As an exercise in my conceptual algebra class we attempted to determine the reason why this theorem holds true in the forward direction. (Note we decided not to tackle the opposite direction) I wrote out a proof of it for the discussion that laid it out rather simply but I would like to make it more formal. However, the more formal version doesn't seem as clear to me as my original did.

Original plain English proof: If we take any rational number p/q and begin to work out the division we see that at each step we have a remainder. This remainder will always be a value between 0 and q-1 because if it was not we would know from the rules of division that our quotie t was wrong.

If our remainder is 0 then the decimal terminates and we are finished. If we never get 0 then there are only a limited set of numbers our remainder can be and eventually one of them will be repeated. As soon as a remainder is repeated the entire decimal will repeat itself giving us a repeating decimal.

Therefore every rational number is represented by a decimal that either terminates or repeats.

Formal proof attempt: Claim: if a number is rational, then it's decimal expansion either terminates or repeats.

Proof: let a/b be a rational number. Then by the definition of rational numbers a/b is a ratio of the integers a and b where b divides a.

Then by the division algorithm a=bq+r for some integers q,r and 0 <= r < b

It follows that if r is 0 the decimal terminates. If r is not 0 then, because r can only be a restricted set of integers, by the pigeonhole principle at some point r must repeat.

Thus the decimal expansion of a rational number either terminates or repeats.

  • $\begingroup$ If $b$ divides $a$, $\frac ab$ is an integer. You'll want to leave that out in the definition of a rational. $\endgroup$ – AlexR Jun 22 '15 at 21:36

This theorem is about the decimal number system, this suggests that the number $10$ might play some role here.

Indeed, for any rational $x=a/b\ $ (with $\gcd(a,b)=1$), if $b|10^n$ then $x$ will have at most $n$ fractional digits.
Else, deduce the problem to the case when $\gcd(b,10)=1\ $ (by multiplying a power of $10$).
Finally, for such a $b$ one needs to find an $n$ that makes $b|(10^n-1)$. Then $n$ will be a period for the infinite sequence.

The classical $1/7=0.(142857)^\bullet$ is a good example to study. Observe that $7|999999$.

  • 2
    $\begingroup$ The theorem is not necessarily "about the decimal number system". It is true that a fraction in [b]any[/b] base corresponds to either a terminating or repeating form. $\endgroup$ – user247327 Sep 7 '17 at 0:18

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