Question

Prove that between every rational number and every irrational number there is an irrational number.

Short answer

Between every rational value and every irrational value there must be an irrational value.

Step-by-step solution

Introduction

A rational value is a value that can be represented in the form of\(\frac{p}{q}\), of two integral values such thatq≠0 .

Any real value which cannot be represented as the quotient of two integral values is said to be an irrational value.

Proof using cases

Let ‘a’ be a rational number and ‘b’ be an irrational number.

Case1- If\(a < b\)

\( \Rightarrow b - a > 0\)

Then there exists\(n\)such that\(n\left( {b - a} \right) > 1\)

\(\begin{aligned}{} &\Rightarrow \left( {b - a} \right) > \frac{1}{n}\\ \Rightarrow a < b - \frac{1}{n} &< b\end{aligned}\)

And clearly\(b - \frac{1}{n}\)is irrational, because it is addition of irrational and rational.

Case 2- If\(a > b\)

\( \Rightarrow a - b > 0\)

Then there exists\(n\)such that\(n\left( {a - b} \right) > 1\)

\(\begin{aligned}{} &\Rightarrow \left( {a - b} \right) > \frac{1}{n}\\ \Rightarrow a > b + \frac{1}{n} &> b\end{aligned}\)

And clearly\(b - \frac{1}{n}\)is irrational, because it is addition of irrational and rational.

Between every rational value and every irrational value there must be an irrational value.