Sum of 2 irrational numbers, rational or irrational, more?

Question

How can I prove that the sum of two irrational numbers is most likely irrational number?

Answer 1

Given a fixed number $z$, you can take each $q\in \mathbb Q$ and find the unique $y_q\in \mathbb R$ such that $z+y_q=q$. Since each $y_q$ is unique, clearly there are only countably many numbers with that can be found.

So out of the entire pool of $|\mathbb R|$ things that can go into the $y$ of "$z+y$", only countably many will yield a rational number, and the rest (uncountably many) will yield an irrational number.

Answer 2

Let $\mathbb{S}$ denote the set $\mathbb{R}\backslash\mathbb{Q}$, then:

• $\forall{x\in\mathbb{S}}:x+(1-x)\in\mathbb{Q}$
• $\forall{x\in\mathbb{S}}:x+(x-1)\not\in\mathbb{Q}$

Therefore, since $\mathbb{S}$ is uncountable:

• The amount of rational results is uncountable
• The amount of irrational results is uncountable