How to prove that the fraction of an irrational number is non-periodic?
Everyone knows that the fraction of an irrational number is an infinite non - periodic one.
But how can we actually prove this non-periodicity? I tried to contact Google, but did not find an answer.
Thank you in advance for your answer.
1 answers
Almost by definition.
If the number is represented as a finite decimal fraction, like
Then it can be written like this:
So it's rational.
If the number is represented as a periodic fraction, then we are dealing with an infinite geometric progression, the sum of which is a rational number:
It remains to prove the opposite - that if the number is irrational, then it is not representable as an infinite periodic fraction. We go from the opposite - let there be an irrational number that can be represented by a finite or infinite periodic fraction. Then, as we have already proved, this number is rational, which means that we get a contradiction. Therefore, an irrational number cannot be represented as either a finite or infinite periodic fraction. Only the infinite non-periodic fraction remains.