Dividing 1 by 4 stops neatly at 0.25, but dividing 1 by 3 gives 0.333… with 3s forever, and dividing 1 by 7 repeats the six digits 142857 as a block.
Some fractions become terminating decimals and others recurring decimals, and what decides it is whether the denominator of the fraction in lowest terms factorises into nothing but 2s and 5s.
Going the other way, multiply a recurring decimal by a power of 10 and subtract the original, and the endless tail vanishes, leaving a fraction.
So terminating and recurring decimals are both ways of writing fractions, that is, rational numbers, as decimals.
What decimal does a division give?
A calculator cuts the digits off wherever its screen runs out, so the screen alone cannot tell you whether a decimal really ends or keeps repeating. This topic turns that question from counting digits into looking at the denominator. Each new decimal digit comes from multiplying by 10, and 10 = 2 × 5, so if the denominator has any prime factor other than 2 and 5, the remainder never reaches 0. Knowing this reason, you can say without dividing that 1/40 terminates and 1/12 recurs.
seegongsik.com
1/n on a number line, the decimal of 1 ÷ n found by long division with its repeating block, and the prime factorisation of the denominator
The number line in the first picture marks the interval from 0 to 1 cut into n equal parts and the position of 1/n, and below it 1 ÷ n is written out by long division. If the decimal recurs, the repeating block gets a bar over it and its length is noted; if it terminates, the picture says at which decimal place it ends. The last two lines show the prime factorisation of n and the verdict, so changing n from 6 to 8 shows the factor 3 disappearing and the recurring decimal becoming terminating. The second picture sets the column subtraction of x from 10x beside a circle shaded by the resulting fraction.
3
👀 What the remainders tell you
①In the long division 1 ÷ 3 the remainder is 1 every time, so the 3s in the quotient never stop.
②In 1 ÷ 7 the remainders go 1, 3, 2, 6, 4, 5 and then return to 1, so the six digits 142857 repeat.
③A remainder can only be a number smaller than the divisor, so unless it becomes 0, a remainder must eventually come back, and from then on the quotient repeats.
Terminating and recurring decimals
🔑 When a fraction terminates
①If the denominator of the fraction in lowest terms has only 2s and 5s as prime factors, the decimal terminates.
②If it has even one prime factor other than 2 or 5, the decimal recurs.
③1/8 = 1/2³ terminates (0.125), while 1/6 = 1/(2 × 3) recurs (0.1666…) because of the factor 3.
Test for a terminating decimal
terminates only when the reduced denominator is of the form 2a × 5b
Always reduce the fraction first, then factorise the denominator.
Turning a recurring decimal into a fraction
It is easy to think that any decimal with a long tail is recurring, but a recurring decimal is only one in which the same block repeats forever. A decimal that goes on forever without repeating, like pi, is irrational and cannot be written as a fraction; that comes in the next year. Also, the terminating test must be applied after reducing the fraction. 3/30 has a 3 in its denominator and looks recurring, but it reduces to 1/10 = 0.1.
seegongsik.com
The column subtraction of x from 10x that turns 0.nnn… into a fraction, and a circle shaded by its size
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One-digit repeating block to a fraction
0.a̅ = a9
If the repeating block has one digit, the denominator is 9.
Two-digit repeating block to a fraction
0.a̅b̅ = ab99
If the block has two digits, you compute 100x − x, so the denominator is 99.
When non-repeating digits come first
Recurring decimal with a non-repeating start
0.ab̅ = ab - a90
Make two equations that move the decimal point to the start of the block and one block later, then subtract.
📝 Writing 0.1666… as a fraction
①Let x = 0.1666….
②Then 10x = 1.666… and 100x = 16.666… have identical digits after the decimal point.
③100x − 10x = 15, so 90x = 15 and x = 15/90 = 1/6.
Worked examples
Example 1
Determine whether the fraction 3/40 is a terminating or repeating decimal.
1
Factor the denominator of the reduced fraction into primes.
40 = 2³ × 5
2
If the only prime factors are 2 and 5, it terminates.
only primes 2, 5 ⇒ terminating (3/40 = 0.075)
▸ terminating (0.075)
A reduced fraction terminates exactly when its denominator has only 2 and 5.
Example 2
Express the repeating decimal 0.444… as a reduced fraction.
1
Let x = 0.444… and form ten times it.
x = 0.444…, 10x = 4.444…
2
Subtracting gives 9x = 4.
10x − x = 9x = 4 ⇒ x = 4/9
▸ 4/9
Multiply a repeating decimal by a power of 10 and subtract to remove the period.
Exam checklist
Rule for recurring decimals to fractions
block of n digits → n nines in the denominator
If k digits after the point do not repeat, put k zeros after the nines.
Grade-8 school exam type
What is the smallest natural number a for which a/30 is a terminating decimal?
①1
②2
③3
④5
⑤6
▸ ③ 3
1
Since 30 = 2 × 3 × 5, the prime factor 3 must cancel for termination.
30 = 2 × 3 × 5
2
a must be a multiple of 3, so the smallest is 3.
a = 3 (3/30 = 1/10 = 0.1)
🎯 Exam points
①To test for a terminating decimal, reduce the fraction and check that the denominator's prime factors are only 2 and 5.
②A recurring decimal is written with dots over the first and last digits of the block or a bar over the whole block; the pictures here use the bar.
③To turn a recurring decimal into a fraction, subtract the original from the equation multiplied by 10ⁿ.
④Every rational number can be written as a terminating or recurring decimal, and the converse also holds.
⑤A non-repeating infinite decimal is irrational (next year).