Prove that the square root of 5 is irrational?

Answer 1

The proof is based on reduction ad absurdum.

Suppose sqrt(5) is rational.

That is sqrt(5) = p/q from some co-prime integers p and q. If they are not co-prime, simply divide both by their GCF.

Multiply by q and square both sides, 5q^2 = p^2.

5 divides the left hand side so 5 must divide the right hand side.

That is, 5 divides p^2 and since 5 is prime, 5 must divide p.

That is p = 5r for some integer r.

Substituting for p gives: 5q^2 = (5r)^2 = 25r^2

Dividing both sides by 5 gives q^2 = 5r^2

5 divides the right hand side so 5 must divide the left hand side.

That is, 5 divides q^2 and since 5 is prime, 5 must divide q.

But that means that p and q are not co-prime: contradiction!

Therefore sqrt(5) is irrational.