Why Every Fraction Terminates or Recurs

Decimals that stop, and decimals that repeat

Every fraction has a decimal, found by dividing the numerator by the denominator. $\frac{1}{8}=1÷8=0.125$1/8 = 1 ÷ 8 = 0.125, and the division stops after 3 decimal places. A decimal that stops is called a terminating decimal.

$\frac{1}{3}=1÷3=0.333\dots$1/3 = 1 ÷ 3 = 0.333…, and the division never stops: the digit 3 repeats forever. A decimal in which a block of digits repeats forever is called a recurring decimal. In print, a dot or a bar over the repeating digits marks the repeat.

These are the only two possibilities: the decimal of every fraction either terminates or recurs, and the denominator decides which.

1 ÷ 8 = 0.125. After three decimal places nothing is left to divide.

Follow the remainders

Long division makes each decimal digit in the same way: take the remainder, multiply it by 10 (bring down a 0), divide by the denominator, and keep the new remainder.

For 1 ÷ 8 the remainder starts at 1. Then 10 ÷ 8 = 1 remainder 2, 20 ÷ 8 = 2 remainder 4, and 40 ÷ 8 = 5 remainder 0. The digits are 1, 2 and 5, and the remainders are 1, 2, 4 and then 0. A remainder of 0 means nothing is left to divide, so every later digit would be 0, and the decimal stops: 0.125.

The remainders of 1 ÷ 8 run 1, 2, 4, then 0. The digit on each arrow is the digit that step writes, and at 0 the digits stop.

Why a remainder must come back

Now divide 1 by 7. The remainders are 1, 3, 2, 6, 4 and 5, and then 1 again. The digits are 1, 4, 2, 8, 5 and 7, so $\frac{1}{7}=0.142857142857\dots$1/7 = 0.142857142857…

The 1 had to come back. Dividing by 7 can only leave the remainders 0, 1, 2, 3, 4, 5 and 6. Here the remainder is never 0, so every remainder is one of the six numbers 1 to 6. The first seven remainders are seven numbers chosen from only six, so two of them must be equal.

Each step depends only on the remainder it starts from. So once a remainder comes back, the same digits and the same remainders follow in the same order, forever. The digits recur, and the repeating block of $\frac{1}{7}$1/7 is at most 6 digits long. It is exactly 6: 142857.

The same argument works for any denominator. Dividing by a number d, either a remainder of 0 comes and the decimal terminates, or it never does. Then every remainder is one of the d − 1 numbers from 1 to d − 1, so within d steps a remainder must come back, and the decimal recurs. Nothing else can happen.

The remainders of 1 ÷ 7 are 1, 3, 2, 6, 4 and 5, and the next one is 1 again, so the digits 142857 repeat.

The repeat does not have to start with the first digit. 1 ÷ 6: 10 ÷ 6 = 1 remainder 4, then 40 ÷ 6 = 6 remainder 4. The remainder 4 has come back after one step, so $\frac{1}{6}=0.1666\dots$1/6 = 0.1666…, with a 1 that happens once and then a 6 that repeats forever.

The remainders of 1 ÷ 6 are 1 and then 4, and 4 comes back at once. The 1 is written once and the 6 repeats: $\frac{1}{6}=0.1666\dots$1/6 = 0.1666…

Which fractions terminate

A terminating decimal is a number of tenths, hundredths, thousandths and so on: $0.125=\frac{125}{1000}$0.125 = 125/1000. So a fraction terminates when it can be written with a power of 10 as its denominator.

Every power of 10 is built from 2s and 5s only, because 10 = 2 × 5. For example, $1000=2^3\times 5^3$1000 = 2³ × 5³. So a denominator made only of 2s and 5s can always be made into a power of 10: multiply the top and the bottom by whatever 2s or 5s are missing. $8=2^3$8 = 2³, so multiply by $5^3=125$5³ = 125: $\frac{1}{8}=\frac{125}{1000}=0.125$1/8 = 125/1000 = 0.125. And $20=2^2\times 5$20 = 2² × 5, so multiply by 5: $\frac{7}{20}=\frac{35}{100}=0.35$7/20 = 35/100 = 0.35.

Denominators built only from 2s and 5s give decimals that stop. A denominator with any other prime factor gives a decimal that recurs.

Why no other prime ever clears

Take $\frac{5}{12}$5/12, which is in lowest terms, and $12=2^2\times 3$12 = 2² × 3. Suppose its decimal terminated. Then $\frac{5}{12}$5/12 would be equal to a whole number over a power of 10, say $\frac{5}{12}=\frac{n}{1000}$5/12 = n/1000. Multiply both sides by 12 and by 1000: 5 × 1000 = 12 × n.

The prime 3 is a factor of 12 × n, so it appears in the prime factorization of that number. But 5 × 1000 is the same number, and its prime factorization is $2^3\times 5^4$2³ × 5⁴, with no 3 in it: 1000 is made of 2s and 5s only, and the numerator 5 has no factor 3, because $\frac{5}{12}$5/12 is in lowest terms. By the Fundamental Theorem of Arithmetic a number has only one prime factorization, so this is impossible. The same happens with 10000 or any other power of 10, so $\frac{5}{12}$5/12 recurs: $\frac{5}{12}=0.41666\dots$5/12 = 0.41666…

Lowest terms matters. In $\frac{3}{12}$3/12 the 3 in the denominator is canceled by the 3 in the numerator, and $\frac{3}{12}=\frac{1}{4}=0.25$3/12 = 1/4 = 0.25, which terminates. So a fraction in lowest terms terminates exactly when its denominator has no prime factors other than 2 and 5.

The slider opens at $\frac{1}{6}=0.1666\dots$1/6 = 0.1666…: 6 = 2 × 3, and the 3 divides no power of 10, so the 6 repeats. Slide d to 16: $16=2^4$16 = 2⁴ divides $10^4=10000$10⁴ = 10000, and $\frac{1}{16}=0.0625$1/16 = 0.0625 stops.

How many decimal places

The number of decimal places in a terminating decimal can be read from the denominator too. To reach a power of 10, the denominator needs as many 5s as 2s. $40=2^3\times 5$40 = 2³ × 5 has three 2s and one 5, so multiply the top and the bottom by two more 5s, which is 25: $\frac{7}{40}=\frac{175}{1000}=0.175$7/40 = 175/1000 = 0.175. The power of 10 is $10^3$10³, so the decimal has 3 places.

$16=2^4$16 = 2⁴ needs four 5s, which is $5^4=625$5⁴ = 625: $\frac{1}{16}=\frac{625}{10000}=0.0625$1/16 = 625/10000 = 0.0625, with 4 places. The number of places is the larger of the two powers of 2 and 5 in the denominator.

The usual mistakes

Deciding before the fraction is in lowest terms. $\frac{6}{15}$6/15 has a 3 in its denominator, but $\frac{6}{15}=\frac{2}{5}=0.4$6/15 = 2/5 = 0.4.

Thinking that an odd denominator means a recurring decimal. 625 is odd, but $625=5^4$625 = 5⁴, so $\frac{9}{625}=0.0144$9/625 = 0.0144 terminates.

Calling a long decimal recurring. $\frac{1}{64}=0.015625$1/64 = 0.015625 has six decimal places and stops, because $64=2^6$64 = 2⁶.