Final answer:
A fraction will have a terminating decimal if its denominator is a power of 2, a power of 5, or a combination of both since the base of our number system is 10. Examples like 1/8 and 5/16 have terminating decimals because their denominators are powers of 2.
Step-by-step explanation:
A fraction will have a terminating decimal expansion if its denominator can be written in the form 2n5m, where n and m are nonnegative integers. This is because the base of our number system is 10, which is 2×5. So any fraction with a denominator that is a factor of a power of 10 will have a terminating decimal expansion.
Example:
- Consider the fraction 1/8. Since 8 is 23, which is a power of 2, 1/8 has a terminating decimal of 0.125.
- For the fraction 5/16, the denominator 16 is 24. Because it contains only the prime factor 2, 5/16 has a terminating decimal of 0.3125.
In contrast, a fraction like 1/3 does not have a prime factorization that matches the form 2n5m and therefore has a non-terminating decimal expansion (0.333...).