TITLE: Can every real number be represented by a (possibly infinite) decimal?
QUESTION [21 upvotes]: Does every real number have a representation within our decimal system? The reason I ask is because, from beginning a mathematics undergraduate degree a lot of 'mathematical facts' I had previously assumed have been consistently modified, or altogether stripped away. I'm wondering if my subconscious assumption that every real number can be represented in such a way is in fact incorrect?
If so, is there a proof? If not, why not?
(Also I'm not quite sure how to tag this question?)

REPLY [10 votes]: Every real number $x$ has a decimal expression
$$x = a_0.a_1a_2a_3\ldots. $$
Proof: Picture $x$ on the real line. Certainly $x$ lines between two consecutive integers; let $a_0$ be the lower of these, so that
$$ a_0 \le x < a_0+1.$$
Now divide the line between $a_0$ and $a_0+1$ into ten equal sections. Certainly $x$ lies in one of these sections, so we can find $a_1$ between $0$ and $9$ such that
$$ a_0 + \frac{a_1}{10} \le x < a_0 + \frac{a_1+1}{10} .$$
Similarly, we can find $a_2$ such that
$$ a_0 + \frac{a_1}{10} + \frac{a_2}{10^2} \le x < a_0 + \frac{a_1}{10} + \frac{a_2+1}{10^2} ,$$
and so on. If we do this enough times, the sum $a_0 + \frac{a_1}{10} + \frac{a_2}{10^2} + \ldots$ gets as close as we like to $x$.

From A Conscise Introduction to Pure Mathematics, Fourth Edition, by Martin Liebeck, pg 22.