How do we know that a non recurring number will not repeat after many digits?

Question

We are told that there are rational numbers that either terminate or repeat and irrationals that neither terminate nor repeat. But how are we so sure that a non terminating non recurring number will not repeat after, say $1000$th or $100000$th place? Has anybody calculated up to such large digits?

Answer 1

First a simple observation; a terminating number is also a repeating number; it repeats the digit $0$ indefinitely. So we might as well say that rational numbers repeat, and irrational numbers do not.

You ask how we can be so sure that a non-repeating number will not repeat after many digits, say $100000$. This is easy; a non-repeating number does not repeat by definition.

If you would like to know how we can be so sure that an irrational number will not repeat after many digits; this is not as easy. Here's the idea: Suppose $x$ is a number that repeats at some point. Let's say that it repeats after $12$ digits, and then repeats in a loop of $50$ digits. Then the decimal parts of $10^{12}x$ and $10^{12+50}x$ are the same; they are both the same repeating loop of $50$ digits. This means that the decimal part of $10^{62}x-10^{12}x$ is $0$, so this is an integer, say $n$. Then a bit of algebra shows that $$n=10^{62}x-10^{12}x=(10^{62}-10^{12})x,$$ which shows that $x=\frac{n}{10^{62}-10^{12}}$. In particular this means that $x$ is rational.

Of course this argument works for all other positive integers as well, not just $12$ and $50$. So this proves that every repeating number is rational. So every irrational number is non-repeating.