How to prove √X is irrational number

[itex]\sqrt{4}[/itex] is irrational?

 

So in other words...

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

 

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

 

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.

 

Although right now you're probably more interested in just getting the right answer, you might want to check out the following Wikipedia entries:
http://en.wikipedia.org/wiki/Square_root
http://en.wikipedia.org/wiki/Square_root_of_2

And for an interesting history of the discovery of irrational numbers look at
http://en.wikipedia.org/wiki/Irrational_number