Question

Prove that \(6 - 4\sqrt{5}\) is an irrational number, given that \(\sqrt{5}\) is an irrational number.

Answer 1

Step 1: Problem Statement:

We are tasked with proving that \( 6 - 4\sqrt{5} \) is irrational, given that \( \sqrt{5} \) is irrational.

Step 2: Proof by Contradiction - Initial Assumption:

Assume, for the sake of contradiction, that \( 6 - 4\sqrt{5} \) is a rational number. This means it can be expressed as a fraction of two integers \( p \) and \( q \), where \( q eq 0 \):
\[ 6 - 4\sqrt{5} = \frac{p}{q} \]

Step 3: Algebraic Manipulation:

Rearrange the equation to isolate \( \sqrt{5} \):
\[ 4\sqrt{5} = 6 - \frac{p}{q} \] \[ \sqrt{5} = \frac{6 - \frac{p}{q}}{4} \] \[ \sqrt{5} = \frac{6q - p}{4q} \] This expression shows \( \sqrt{5} \) as the ratio of two integers, \( 6q - p \) and \( 4q \), implying \( \sqrt{5} \) is rational.

Step 4: Identifying the Contradiction:

This result directly contradicts the given premise that \( \sqrt{5} \) is an irrational number. Therefore, the initial assumption that \( 6 - 4\sqrt{5} \) is rational must be incorrect.

Step 5: Conclusion:

As assuming \( 6 - 4\sqrt{5} \) is rational leads to a contradiction, it is definitively concluded that \( 6 - 4\sqrt{5} \) is an irrational number.