- #1

Simfish

Gold Member

- 818

- 2

## Homework Statement

Prove that [tex]\sqrt{2}+\sqrt{3}[/tex] is irrational.

## Homework Equations

## The Attempt at a Solution

So we know that [tex](\sqrt{2}+\sqrt{3})(\sqrt{3}-\sqrt{2}) = 1[/tex]. But a rational number must be of the form a/b, and if (a/b)c = 1, the only number c that works (for rational numbers) is c = b/a in reduced form due to unique inverses for rational numbers. But here we have a value of c that is NOT of the form c = b/a. And so once we prove that [tex]\sqrt{2}+\sqrt{3}[/tex] is NOT [tex]\frac{1}{\sqrt{3}-\sqrt{2}}[/tex], we can only conclude that [tex](\sqrt{2}+\sqrt{3})[/tex] is irrational.