2.1 A first glimpse of sequences: Decimal expansions

In order to motive what is to follow, we return to a topic discussed in detail in IMU. Many rational numbers can be easily expressed in terms of decimals: for instance, we can easily write $1/5$ in decimals as $1/5=0.2$. However, if we try to do the same thing with $1/3$, the situation is more complicated. Here we have an infinite, or recurring, decimal expansion $1/3=0.333\dots$, which is sometimes written as $1/3=0.\dot{3}$. But what does this equality really mean?

We can easily make sense of any finite decimal expansion, such as $0$, $0.3$, $0.33$ and $0.333$. Indeed, these are all just rational numbers

If we multiply these rationals by $3$, we see that

Thus, we can see that the values $3\times 0$, $3\times 0.3$, $3\times 0.33$, $3\times 0.333$, $\dots$ are all getting closer and closer to $1$. In this way, we can think of $0$, $0.3$, $0.33$, $0.333$, $\dots$ as getting closer and closer to $1/3$. This is the intuitive idea that the infinite decimal expansion $0.333\dots=1/3$ is trying to capture.

The same basic idea also holds for irrational numbers. For instance, we have frequently referred to the famous expansion $\sqrt{2}=1.414213\dots$. If we truncate this expansion and consider the rational numbers $1$, $1.4$, $1.41$, $1.414$, $\dots$, then we see that

Thus, we can see that the values $1^{2}$, $1.4^{2}$, $1.41^{2}$, $1.414^{2}$, $\dots$ are getting closer and closer to $2$. In this way, we can think of $1$, $1.4$, $1.41$, $1.414$, $\dots$ as getting closer and closer to $\sqrt{2}$.¹¹ 1 For the purpose of this introduction, we take a few things for granted. For instance, whilst its not too difficult to see where the digits in the expansion $1/3=0.333\dots$ come from (since here there is a simple repeating pattern), the digits in the expansion of $\sqrt{2}$ don’t appear to have any pattern: $$\sqrt{2}=1.41421356237309504880168872420969807856967187537694807317667973799\dots$$ Where on Earth does this come from? How can we be sure that these values in the corresponding sequence are getting closer and closer to $\sqrt{2}$?

The key takeaway here is that we shouldn’t think of an infinite decimal expansion such as $0.333\dots$ or $1.414213\dots$ just as a single number, but a whole sequence of (rational) numbers. In particular:

In each case, the terms in the sequence get closer and closer to the ‘true’ value of $1/3$ or $\sqrt{2}$.

We still haven’t fully answered what the identities $1/3=0.333\dots$ or $\sqrt{2}=1.414\dots$ really mean, since the details of the above discussion are rather vague and imprecise (what do we really mean by ‘closer and closer’, for instance?).

In this section, we shall systematically develop the theory of sequences, which are just ordered lists of numbers. This includes sequences such as $(0,0.3,0.33,0.333,\dots)$ above, but the theory is very general and we shall see many more interesting examples which arise in different contexts. In the next chapter, we use our theory to study series (a special kind of sequence). Then, and only then, we shall finally be able to make complete sense of decimal expansions.