Question

underoot 3 is irrational

Answer 1

Let sqrt3 be a rational number. Let p and q be coprime integers.

Let p/q = sqrt(3) = reduced form of rational number. .
p * p = 3 * q *q.
So on the LHS, p must have a factor 3. Let p=3n.

So 3 n*n = q*q
Now on the RHS q must have a factor 3. So q=3m.

n*n = 3*m*m.
So it turns out that p and q are not coprime.
It's proved by contradiction that sqrt3 is irrational .