Terminating and repeating decimals

What is it about?

A fraction is a division: $\frac{3}{8} = 3 \div 8$. In long division you write the decimal point after the whole-number part and bring down a zero to every remainder:

So $\frac{3}{8} = 0.375$. Because the remainder eventually becomes $0$, this is a terminating decimal.

For $\frac{3}{7}$ the remainder is never $0$. The remainders are $3, 2, 6, 4, 5, 1$ – and then $3$ again. From there the whole calculation repeats, and so do the digits. This gives a repeating decimal. The digits that repeat are called the period and get a bar over them:

If the period does not start right after the decimal point, the digits before it form the pre-period: $\frac{1}{6} = 0.1\overline{6}$. We distinguish purely repeating decimals ($0.\overline{3}$) and decimals repeating after a pre-period ($0.1\overline{6}$).

Why must something repeat? When dividing by $n$, every remainder is less than $n$. Apart from $0$ there are only the remainders $1, 2, \ldots, n-1$. After at most $n - 1$ steps a remainder must appear for the second time. That is why the period of a fraction with denominator $n$ has at most $n - 1$ digits. The remainder circle shows this: each arrow leads from one remainder to the next; the path ends at $0$ or runs in a cycle.

Terminating or repeating – without dividing: Reduce the fraction fully and factorise the denominator into primes.

• If only the prime factors 2 and 5 occur, you can expand to $10$, $100$, $1000$, …; the decimal terminates: $\frac{3}{8} = \frac{3 \cdot 125}{8 \cdot 125} = \frac{375}{1000} = 0.375$.
• If neither 2 nor 5 occurs, it is purely repeating: $\frac{5}{11} = 0.\overline{45}$.
• If 2 or 5 and other prime factors occur, it repeats after a pre-period: $\frac{5}{12} = 0.41\overline{6}$, because $12 = 2 \cdot 2 \cdot 3$.

Important: reduce first! $\frac{3}{6}$ has the denominator $6 = 2 \cdot 3$, but it equals $\frac{1}{2} = 0.5$.

The number line zoom shows what a digit means: the first decimal place tells you in which tenth between two whole numbers the fraction lies. Magnify this tenth ten times and you read off the hundredths – and so on.

Try it

1. Work through 3/7 step by step with “Next digit”. Write down the remainders. When do you notice that the digits repeat?
2. Compare with 3/8. How can you see on the paper that the division ends?
3. For 1/6 the remainder $4$ repeats at once. Why is the $1$ not under the bar?
4. Look at 1/13 with all remainder arrows. Then tap remainder $2$ on the circle. How many cycles are there for $13$?
5. Compare 1/17 with 1/16. The denominators are almost the same – the results are completely different. Why?
6. Open the number line zoom for 1/3. Why does every level look the same? Compare with the zoom for 3/8.
7. In the table of all denominators the terminating fractions are green. Which denominators are they? Then set the numerator to 3: why do some tiles turn green?
8. Is 22/7 equal to $\pi = 3.14159\ldots$? From which digit on do they differ?

Tasks

Write as a decimal without long division: a) $\frac{7}{20}$ b) $\frac{3}{16}$ c) $\frac{9}{12}$

Task 2: Terminating or repeating?

Decide using only the prime factors of the denominator: $\frac{7}{40}$, $\frac{5}{6}$, $\frac{9}{12}$, $\frac{4}{15}$, $\frac{11}{22}$, $\frac{3}{14}$. Table for numerator 3

Work out $\frac{5}{12}$ by long division. Write down all remainders and give the result with a bar.

Task 4: Two cycles

Work out $\frac{1}{13}$ and $\frac{2}{13}$. Both periods have $6$ digits. Why not $12$, although there are $12$ remainders other than $0$?

Task 5: Back to a fraction

a) Write $0.35$ and $0.125$ as fully reduced fractions. b) We have $\frac{1}{9} = 0.\overline{1}$ and $\frac{1}{99} = 0.\overline{01}$. Use this to write $0.\overline{4}$ and $0.\overline{36}$ as fractions.

Task 6: Record (puzzle)

Which fraction with a denominator up to $50$ has the longest period? Explain why it cannot be longer.

Ideas for the classroom

• Use: About grade 6, decimals: fractions as terminating or repeating decimals, converting. Suitable for introducing the topic (step by step with show=0) and for consolidation (remainder circle, table). The prime factorisation from the simulation “Prime factorisation and divisibility” is applied here.
• Two views: Long division (squared paper, with either the remainder circle or the number line zoom) and Which denominators? (all fractions $\frac{z}{n}$ with $n \leq 50$ as coloured tiles, optionally with prime factors; factors that cancel are crossed out).
• Reasoning instead of memorising: The remainder circle makes the pigeonhole principle visible: there are only finitely many remainders, so one must repeat. The tags next to the remainders show that each remainder fixes its next digit – that is why the digits repeat along with the remainder.
• Going further: The length of the pre-period is the larger of the two exponents of $2$ and $5$ in the reduced denominator ($12 = 2^2 \cdot 3$ → $2$ digits). Converting decimals with a pre-period back into fractions is not covered here.
• Common stumbling blocks:
  • “The period starts as soon as a digit repeats.” – In $\frac{1}{17} = 0.0588\ldots$ the $8$ appears twice, yet the period has $16$ digits. What matters is the remainder, not the digit.
  • “$0.\overline{3}$ is roughly $\frac{1}{3}$.” – $0.\overline{3}$ is exactly $\frac{1}{3}$; only $0.333$ is rounded. Calculators show rounded values such as $0.6666666667$.
  • “The denominator $6$ contains a $2$, so $\frac{1}{6}$ terminates.” – Only $2$ and $5$ may occur.
  • “$\frac{3}{6}$ repeats because $6$ has the factor $3$.” – Reduce first: $\frac{3}{6} = \frac{1}{2}$.