Question

the sum of a rational number and an irrational number equals:

a terminating decimal.
a rational number.
a fraction.
an irrational number.

Explanation:

Define the algebraic representation of the sum

Let \(r\) be a rational number and \(i\) be an irrational number. Assume their sum is a rational number \(q\):

$$r + i = q$$

Analyze the assumption using algebraic properties

Subtract \(r\) from both sides to isolate \(i\):

$$i = q - r$$

Since the difference of two rational numbers (\(q\) and \(r\)) must be rational, this implies \(i\) is rational, which contradicts the given fact that \(i\) is irrational.

Determine the correct classification

The assumption that the sum is rational leads to a contradiction. Therefore, the sum must be an irrational number.

Answer:

• a terminating decimal.
• a rational number.
• a fraction.
• an irrational number. (Correct answer)