prove that Square root 2 + square root 3 is irrational. step by step

Question

anonymous · Answer

well, you are adding two irrational numbers, so why would the sum be rational? it wouldn't be.

kinggeorge · Answer

@student_basil Both $\sqrt{2}$ and $1-\sqrt2$ are irrational, yet their sum is 1, which is rational. So you can't simply say that $\sqrt2+\sqrt3$ is irrational because both $\sqrt2$ and $\sqrt3$ are irrational.

What I would do instead is suppose that $\sqrt2+\sqrt3$ were rational. Then$$\sqrt2+\sqrt3=\frac{a}{b}$$for some integers $a,b$. Then we square both sides to get$$5+2\sqrt6=\frac{a^2}{b^2}.$$So$$\sqrt{6}=\frac{a^2-5b^2}{2b^2}.$$This would imply that $\sqrt6$ is rational, which is clearly a contradiction. So $\sqrt2+\sqrt3$ is irrational.

anonymous · Answer

Maybe use a lemma to prove that $$\sqrt{6}$$is indeed irrational. Use the same technique you would use to prove any number is irrational, which is pretty much what KingGeorge used.