Intro to Number Sets

There are several common sets of numbers that mathematicians use in their studies. In order from simple to complex, they are:

1. The natural numbers \(\mathbb{N}\)

2. The integers \(\mathbb{Z}\)

3. The rational numbers \(\mathbb{Q}\)

4. The real numbers \(\mathbb{R}\)

5. The complex numbers \(\mathbb{C}\)

Each set is constructed in some way from the previous one, and this path will show you how they are constructed from the most basic numbers. You may have come across these terms in a math class that you attended, and may have had other definitions given to you. In this path, you will obtain a firm, complete understanding of these sets, how they are constructed, and what they mean in mathematics.

Why are number sets important?

Before we go any further though, it would be nice to know the motivation behind defining the number sets first.

A set is a fancy name for a collection of objects. Some collections of objects have special properties — such as the set of all blue things, which are special in that they’re all blue. In math, if a set of objects all have a certain property, we can make inferences about them — that is, there are certain things we can say about them that you can deduce logically. For example:

In a field, every nonzero number has a multiplicative inverse.

You don’t need to know what a field is yet (it’s a special type of set), but now you can make inferences about them without restricting yourself to a specific example when talking about them. For example, you know that if a set is a field, then every number in that set that isn’t zero can divide into another number in that set (by multiplying by its “multiplicative inverse”) and produce yet another number in that set.

Conversely, you can also tell when a set is or isn’t a field based on whether it satisfies the properties a field has. For example, since you can’t divide \(3\) by \(2\) (because the result is \(1.5\) which is not a natural number), you now know that the natural numbers are not a field.

Now let’s turn to our first set: the natural numbers.