Prove that the sum of a rational number and an irrational number is always irrational. - AQA - A-Level Maths Pure - Question 9 - 2019 - Paper 1

We can express the rational number $m$ in the form of a fraction:

$m = \frac{a}{b}$

where $a$ and $b$ are integers and $b \neq 0$. The number $n$, being irrational, cannot be expressed as such.

From our previous assumption, we can rearrange the equation for $n$:

$n = s - m = s - \frac{a}{b}$

This indicates that $n$ can be written in the form:

$n = \frac{bs - a}{b}$

Since $s$ is assumed to be rational and both $a$ and $b$ are integers, the expression $bs - a$ will also be an integer. Hence, $n$ is presented as a fraction of integers, leading us to conclude that $n$ is rational.

This leads to a contradiction because we initially stated that $n$ is irrational. Therefore, our initial assumption that $s = m + n$ is rational must be false. Hence, we can conclude that the sum of a rational number and an irrational number is always irrational.