Proof: The square roots of non perfect squares are irrational
Let's assume the square roots of non perfect squares are rational and then disprove it
c and c+1 are the roots of two perfect squares
think c=3 and c+1=4 3^2 and 4^2
we're trying to say that the square roots of all numbers between 3^2 and 4^2 are irrational. Anyway carrying on
a is a rational number between c and c+1:
c<a<c+1 and c^2<a^2<(c+1)^2
we're asserting a^2 has a rational root a
Then: 0<a-c<1
If a is a rational number then s is the smallest number such that sa is an integer
ex: a=3/4 s=4 sa=3 get it? ok
0<s(a-c)<s
let r=s(a-c) r is an integer
0<r<s
let's consider ra
ra=sa^2 -sac ra is an integer as well oh shit!!
we said s was the smallest integer that would make the rational a an integer when made into sa
and by definition r is greater than 0 and less s ... so our assumption must be false about the existence of rational root a
Therefore the square roots of non-perfect square numbers are all irrational.
w00t w00t