Question

The difference of a rational number and an irrational number is:

• A. Always an irrational number
• B. Always a rational number
• C. Both rational and irrational numbers
• D. Zero

Hint:

Rational ± irrational = irrational (always). The rational part never cancels the non-repeating, non-terminating nature of the irrational part.

Answer 1

The Correct Option is A

Step 1: Recall the definitions.
A rational number is one that can be expressed as \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
An irrational number cannot be expressed as a simple fraction (for example, \( \sqrt{2}, \pi, e \)).

Step 2: Consider the difference.
Let the rational number be \( r \) and the irrational number be \( i \).
Then, the difference is \( r - i \).

Step 3: Analyze the result.
Suppose \( r - i \) were rational. Then, rearranging: \[ i = r - (\text{rational number}) \] Since the subtraction of two rational numbers is always rational, this would make \( i \) rational — which is a contradiction.

Step 4: Conclusion.
Therefore, the difference of a rational and an irrational number is always irrational.