If r is rational and r^2 is an integer, prove that r is an integer....Math Proofs- irrational and integers?
If r is rational and r^2 is an integer, prove that r is an integer.
please help me figure this out
- Meng tianLv 79 years agoFavorite Answer
r is rational, so r = a / b for some integers a and b, where b is not zero and GCD(a,b) = 1.
r^2 = a^2 / b^2 is an integer.
Thus, a^2 = kb^2 for some integer k.
GCD(a,b) =1 implies GCD(a^2 , b^2) = 1
We have: GCD(a^2 , b^2) = GCD(kb^2 , b^2) = b^2 = 1
This implies b = 1 or = -1
Since r = a / b, it is the case that r = a / 1 = a or r = a / -1 = -a
Either way, since a is an integer, it follows that r = a and r = -a are also integers.