How to prove?
Hi! I'm new on this forum and my English is not so good so i might be posting in the wrong forum, I'm sorry for that. Can anyone please help me solve this problem:
Prove that a rational number in the form of p/q, can be written as a periodical decimal number. (we can also use 0,000000000 and similar as a periodical decimal number).
Thanks in advance. =)
I think the main idea is as follows:
To write p/q as a decimal you perform long division, first rewriting p as p.00000000...
When dividing by q there are at most q possible remainders. If the remainder is ever 0, then the decimal terminates. Since there are only finitely many possible remainders, a remainder must eventually come out twice. The first time that this happens is when repetition will occur (you probably have to exclude the very first division here since it is different from the others).
I would never come up with this on my own ^^ thank you!