Is zero a natural number?

[polymath257]

In set theory, the natural numbers are defined as the smallest set N such that

1. the empty set is an element of N
2. Whenever x is an element of N, so is xu{x} (the union). The set xu{x} is called the successor of x.

Then, 0 is defined to be the empty set, 1 is the successor of 0, 2 is the successor of 1, etc.

So, at the most fundamental level, it makes sense to have 0 be a natural number.

In computer science, it is common to index into a collection of data point using what is known as an 'offset'. When this is done, the
first element of the data set has an offset of 0. So, it is convenient for computer scientists to start counting at 0.

In any algebraic system, identity elements are important. So, 0 is an additive identity, meaning that x+0=x=0+x for all x. It is usually desired that number systems have such identity elements (so, 1 is an identity element for multiplication: x*1=x=1*x). So it is convenient to have 0 be a natural number when studying the algebraic properties of the set of natural numbers.

That said, most other areas of math start indexing with 1. So, we generally (but not always) see sequences and series start with an index of 1 and not 0. For this reason, most areas of math have 0 NOT a natural number.

The upshot is that it is purely a convention. Pick one or the other, let your reader know which you are using, and continue about your business.