Exercise 495Logic, sets and proofs › Proof techniquesHigh schoolAdded 3 Nov 2026

Proof by contradiction

#contradiction#irrationality#sqrt2

Statement

Prove by contradiction that $\sqrt2$ is irrational.

Hint

Suppose $\sqrt2=p/q$ with $\gcd(p,q)=1$ . Derive that both $p$ and $q$ are even.

Show solutionHide solution

Assume. $\sqrt2=p/q$ , $p,q$ coprime positive integers.

Square. $2q^2=p^2$ , so $p^2$ is even, hence $p$ is even: $p=2k$ .

Substitute. $2q^2=4k^2$ , so $q^2=2k^2$ , hence $q$ is even.

Contradiction. $p,q$ both even contradicts $\gcd(p,q)=1$ .

Conclusion. $\sqrt2$ is irrational.