Question

Prove that if x is irrational, then 1/x is irrational.

Short answer

“If\(x\)is irrational then \(\frac{1}{x}\) is irrational”

Step-by-step solution

Introduction

Consider the statement,

“If \(x\)is irrational then \(\frac{1}{x}\) is irrational”

The purpose is to show this statement using contraposition proof.

Statements for contraposition proof

Proof by contraposition:

Contraposition is \( - q \to - p\)when the statement is\(p \to q\).

In the above statement,

\(p\): \(x\) is an irrational

\( - p\):\(x\) is a rational

\(q\):\(\frac{1}{x}\)is irrational

\( - q\): \(\frac{1}{x}\)is rational

Proof by contraposition

Contraposition statement\(\left( { - q \to - p} \right)\): If \(\frac{1}{x}\)is rational then\(x\)is rational.

Proof:

Suppose that\(\frac{1}{x}\) is rational.

That is, \(\frac{1}{x} = \frac{p}{q}\), for some integers \(p\)and \(q\)with \(q \ne 0\).

Clearly \(\frac{1}{x}\)cannot be 0 (if it was, then there is contradiction \(1 = x.0\)by multiplying both sides by x).

Since,\(\frac{1}{x} \ne 0\)the integer\(p \ne 0\).

Considering \(x\)for final proof

Consider

\(x = \frac{1}{{\frac{1}{x}}}\)

\(\begin{aligned}{l} = \frac{1}{{\frac{p}{q}}}\\ = \frac{q}{p}\end{aligned}\)

Therefore,\(x = \frac{q}{p}\)can be written as the quotient of two integers with the denominator\(p \ne 0\)non-zero.

Hence, the number x is rational.

Therefore, the statement\(\left( { - q \to - p} \right)\): If \(\frac{1}{x}\)is rational then\(x\)is rational is true.

Thus, by the contraposition, the statement \(p \to q\)is true.

That is, “If \(x\) is irrational then\(\frac{1}{x}\)is irrational”