ACARA v9 CONTENT DESCRIPTION “recognise terminating and recurring decimals, using digital tools as appropriate”
Two ways a decimal can behave
Every fraction made from whole numbers can be written as a decimal, and that decimal does one of just two things: it stops, or it repeats. A decimal that stops is called a terminating decimal. A decimal that repeats a fixed block of digits forever is called a recurring decimal, or a repeating decimal. For example, one quarter is 1/4 = 0.25, which stops neatly after two places, while one third is 1/3 = 0.333..., where the digit 3 carries on without end. Both kinds are rational numbers, because both come from a fraction of two whole numbers. The useful question is simply which of the two behaviours a given fraction will show, and the rest of this unit gives you reliable ways to tell before and after you divide.
Two ways a fraction can turn out
Some fractions stop; others repeat a block forever.
A fraction becomes either a terminating decimal that stops or a recurring decimal that repeats a block forever.
Terminating decimals that stop
A terminating decimal happens when the division of the top number by the bottom number eventually reaches a remainder of 0. Once the remainder is 0 there is nothing left to bring down, so the decimal simply stops. Take 3/8: dividing 3 by 8 gives 0.375, and after three decimal places the remainder is 0, so the answer ends there. The same thing happens with 1/4 = 0.25, with 1/5 = 0.2, and with 1/8 = 0.125. Each of these has a finite number of decimal places, which is exactly what the word terminating describes. You can always confirm it by doing the short division by hand and watching for the moment the remainder becomes 0.
A terminating decimal
The division reaches a remainder of 0 and stops.
A terminating decimal stops because the division eventually reaches a remainder of 0.
Recurring decimals that repeat forever
A recurring decimal happens when the division never reaches a remainder of 0. Instead a remainder you have already met comes back, and from that point the same digits cycle round again and again. With 1/3 the division keeps giving a remainder of 1, so the result is 0.333... with the digit 3 repeating forever. With 1/6 you get 0.1666..., where only the 6 repeats. With 1/7 a whole block repeats: 1/7 = 0.142857142857..., and the six digits 142857 return in the same order without end. To write a recurring decimal clearly in plain text, show enough of the pattern and add three dots, as in 0.1666..., or state in words that the block 142857 repeats. A recurring decimal still follows a fixed pattern, and that is what separates it from an irrational number such as the square root of 2, whose digits never settle into a repeating block. Recurring decimals are rational; irrational numbers are not.
A recurring decimal
A fixed block of digits repeats without end.
A recurring decimal repeats a fixed block of digits forever; here 142857 repeats, and in 1/6 the digit 6 repeats.
Reading the denominator to predict the decimal
There is a quick test that predicts the result before you divide, as long as the fraction is written in lowest terms. Factor the denominator into primes. If the only prime factors are 2 and 5, the decimal terminates; if any other prime factor appears, such as 3, 7 or 11, the decimal recurs. The reason is that our place-value system is built on ten, and 10 = 2 x 5, so a denominator made only of 2s and 5s can be rewritten over a power of ten, which always gives a decimal that stops. For example, 7/40 terminates because 40 = 2 x 2 x 2 x 5, which is made only of 2s and 5s. By contrast, 5/12 recurs because 12 = 2 x 2 x 3 carries a factor of 3, and that 3 cannot be folded into a power of ten. Always reduce the fraction first, because a hidden common factor can change the denominator you are testing.
The denominator test
Only 2s and 5s means it stops; any other prime makes it recur.
In lowest terms, if the only prime factors of the denominator are 2 and 5 the decimal terminates; any other prime factor makes it recur.
Using digital tools to check
A calculator or other digital tool is a fast way to confirm whether a fraction terminates or recurs, and the curriculum expects you to reach for one where it helps. There is one trap worth remembering: the display holds only a limited number of digits, so it rounds the last one. A recurring decimal can therefore look as though it ends. A calculator might show 1/7 as 0.1428571, which appears finite, yet the true value keeps repeating the block 142857 forever. The safe habit is to use the tool to confirm a result you can also reason about: carry out a little of the division by hand, or spot the repeating block, and let the screen agree with your thinking. Digital tools are there to confirm the reasoning, not to replace it.
Checking with a digital tool
The screen is fast but rounded, so read it with care.
A digital tool quickly shows the decimal; remember the screen rounds, so a recurring decimal may look like it stops when it does not.
Why this matters
Knowing in advance whether a fraction will terminate or recur is a genuinely useful skill. It lets you predict and check answers quickly, so an unexpected run of repeating digits no longer looks like a mistake. It helps you decide when to keep an exact fraction and when a rounded decimal is good enough for the job at hand. It also sharpens the larger picture of the number system: terminating and recurring decimals are exactly the rational numbers, and telling them apart from the irrational numbers, which neither stop nor repeat, is the heart of this part of Year 8. In this unit the goal stays on recognising and classifying the two kinds; turning a recurring decimal back into a fraction by algebra is a later topic and is not needed here.
Quick self-check
1. Which decimal is a terminating decimal?
2. What is 1/3 as a decimal?
3. A fraction in lowest terms has denominator 8 (8 = 2 x 2 x 2). Its decimal will...
4. A fraction in lowest terms has denominator 6 (6 = 2 x 3). Its decimal will...
5. In the recurring decimal 0.142857142857..., what repeats?