How do we know irrational numbers don’t repeat?

In: Mathematics

A number is rational if it is the ratio of two integers. It is irrational if it can’t be written as such.

Let’s say you have a number that does repeat. For example x=0.123123123…

So 1000x = 123.123123123…

Subtract x from both sides and you get 999x = 123, or x = 123/999, i.e. x is rational.

Therefore if x is repeating then it is rational.

There’s another proof that shows that if x is rational then it is repeating (or finite).

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.

Well that’s their definition… rational numbers repeat (or just stop), irrational numbers do not. More specifically a rational number can be written as a fraction where the top and bottom are both whole numbers, irrational numbers can’t do that.

Maybe you mean, “How do we know number X is irrational?”, like the square root of 2, or Pi, or e. These are questions answered here already. Generally you prove that a fraction of whole numbers is somehow impossible and that’s good enough.