terminating decimal

A terminating decimal is one that politely comes to a stop. After finitely many digits past the decimal point, it simply ends — no “and so on,” no trailing dots. The price 4.99, the measurement 0.125, and the whole number 7.0 are all terminating decimals.

Because it ends, a terminating decimal can always be written exactly as a fraction whose denominator is a power of 10. For example 0.125 = 125/1000, which reduces to 1/8. That makes every terminating decimal a rational number — there is nothing approximate about it.

There is a neat rule for which fractions terminate. Write the fraction in lowest terms; it gives a terminating decimal exactly when its denominator’s only prime factors are 2 and 5 (the primes that divide 10). So 3/8 = 0.375 terminates because 8 = 2 × 2 × 2, but 1/3 does not, because 3 is neither 2 nor 5 — it gives a repeating decimal instead.