# Thread: ir/rational expansion

1. ## Tricky one

Which rational numbers between 0 and 1 have finite decimal expansions? Give a brief description, with some explanation in your answer.

Example 3/40 = 0.075, however 2/3 = 0.66666666666............

Help I am

2. Which rational numbers? There are infinitely many. Is that everything the question says?

3. The question I have been given is essencially that:

Give a brief description, with some explanation, of which rational numbers between 0 and 1 have finite decimal expansions? [An example is 3/40 = 0.075, however 2/3 = 0.66666666666............]

I am truly

4. Question - what kind of course is this for? Do you think your teacher/professor wants a rigorous proof or a general explanation?

5. I think a general proof would do, obviously the more I put down the better

6. Check out this page from MathWorld.

http://mathworld.wolfram.com/DecimalExpansion.html

Tell me if it helps. If not, I'll post more.

7. I had already looked at this site up, thanks anyway.

I always use google before mathhelpforum actually. It does help a little but I think I need more... sorry for being awkward.

8. They are the numbers of the form m/n, where m<n, n=2^j*5^k for some nonnegative integers j and k, and m is relatively prime to n. The reason for this is that the algorithm for division will terminate after j of k steps, whichever is greater, because 2 and 5 are factors of 10 and therefore 10^max(j,k) is divisible by n, but if n contains any other prime factor, it will not divide 10^a for any integer a, and the algorithm for division will repeat itself (it has to repeat because of the finite sets involved; the numbers that you get at each stage of the division come from a finite set, and each number determines the next one).