Exercise 61Number theory › IrrationalityHigh schoolAdded 3 Nov 2026

Ratios involving irrationals

#ratio#irrationality#algebraic

Statement

Show that $\frac{\sqrt2+1}{\sqrt2-1}$ is irrational.

Hint

Rationalize: $\frac{\sqrt2+1}{\sqrt2-1}=\frac{(\sqrt2+1)^2}{2-1}=3+2\sqrt2$ .

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Simplify. $\frac{\sqrt2+1}{\sqrt2-1}=3+2\sqrt2$ .

Irrational. If $3+2\sqrt2\in\mathbb{Q}$ , then $\sqrt2\in\mathbb{Q}$ , contradiction.

Conclusion. The ratio is irrational.