Question

Prove or disprove that the product of a nonzero rational number and an irrational number is irrational.

Short answer

The multiplication of a non-zero rational number and an irrational number gives irrational number.

Step-by-step solution

Introduction

This is true, i.e., the multiplication of a non-zero rational number and an irrational number gives irrational number.

Show the result using a proof by contradiction.

Prove using contradiction

Assume \(x\) to be a nonzero rational number and \(y\) to be an irrational number.

And assume\(k = x.y\)

Suppose that\(k\) is a rational number.

Then by definition there exist integers p and q with\(q \ne 0\)such that

\(k = \frac{p}{q}\)

Now\(x.y = \frac{p}{q}\)

\( \Rightarrow y = \frac{p}{{xq}}\)

Since\(x\)is a nonzero rational number and\(q\)is a nonzero integer,\(xq\) is a nonzero rational number.\(p\) is an integer.

So\(\frac{p}{{xq}}\)is a rational number.

But,\(y\)is an irrational number.

This is a contradiction.

i.e.,\(x.y\)is irrational.

Thus, the multiplication of a non-zero rational number and an irrational number is irrational number.