Question

The decimal expansion of a rational number cannot be

• A. terminating
• B. non-terminating
• C. non-terminating, repeating
• D. non-terminating, non-repeating

Answer 1

The correct option is D non-terminating, non-repeating

For all rational numbers of the form $\frac{p}{q}(q\ne 0)$, on division of $p$ by $q$, two things may happen:
1) The remainder becomes zero and the decimal expansion terminates or ends after a finite number of steps.

For example, $\frac{1}{4}=0.25$
2) The remainder never becomes zero and the remainders repeat after a certain stage forcing the decimal expansion to go on for ever. In such a case, we have a repeating block of digits in the quotient.

For example, $\frac{1}{6}=\mathrm{0.16666...}$
From above, we can see that the decimal expansion of a rational number is either terminating or non-terminating repeating (recurring). Hence, we can safely conclude that the decimal expansion of a rational number should be terminating or non-terminating repeating. Hence, it can't be non-terminating non-repeating.