Irrational and rational proofs

Hey, I was wondering about proving that square roots of numbers are rational or irrational.
I know about proving this for the numbers 1(duh), 2 and all multiples of 2 and 4, but what about 3?
How do i prove that 3(or any other number) is irrational or rational?

Use the uniqueness of prime decomposition


if there exists natural numbers p/q s.t p^2/q^2 = n, some n in the naturals

=> p^2 = n.q^2

If we decompose n,q,p into primes, we see that unless n = a^2 for some a in the naturals, then one of the prime exponents on the RHS of the equations is odd, and by the uniqueness of prime factorisation, this cannot be true as the exponents on the LHS are all even.


You can prove the uniqueness of prime decomposition quite easily, wiki it.