Question

Prove that the sum of a rational number and an irrational number is always irrational.

Answer 1

Assume that $a$ is rational, $b$ is irrational, and $a+b$ is rational. Since $a$ and $a+b$ are rational, we can write them as fractions.

Let $a=\frac{c}{d}$ and $a+b=\frac{m}{n}$

Substituting $a=\frac{c}{d}$ in $a+b=\frac{m}{n}$ gives the following:

$\frac{c}{d}+b=\frac{m}{n}$

Now, let's subtract $\frac{c}{d}$ from both sides of the equation.

$b=\frac{m}{n}-\frac{c}{d}$, or

$b=\frac{m}{n}+(-\frac{c}{d})$

Since the rational numbers are closed under addition, $b=\frac{m}{n}+\left(\frac{-c}{d}\right)$ is a rational number. However, the assumptions said that $b$ is irrational, and $b$ cannot be both rational and irrational. This is our contradiction, so it must be the case that the sum of a rational and an irrational number is always irrational.