Question

Use a proof by contradiction to prove that the sum of an irrational number and a rational number is irrational.

Short answer

The sum \(k = m + n\) is irrational.

Step-by-step solution

Introduction

The purpose is to find that the sum of an irrational number and a rational number is an irrational number by using the proof of contradiction.

Proving with the help of contradiction method

Assume \(m\)is rational and \(n\) is irrational.

It is enough to prove that the number\(k = m + n\)is an irrational number.

For that,

Conversely, assume that the sum of\(k = m + n\)is rational.

Clearly if\(m\)is rational then\( - m\)is rational.

Since the addition of the rational numbers are rational,

Consider,

\(k + \left( { - m} \right) = m + n + \left( { - m} \right)\)

\(\begin{aligned}{l} = m + n - m\\ = n\end{aligned}\)

Therefore,\(k + \left( { - m} \right)\)is rational but by the supposition n is irrational.

This is contradiction to addition of the rational numbers are rational.

Hence, it is not true for sum of\(k = m + n\)is rational.

Thus, the sum \(k = m + n\)is irrational.