The naturals are...
the basic counting numbers that come up naturally all of the time.
That is how they get their name.
The problem is that zero is special. Whether or not it comes up depends strongly on what you are doing.
For instance in logic, everything in math is built out of the empty set, aka 0. Therefore 0 comes up naturally, and what you get first are 0, 1, 2, ... as the numbers of things in sets. So logicians think of 0 as a natural number.
By contrast in combinatorics (the study of how many ways of producing specific kinds of combinations of things) 0 more often than not either doesn't make any sense, or else is a weird exception. So people who work in combinatorics think of 1, 2, 3, 4,... as natural.
In analytics (you can think of this as the high-falutin' cousin of Calculus) there are a lot of infinite series. These are functions of x written out like this:
a0 * x^0 + a1 * x^1 + a2 * x^2 + ...
As you see, they start with the 0'th term, which is your constant. So any analyst thinks of the natural numbers as 0, 1, 2, 3,...
But if you are a number theorist (ie the kind of person who thinks about things like factoring, primes, etc) then your life is devoted to working with positive integers. So to you the natural numbers look like 1, 2, 3, 4,...
And so it goes. In every area of math you use numbers differently. So whether or not 0 seems like a natural number that keeps on turning up will depend on what you are doing. And really, it doesn't matter. The highschool text books that lecture away on what whole numbers are, versus what natural numbers are, are just BS. The textbook writers needed a distinction, they invented one, and no mathematician really gives a damn.
The fact is that if you can't look beyond terminology to the actual ideas, then math isn't going to be your bag. There are other places where you can throw around lots of terminology without worrying about whether it matters and without trying to figure out what you are talking about. Deconstructionist philosophy for instance. But if you miss the idea for the terminology, mathematicians are likely to notice and wonder WTF you are thinking...
Cheers,
Ben
