Is it possible to prove a repeating decimal is infinite

Question

How would one prove that a decimal representation of a fraction is infinite for example 1/3.

Apologizes in advance if this question is trivial to all of you mathematics wizards.

Answer 1

A fractioon ${p\over q}$, $p\leq q$ represented as decimal $0.a_1...a_n$ I believe you want to show that there exists a repeating pattern.

$a_1$ is the quotient of $10p$ by $q$, the rest is $r_1$, $a_2$ is the quotient of $10r_1$ by $q$ the rest is $r_2$, recursively, you define $a_i,r_i$, at some point one rest will repeat, say $r_l=r_{n+l}$, then $a_{n+l+1}=a_{l+1}$ and the sequence repeats $a_{l+1}...a_{n+l}a_{l+1}...a_{n+l}...$.