Step 1: Understanding the given problem:
We are asked to prove that \( 6 - 4\sqrt{5} \) is an irrational number, given that \( \sqrt{5} \) is an irrational number.Step 2: Assume the contrary:
We begin by assuming the opposite, that \( 6 - 4\sqrt{5} \) is a rational number. If it were rational, we could express it as a fraction of two integers, say:Step 3: Solving for \( \sqrt{5} \):
Rearranging the equation above to isolate \( \sqrt{5} \):Step 4: Contradiction:
However, this contradicts the given fact that \( \sqrt{5} \) is an irrational number. Therefore, our assumption that \( 6 - 4\sqrt{5} \) is rational must be false.Step 5: Conclusion:
Since assuming \( 6 - 4\sqrt{5} \) is rational leads to a contradiction, we conclude that \( 6 - 4\sqrt{5} \) must be an irrational number.