A fraction in lowest terms has a terminating decimal exactly when its denominator has no prime factors other than 2 and 5. Otherwise the decimal repeats without end.
Some fractions give a decimal that ends after a finite number of digits: 1/2 = 0.5, 3/8 = 0.375, 7/20 = 0.35. These are terminating decimals.
Others give a decimal in which a block of digits repeats without end: 1/3 = 0.333…, 1/7 = 0.142857142857…. These are non-terminating recurring decimals.
Which kind a fraction gives can be decided before dividing, from the prime factorisation of its denominator.
First write the fraction in lowest terms: numerator and denominator have no common factor other than 1. Then factorise the denominator.
The decimal terminates if and only if the denominator has no prime factors other than 2 and 5. If any other prime is a factor, the decimal recurs.
Why are 2 and 5 the primes that matter, and not 3 or 7?
Our number system uses base ten, and 10 = 2 × 5. A terminating decimal is a fraction with a power of ten as its denominator: 0.35 = 35/100, and 100 = 2² × 5².
A fraction can be rewritten over a power of ten only if its denominator divides some power of ten. The powers of ten contain only the primes 2 and 5. So a denominator with a factor 3 or 7 can never divide a power of ten, and the fraction can never be written with a finite number of decimal digits.
In long division by a denominator q, each step leaves a remainder between 0 and q − 1. If a remainder of 0 appears, the division stops. If it never appears, there are only q − 1 other possible remainders. Within q steps, some remainder must occur a second time. From then on, the division repeats the same steps, so the digits repeat.
So every fraction gives either a terminating or a recurring decimal. Which one depends on whether the denominator’s primes are only 2s and 5s.
7/20: the denominator is 20 = 2² × 5. Only 2 and 5, so the decimal terminates. 7/20 = 35/100 = 0.35.
1/6: the denominator is 6 = 2 × 3. The factor 3 means the decimal recurs. 1/6 = 0.1666…
13/125: the denominator is 125 = 5³. Only 5, so it terminates. 13/125 = 104/1000 = 0.104.
Check the denominator. After reducing the fraction, a denominator made only of 2s and 5s gives a terminating decimal. Any other prime factor gives a recurring one.
Does 17/8 give a terminating decimal? Decide from the primes before dividing.
17 is prime and does not divide 8, so the fraction is in lowest terms. 8 = 2³, which has no prime factor other than 2.
So it terminates. Multiplying top and bottom by 5³ = 125 gives 2125/1000 = 2.125.
The rule applies only to a fraction in lowest terms. 3/6 has denominator 6 = 2 × 3, which contains 3. But 3/6 = 1/2, and 1/2 = 0.5 terminates.
The common factor 3 cancelled, and the 3 in the denominator went with it. So cancel common factors before examining the denominator.
1A fraction in lowest terms has a terminating decimal exactly when its denominator…
2Why are 2 and 5 the special primes?
3Terminating or recurring? Each fraction is in lowest terms.
417/8 gives a ____ decimal, because 8 = 2³.
5Why must a fraction be in lowest terms before the denominator is checked?
6For a fraction in lowest terms, which denominator gives a recurring decimal?
Next: proving that some numbers cannot be written as a fraction at all.
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