large stick. Similarly, 102 is 2 sticks and 1 large stick. But when we start writing this in "short-hand" we run into a problem which is only solved by thinking of 102 as 2 sticks, 0 big sticks and 1 large stick.
So the use of 0 in symbolic place notation is essential.
Now, it is clear that any bunch of people involved in doing basic algebra or non-basic geometry (or algebraic geometry) will be led to all these concepts sooner or later.
However, the Greek school of Geometry led by Euclid does not seem to have had any interest in algebra because of their philosophical bias towards geometry.
The first major mathematical school that worked on Algebra appears to have developed in India; this is why these concepts (and the important concept of an algebraic variable "x") are likely to have evolved in India first.
Coming to the question of who (or which civilisation) should be given the "credit" for inventing the 0. This is not a question in mathematics but a question for historians (or economic historians, given the WTO-led desire to patent all ideas). In particular, it is not a question that interests me.