Question

Prove that 5 -2√3 is irrational, given that √3 is irrational.

Answer 1

I'll do a proof by contradiction.

Assume 5-2*sqrt(3) is rational. If so, then it's of the form p/q for some integers p and q where q is nonzero.

5-2*sqrt(3) = p/q

-2*sqrt(3) = (p/q) - 5

-2*sqrt(3) = (p/q) - (5q)/q

-2*sqrt(3) = (p-5q)/q

sqrt(3) = (p-5q)/(-2q)

The numerator p-5q is some integer, and -2q is some nonzero integer. This means (p-5q)/(-2q) is rational.

But this contradicts sqrt(3) being irrational. This contradiction happens because we made the assumption that 5-2*sqrt(3) is rational.

Therefore 5-2*sqrt(3) must be irrational.