Question

What do you understand by irrational number? Prove that \( 5 - 3\sqrt{2} \) is an irrational number.

Hint:

To prove that a number is irrational, assume that it is rational and derive a contradiction.

Answer 1

Step 1: Define an irrational number.}
An irrational number is a number that cannot be expressed as the ratio of two integers. It has non-terminating and non-repeating decimal expansions. Examples of irrational numbers include \( \pi \), \( \sqrt{2} \), etc.
Step 2: Assume \( 5 - 3\sqrt{2} \) is rational.}
Assume that \( 5 - 3\sqrt{2} \) is a rational number. That is, we assume that: \[ 5 - 3\sqrt{2} = \frac{p}{q} \] where \( p \) and \( q \) are integers, and \( q \neq 0 \).
Step 3: Isolate \( \sqrt{2} \).}
Rearranging the above equation: \[ 3\sqrt{2} = 5 - \frac{p}{q} \] \[ 3\sqrt{2} = \frac{5q - p}{q} \] \[ \sqrt{2} = \frac{5q - p}{3q} \]
Step 4: Conclude that \( \sqrt{2} \) is rational.}
Since \( \frac{5q - p}{3q} \) is a ratio of integers, \( \sqrt{2} \) must be rational. But we know that \( \sqrt{2} \) is an irrational number, which leads to a contradiction.

Step 5: Conclusion.}
Therefore, our assumption that \( 5 - 3\sqrt{2} \) is rational is false. Hence, \( 5 - 3\sqrt{2} \) must be an irrational number. % Final Answer Final Answer:
Thus, \( 5 - 3\sqrt{2} \) is an irrational number.