- #1
Gear300
- 1,213
- 9
I am asked to prove that a real number is rational if and only if it has a periodic decimal expansion.
I have shown that any periodic decimal expansion has an integer p such that multiplication returns an integer. For the case of showing that all rational numbers have a periodic decimal expansion, I have shown that the expansion can eventually become periodic (repeated 9's being a trivial case)...but I'm not sure if this is what is being. If it isn't, then is it true that any rational number has an entirely periodic decimal expansion (I can't really come up with an example for some numbers)?
I have shown that any periodic decimal expansion has an integer p such that multiplication returns an integer. For the case of showing that all rational numbers have a periodic decimal expansion, I have shown that the expansion can eventually become periodic (repeated 9's being a trivial case)...but I'm not sure if this is what is being. If it isn't, then is it true that any rational number has an entirely periodic decimal expansion (I can't really come up with an example for some numbers)?