You know what numbers are. Numbers are probably the first aspect of mathematics most of us learn about. We learn about them long before we start being rigorous.

The result is that “numbers” as a concept is so intuitive to us, that it takes an active mental effort to make our assumptions about them explicit. This is necessary, however, to define them properly. We must, as they say, unlearn what we’ve learned.

To understand basic university-level mathematics, we don’t need to dig too deep into the different types of numbers that exist. We just need four families of numbers. For each we describe the set of all possible numbers with a capital letter in blackboard script like $\mathbb N$, $\mathbb Z$, $\mathbb Q$ or $\mathbb R$.

We will discuss:

The main thing you’ll need to remember is what these four families are, how they are defined and some of the subtleties about them. One thing we’ll give away right now is that these are all subsets of each other: every natural number is an integer, every integer is a rational number and every rational number is a real number.

They are also strict subsets: there are real numbers that aren’t rational numbers, there are rational numbers that aren’t integers and there are integers that aren’t natural numbers.

Or, more compactly:

\[\oc{\mathbb N} \subset \gc{\mathbb Z} \subset \bc{\mathbb Q} \subset \rc{\mathbb R} \p\]

The natural numbers

The natural numbers are also called the counting numbers. They are a set of numbers we use for talking about amounts of things. Apples seem to be the canonical example. I can have $0$ apples, or $1$ apple, or $2$ apples and so on.

They are also called the whole numbers, because we do not, for the moment, cut our apples into chunks. The numbers can only be used to count quantities of whole objects. Other than that, any amount you can have of something is a natural number.

If you’re writing your own math where this is relevant, it’s best to define $\oc{\mathbb N}$ explicitly to avoid any ambiguity.

In the article on set theory, we explained how to represent the natural numbers using just the language of set theory and nothing else. To summarize:

You can use natural numbers perfectly well without thinking of them as sets, but it’s nice to know that there is a rigorous definition for them that makes sense. With this we can start to define the operations on the natural numbers we alrady know.

The key thing about these operations is that they are always defined on natural numbers. The successor function is always defined by definition: if we apply it to a natural number, the result is always another natural number. The addition is just repeated application of the successor function, so it should also always result in another natural number. Multiplication is repeated addition, so it’s always defined and so on for exponentiation and tetration.

The integers

The rationals

The reals

The real numbers are where our definitions get a little hairy. It took mathematicians a while to work out exactly how to construct the real numbers, even though people had already been using them implicity for hundreds of years.

The simplest way to think of them is in terms of decimal expansions. For instance, the number $\pi$ can be written as

\[3.141592 \ldots\]

The sequence of digits after the decimal can be expanded further to make this representation more precise. If we allow ourselves to use infinite sequences of digits, then the reals are those numbers that can be written as a finite sequence of digits before the decimal point, and an infinite sequence after.

Like with the rationals, we do end up with some doubles: the number 1 can be written as $1.000\ldots$ or as $0.999\ldots$. If we inspect these carefully, after making rigorous definitions, we find that they refer to the same number. However, if we accept that such equivalent representations exist, we can think of set of reals very naturally as all numbers that can be expressed in this manner.

If it makes you uncomfortable to define a set of numbers based on something as unpredictiable and poorly defined as an infinite sequence, then you are in good company. A proper definition of the reals would be more careful than this. As a result, it’s also a bit more involved.

To define the reals more rigorously, we can work with approximations. Take, for instance, the number $\sqrt{2}$. In a room with sides of 1 meter, this is the distance from one corner to the corner opposite.

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First things first: can we express $p$ as a rational number? This is one of those classic problems. For many people, the first proof they ever see. If you haven’t seen it before, here it is.

$\sqrt{2} \notin \bc{\mathbb Q}$.

Assume towards a contradiction that $\sqrt{2}$ can be written as a rational. That is we assume that there are integers $\bc{p}$ and $\rc{q}$ so that

\[\frac{\bc{p}}{\rc{q}} = \sqrt{2}\]

or, equivalently

\[\frac{\bc{p}^2}{\rc{q}^2} = 2 \p\]

We can also assume that whoever provided us with this rational expression of $\sqrt{2}$ took the trouble to give us an irreducible fraction. That is, $\bc{p}$ and $\rc{q}$ have no common divisor: we cannot simplify the fraction $\frac{\bc{p}}{\rc{q}}$.

We can now rewrite:

\[\begin{align*} \bc{p}^2 &= 2\rc{q}^2} \\ \frac{1}{2}\bc{p}\bc{p} &= \rc{q}\rc{q}} \p \\ \end{align*}\]

Breaking out of the reals

If you’ve been paying attention, you should have noticed a pattern. We defined the natural numbers, and some operations that were defined on all of them. Then, looking at the inverses of these operations, we were forced to “break out” of the natural numbers. The inverse of addition broke us out into the integers. The inverse of multiplication broke us out into the rationals.

Finally, the inverse of the square root broke us out of the rationals. Here, we took a shortcut. Instead of just adding the radicals to the rationals, we defined a number system from scratch that contained the radicals, the rationals and many other numbers.

Is this the end of the line? Are there inverses that will break us out of the reals, into a new, even greater universe? The answer is yes, and we don’t even need a new operation. We can stick with the humble square, but force it to result in a negative number. That is, what is the number $x$ such that

\[x \times x = -1 \p\]

With high-school math, your intuition will tell you that there is “no such number”.

But with the benefit of the more precise definitions we’ve developed above, we can say that “there is no such real number”. What if we just assume that $x$ does exist, in some larger family of numbers? We can give it a name, and start doing some math to see if this assumption holds, and leads to a working set of numbers.

The conclusion is that it does. The resulting set of numbers are called the complex numbers $\mathbb Q$, which we will discuss in a later article.