Fundamental questions

Growing up, we learn about increasingly advanced systems of numbers:

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  The natural numbers $\mathbb{N}=\{1,2,3,\dots\}$ (and $\mathbb{N}_{0}=\{0,1,2,3,\dots\}$);

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  The integers $\mathbb{Z}=\{\dots,-2,-1,0,1,2,\dots\}$;

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  The rationals $\mathbb{Q}:=\{p/q:p\in\mathbb{Z},\,q\in\mathbb{N}\}$.

These systems of numbers are introduced to analyse increasingly advanced mathematical problems: we use $\mathbb{N}$ to count apples (and $\mathbb{N}_{0}$ to admit the sad possibility of there being no apples); we use $\mathbb{Z}$ to express that Jemima owes Tim $\pounds 10$ and, finally, if we want to divide a cake amongst 5 friends, then $\mathbb{Q}$ comes to our aid. In particular, as we pass from $\mathbb{N}$ to $\mathbb{Z}$ to $\mathbb{Q}$ we are free to perform more and more arithmetic operations. In $\mathbb{N}$ we can only add and multiply; within $\mathbb{Q}$ we are free to perform all four arithmetic operations $+$, $-$, $\cdot$ and $\div$ (provided we avoid dividing by $0$).

As useful as it is to be able to count apples and divide cakes, the system of rational numbers $\mathbb{Q}$ is insufficient for analysing many other simple mathematical problems.

The above examples express the fact that the famous constants $\sqrt{2}$, $\pi$ and $e$ are not rational numbers. But if they are not rational, then what are they?

In order to answer this question, we need to introduce a new system of numbers which is rich enough to allow us to define, amongst many other things, $\sqrt{2}$, $\pi$ and $e$. This is the system of real numbers $\mathbb{R}$.

You encountered real numbers in IMU as infinite decimal expansions. Some famous examples are

But what do these expressions really mean? Do you know how a method to compute the digits of $\sqrt{2}$ or $\pi$ by hand? How do we make sense of addition and multiplication of infinite decimal expansions? For example, let’s ask the seemingly simple question:

To perform long multiplication of integers, we start from the right:

However, for an infinite decimal expansion $\pi=3.14159\dots$ there is no right! So how do we make sense of $\pi^{2}$?

Matters are even more mysterious when we consider operations such as irrational exponentiation. We know how to make sense of $2^{n}$ for $n\in\mathbb{N}$ by repeated multiplication, and $2^{p/q}$ for $p\in\mathbb{Z}$ and $q\in\mathbb{N}$ by taking reciprocals and roots. However, using our imagination we can ask:

Does this expression make sense? My calculator seems to think so: if I plug it in, then I get something like $2.66514\dots$, but where do these digits come from and what do they mean? How could I compute it by hand? How does $2^{\sqrt{2}}$ relate to our existing notations of exponentials $2^{n/m}$?

Underlying all of the above are yet deeper question about what real numbers actually are.

As explored in IMU, it is often useful to think about number systems geometrically in terms of the number line. That is, using the order $\leq$ on $\mathbb{Q}$, we represent the rational numbers as points on a line with $x$ lying to the left of $y$ if $x<y$. If we only think about integers $\mathbb{Z}$ in this way, then our number line has many large holes. When we consider $\mathbb{Q}$, the points on our number line become a lot more dense, but there are still lots of holes: for instance, there are holes where we expect $\sqrt{2}$, $\pi$ and $e$ to be.

Geometrically, $\mathbb{R}$ is the system of numbers we obtain once we ‘fill in all the holes’ in the number line formed from $\mathbb{Q}$. Thus, if we think of $\mathbb{R}$ geometrically, then our real number line is a continuum: a continuous, unbroken line.

In principle, the idea of a continuous number line is very appealing, but taken on its own it doesn’t seem very precise or give any indication of how to work with real numbers in practice.

This is a difficult question. Whilst it had been known since antiquity that $\sqrt{2}$ is not rational, it was only in the late 19th century, more than 2000 years later, that mathematicians began to understand what real numbers and the continuum actually are.