Question

The sum of a rational number and an irrational number is which type of number?

• A. An integer
• B. Irrational number
• C. Natural number
• D. None of these

Hint:

Rationals are closed under addition/subtraction; irrationals are not. Adding any nonzero rational to an irrational keeps it irrational.

Answer 1

The Correct Option is B

Step 1: Let \(r\) be rational and \(i\) be irrational.
Assume for contradiction that \(r+i\) is rational.
Step 2: Use closure of rationals under subtraction.
If \(r+i\) were rational, then \(i=(r+i)-r\) would be the difference of two rationals, hence rational—contradiction.
Step 3: Conclude.
Therefore, \(r+i\) must be irrational.