The set of integers is traditionally given the label Z.

When practical,

this is usually written using what is known as a double-struck capital font.

That's true for most of the labels used for sets of numbers.

While a set of integers is infinite in size,

it does not include a value known as infinity, either positive or negative.

This is because infinity is a concept that is beyond the definition of an integer.

If for no other reason, then its properties are not the same

as those of just a really, really big integer.

A very common subset of integers are the natural numbers,

sometimes called the counting numbers.

The usual symbol for this is a double struck capital N but

is also often referred to as Z+, which is the more common notation in

cryptography and referred to as the positive integers.

This is as good a point as any to say that there's no international governing body

that regulates what various number sets are, let alone the label used for them.

So, while there are some pretty widespread conventions that are used,

there are also some quite a few common alternatives.

For instance, does the set of natural numbers include zero or not?

Depends on who you're talking to.

Usually it does not.

And while that's how we will use it,

we need to be aware that some authors do include it.

What's important are not the specific definitions, but rather that they are used

consistently, and the results and claims are likewise consistent with them.

Other sets that we will see and use from time to time include negative integers,

non-negative integers and non-positive integers.

For the most part, the meanings are obvious, but

this apparent obviousness itself causes miscommunication.

The fine print deals with whether or not zero is included in the set and

unfortunately, authors not to mention people in general, can get a bit sloppy.

In particular, many people will assume that zero is included in the set of

positive integers since we almost never write it with a minus sign.

Others will assume it is in both the positive and

negative integers, since we can write it with or without a minus sign.

But these are purely notational distinctions and

hardly form a sound basis for defining number sets.

But humans are only human.

We are often not good at distinguishing the difference

between how we choose to represent a concept and the concept it represents.

This is actually a very powerful survival trait, but

it can get in our way from time to time.

So you will need to always be on the lookout to determine

what is meant in a particular context.