How to know that irrational numbers never repeat?

Question

Izabella Henson · Accepted Answer

We know that irrational numbers never repeat by combining the following two facts:
every rational number has a repeating decimal expansion, and
every number which has a repeating decimal expansion is rational.
Together these facts show that a number is rational if and only if it has a repeating decimal expansion.
Decimal expansions which don't repeat are easy to construct; other answers already have examples of such things.
The irrationality of a number by examining its decimal expansion. While it is true that an irrational number has a non-repeating decimal expansion, you don't need to show a given number has a non-repeating decimal expansion in order to show it is irrational. In fact, this would be very difficult as we would have to have a way of determining all the decimal places. Instead, we use the fact that an irrational number is not rational; in fact, this is the definition of an irrational number (note, the definition is not that it has a non-repeating decimal expansion, that is a consequence). In particular, to show a number like

\sqrt{2}

is irrational, we show that it isn't rational. This is a much better approach because, unlike irrational numbers, rational numbers have a very specific form that they must take, namely

\frac{a}{b}

where

a

and

b

are integers,

b

non-zero. The standard proofs show that you can't find such

a

and

b

so that

\sqrt{2} = \frac{a}{b}

thereby showing that

\sqrt{2}

is not rational; that is,

\sqrt{2}

is irrational.

How to know that irrational numbers never repeat?

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