How do we know irrational numbers don’t repeat?

Question

How do we know irrational numbers don’t repeat?

In: Mathematics

Answer 1

The key is to show that all repeating decimals can easily be exactly represented as rational numbers, i.e., as fractions. If you have a decimal that goes 0.abc…xyzabc…xyz… then that’s always equal to abc…xyz/999…999 where the number of nines equals the number of digits above. For cases where the repeating doesn’t start until later after the decimal point some simple arithmetic can be used.

So, if every repeating decimal is rational, no irrationals can be repeating. QED.