# How to prove √X is irrational number

when X is even number,it's easy to prove
but how about the condition which X is odd number?
I have no idea of this

Hurkyl
Staff Emeritus
Gold Member
$\sqrt{4}$ is irrational?

$\sqrt{4}$ is irrational?

Char. Limit
Gold Member
So in other words...

$$\sqrt{x}$$ is irrational iff x=/=n^2 for n belonging to the integer set.

So in other words...

$$\sqrt{x}$$ is irrational iff x=/=n^2 for n belonging to the integer set.
yes, but how to prove?

Fundamental theorem of arithmetic. Assume p^2/q^2=x with gcd(p,q)=1, and see what has to divide what.

Fundamental theorem of arithmetic. Assume p^2/q^2=x with gcd(p,q)=1, and see what has to divide what.

what does gcd mean?

Greatest common divisor. If gcd(p,q)=1, it means the fraction p/q is in lowest terms.

Look at the proof for sqrt(2), and adapt it. Remember that "even" just means "is divisible by 2", so that if you're checking a number other than 2, you won't be thinking about "even" anymore.