Rational Number plus Irrational Number is Irrational

Theorem
Rational number plus irrational number is irrational.

That is, let $x \in \Q$, $y \in \R \setminus \Q$ and $x + y = z$.

Then $z \in \R \setminus \Q$.

Proof
$z \in \Q$.

By definition of rational numbers:


 * $\exists a, b \in \Z, b \ne 0: x = \dfrac a b$


 * $\exists c, d \in \Z, d \ne 0: z = \dfrac c d$

Then:

This shows that $y$ is rational, which is a contradiction.

By Proof by Contradiction, $z$ is irrational.