Mathematics

The key is proof by contradiction: assume $\sqrt{7}$ is rational, write it as a fraction in lowest terms, then square both sides to derive an equation. This equation will force both the numerator and denominator to share a common factor of 7, contradicting the assumption that the fraction was fully reduced. This logical impossibility proves $\sqrt{7}$ must be irrational.

The key is proof by contradiction: assume $\sqrt{7}$ is rational, write it as a fraction in lowest terms, then square both sides to derive an equation. This equation will force both the numerator and denominator to share a common factor of 7, contradicting the assumption that the fraction was fully reduced. This logical impossibility proves $\sqrt{7}$ must be irrational.