Question

is a rational terminating decimal

Answer 1

Yes, a rational number can be represented as a terminating decimal. A rational number is defined as a number that can be expressed as the fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \).

A rational number will have a terminating decimal representation if, in its simplest form, the denominator \( b \) (after removing all common factors with the numerator) has no prime factors other than 2 and/or 5. For example:

- \( \frac{1}{4} = 0.25 \) (terminating)
- \( \frac{3}{10} = 0.3 \) (terminating)
- \( \frac{5}{8} = 0.625 \) (terminating)

In contrast, if the denominator has prime factors other than 2 or 5, the decimal representation will be repeating. For example:

- \( \frac{1}{3} = 0.333...\) (repeating)
- \( \frac{2}{7} = 0.285714285714... \) (repeating)

So, to determine if a rational number has a terminating decimal, check the prime factorization of the denominator after simplifying the fraction. If it consists solely of the primes 2 and/or 5, then it is a terminating decimal.