More on Real Numbers

Subsubsection 0.1.1 More on Real Numbers

In the preceding paragraphs we have talked about the decimal expansions of real numbers and there is just one more point that we wish to touch on. The decimal expansions of rational numbers are always periodic, that is the expansion eventually starts to repeat itself. For example

\begin{align*} \frac{2}{15} &= 0.133333333\dots\\ \frac{5}{17} &= 0.\underline{ 2941176470588235}2941176470588235\underline{2941176470588235}294117647058823\dots \end{align*}

where we have underlined some of the last example to make the period clearer. On the other hand, irrational numbers, such as \(\sqrt{2}\) and \(\pi\text{,}\) have expansions that never repeat.

If we want to think of real numbers as their decimal expansions, then we need those expansions to be unique. That is, we don't want to be able to write down two different expansions, each giving the same real number. Unfortunately there are an infinite set of numbers that do not have unique expansions. Consider the number 1. We usually just write “1”, but as a decimal expansion it is

\begin{gather*} 1.00000000000\dots \end{gather*}

that is, a single 1 followed by an infinite string of 0's. Now consider the following number

\begin{gather*} 0.99999999999\dots \end{gather*}

This second decimal expansions actually represents the same number — the number \(1\text{.}\) Let's prove this. First call the real number this represents \(q\text{,}\) then

\begin{align*} q &=0.99999999999\dots \end{align*}

Let's use a little trick to get rid of the long string of trailing 9's. Consider \(10q\text{:}\)

\begin{align*} q &=0.99999999999\dots\\ 10q &=9.99999999999\dots \end{align*}

If we now subtract one from the other we get

\begin{align*} 9q &= 9.0000000000\dots \end{align*}

and so we are left with \(q=1.0000000\dots\text{.}\) So both expansions represent the same real number.

Thankfully this sort of thing only happens with rational numbers of a particular form — those whose denominators are products of 2s and 5s. For example

\begin{align*} \frac{3}{25} &= 0.1200000\dots = 0.119999999\dots\\ -\frac{7}{32} &= -0.2187500000\dots = -0.2187499999\dots\\ \frac{9}{20} &= 0.45000000\dots = 0.4499999\dots \end{align*}

We can formalise this result in the following theorem (which we haven't proved in general, but it's beyond the scope of the text to do so):

Theorem 0.1.2

Let \(x\) be a real number. Then \(x\) must fall into one of the following two categories,

\(x\) has a unique decimal expansion, or
\(x\) is a rational number of the form \(\frac{a}{2^k 5^\ell}\) where \(a\in \mathbb{Z}\) and \(k,l\) are non-negative integers.

In the second case, \(x\) has exactly two expansions, one that ends in an infinite string of 9's and the other ending in an infinite string of 0's.

When we do have a choice of two expansions, it is usual to avoid the one that ends in an infinite string of 9's and write the other instead (omitting the infinite trailing string of 0's).