0 as an element of the natural numbers

Question

For what reasons would or wouldn't one want 0 to be the start of the natural numbers as opposed to 1? Why would one want it to be 1, or why wouldn't one?

Answer 1

There’s no agreement on whether zero is included in common sets of natural numbers. Inclusion of 0 of a natural number was a definition that occurred a long time ago – I think it was the 19th century.

One math professor once said that one should think of natural numbers to fill in the blank of the following sentence: I have ___ pieces of cake. It is easy to see that only integers count, and also that only positive numbers count as natural. However, it is possible to have no pieces of cake, so zero must be included as a natural number.

Therefore, it might be wise to start the natural number set when teaching children, for example, as they must know the important distinction when counting objects, i.e. having no pieces of cake versus having multiple pieces of cake.

Some advantages of considering 0 to be a natural number:

1. The starting point for set theory is the empty set. The number n can be identified as the set of the first n natural numbers
2. Programming and computers usually start counting by 0
3. It is easier to exclude defined elements if we need naturals without zero. It is complicated to define a new element if we don’t already have it
4. Integer, real, and complex numbers include zero, which seems much more important than 1 in those sets since those sets are symmetric with respect to 0
5. The degree of a polynomial can be zero, as can be the order of a derivative

Disadvantages of considering 0 to be a natural number:

1. Generally speaking, 0 is not considered a natural number and is excluded when talking about positive/negative numbers
2. People commonly start counting from 1, 2, 3, etc. and exclude 0
3. The 1st number is 1, not 0
4. When defining limits, 0 plays a role which is symmetric to infinity, and the latter would not be considered a natural number

Answer 2

In Set Theory, the way one typically defines natural numbers is starting from $0$, so it is natural to have the set of natural numbers include $0$.

In Analysis, for example, it is typical to work with sequences, and as one usually counts from $1$, and not from $0$, the notation is that natural numbers omit $0$. Such a convention is used in Arithmetic as well, since one often works with the set of positive integers.

Anytime you see $\mathbb{N}$ used by an author you should understand that it is not a universally convention.