Number: Recurring decimals, or just the wrong base?
View Sequence overviewWhether a simplified fraction has a terminating representation depends on the relationship between its denominator and the base in which it is written.
A simplified fraction has a terminating representation in a given base when every prime factor of its denominator is also a prime factor of the base.
Each group
Sometimes always never Student sheet
Blank fraction sort Student sheet
Completed cards from Fraction sort Student sheet from Task 1
Task
Remind the class of the two conjectures developed in the previous lessons:
- In base 10, a simplified fraction has a terminating decimal when its denominator contains no prime factors other than 2 and 5.
- In binary, a simplified fraction has a terminating representation when its denominator is a power of two.
Ask: Why do these rules seem so different? Could there be a single rule that explains both?
Discuss briefly how base 10 and binary are only two of many possible counting systems. For example, quantities can be grouped in twelves (a dozen), the Maya used a base-20 system, and traces of a base-60 system still appear today in the way we measure time and angles.
Explain that mathematicians often test a conjecture by looking at new cases. If a rule works for base 10 and binary, will it also work in base 6, base 12, or base 20?
Pose the problem: Which fractions have terminating representations in different bases, and what determines whether a fraction terminates or recurs?
Introduce base 12 as the next system to test. Draw a place value chart on the board, leaving the columns after the point blank.
| 144s | 12s | 1s | . | $\frac{1}{12}$s | $\frac{1}{144}$s |
|---|---|---|---|---|---|
Ask: What should the first column to the right of the point be worth?
Since it is one place to the right of the ones column, it is worth $\frac{1}{12}$.
Ask students to use the same reasoning to determine the value of the next column.
Check understanding with a few simple examples. For example:
- What fraction is represented by $0.3_{12}$?
- What fraction is represented by $0.6_{12}$?
Return to the question: Which fractions have terminating representations in different bases?
Students should work in pairs and use Wolfram Alpha to support the investigation.
If students test a base larger than 10, Wolfram Alpha’s results may include letters. For example, $\frac{8}{9}$ in base 12 might be written as 0.A8, where A represents the digit value 10. When more than ten digit symbols are needed, letters are used after 9.
If pairs aren’t sure where to start, offer one of the following strategies.
Fix the fraction, vary the base
Choose a fraction and test it across as many bases as possible, recording whether its representation terminates or recurs in each base.
Good fractions to start with include: $\frac{1}{3}$, $\frac{1}{6}$, and $\frac{1}{7}$.
Ask:
- In which bases does the fraction terminate?
- In which bases does it recur?
- What do the successful bases have in common?
Fix the base, vary the fraction
Choose a base and test a range of fractions within that base.
Good bases to start with include base 7 and base 12.
Ask:
- Which fractions terminate?
- Which recur?
- What do the fractions that terminate have in common?
Group by denominator type
When exploring a new problem, students often begin by sorting examples into familiar categories. One common approach is to group fractions according to the type of denominator they have, such as prime denominators ($\frac{1}{5}$, $\frac{1}{7}$, $\frac{1}{11}$) and composite denominators ($\frac{1}{4}$, $\frac{1}{9}$, $\frac{1}{15}$).
Students can test each group across the same set of bases and compare the results.
Ask: Did every fraction in a group behave the same way? For any that didn’t, what was different about their denominators?
This approach often produces useful observations, but it may also reveal that whether a denominator is prime or composite is not, by itself, enough to explain why a fraction terminates. Students may need to look more closely at the prime factors of both the denominator and the base.
Checkpoint: Pause the class and ask a few groups to briefly explain how they have been organising their investigation. Possible prompts:
- Did you fix the fraction and vary the base, or fix the base and vary the fraction? Why?
- What strategy has helped you find useful patterns so far?
- Has anyone changed their approach partway through the investigation? What prompted the change?
Record different investigation strategies where students can see them.
Once several approaches have been shared, shift attention to emerging ideas:
- What conjectures are we starting to generate?
- Does anyone have a result that supports or challenges another group’s conjecture?
- Could a single rule explain all of the results collected so far?
Return students to the investigation and encourage them to test examples that support, challenge, or refine their emerging conjectures.
Hand out the Sometimes, always, never Student sheet. Ask students to record a conjecture, along with examples that support it and any counterexamples they have encountered.
Ask: You’ve now investigated fractions in a range of different bases. What have you learnt?
Students might suggest that:
- the same fraction can terminate in one base and recur in another.
- different bases seem to favour different fractions.
- whether a fraction terminates depends on both the denominator and the base.
- the base 10 and binary rules appear to be connected somehow.
Ask: Is having lots of examples the same thing as knowing why something is true?
Discuss the difference between a conjecture and a justification.
- A conjecture is a claim based on patterns that have been observed. Evidence can increase confidence in a conjecture, but another example could still prove it wrong.
- A justification is different. A justification explains why a mathematical statement must be true.
Students may have developed different conjectures during the investigation. Rather than trying to reach a single class rule, focus discussion on what kinds of explanations might account for the patterns observed.
Try the following examples:
- Take $\frac{1}{4}$ in base 10. Can it be rewritten as an equivalent fraction with 10, 100, or 1000 on the denominator?
- What does that tell us about its decimal representation?
- Now try $\frac{1}{6}$. Can it be rewritten with 10, 100, or 1000 on the denominator? Why or why not?
- What about $\frac{1}{4}$ in base 3? Can it be rewritten with 3, 9, or 27 on the denominator? What does that suggest about its representation in base 3?
Guide students towards the idea that terminating representations seem to be connected to powers of the base being used.
For example:
- $\frac{1}{4}=\frac{25}{100}$, so it terminates in base 10.
- $\frac{1}{8}=\frac{125}{1000}$, so it terminates in base 10.
- $\frac{1}{6}$ cannot be written with a power of ten on the denominator, so it recurs in base 10.
Conclude by noting that mathematicians often move from noticing patterns, to making conjectures, to developing explanations. The investigations in this sequence have produced increasingly powerful conjectures about terminating representations. The professional learning section below provides a fuller mathematical explanation of why these patterns occur.
Why does whether a fraction’s decimal recurs depend on the base?
A fraction terminates in a given base when it can be rewritten as an equivalent fraction whose denominator is a power of that base.
In base 10, terminating decimals are fractions with denominators such as 10, 100, 1000, and so on. For example: $\frac{25}{100}=0.25$ because 100 is a power of ten.
To determine whether a fraction will terminate, we can ask a simple question: Can the denominator be turned into a power of the base by multiplying by a whole number?
Take $\frac{1}{4}$ in base 10. Because $100 \div 4 = 25$ we can write $\frac{1}{4}=\frac{25}{100}$. Since $\frac{1}{4}$ can be rewritten with a power of ten on the denominator, its decimal representation terminates.
Now consider $\frac{1}{6}$. Neither 10, nor 100, nor 1000 is divisible by six, so there is no way to rewrite $\frac{1}{6}$ with a power of ten on the denominator. As a result, its decimal representation cannot terminate and must recur.
The same idea works in any base. For example, in binary the denominators of terminating representations are powers of two. A fraction such as $\frac{1}{8}$ terminates in binary because its denominator is already a power of two.
A fraction such as $\frac{1}{10}$ does not terminate in binary because no power of two is divisible by ten.
This is where prime factors become important. Every power of ten is made entirely from the prime factors two and five: $10=2\times5$, $100=2^2\times5^2$, $1000=2^3\times5^3$, and so on.
A denominator can divide a power of ten only if it is also built entirely from twos and fives.
Similarly, every power of two is built entirely from the prime factor two, so a fraction terminates in binary only when its denominator is also built entirely from twos.
This leads to the general rule: A simplified fraction has a terminating representation in a base when every prime factor of its denominator is also a prime factor of the base.
A fraction terminates in a given base when it can be rewritten as an equivalent fraction whose denominator is a power of that base.
In base 10, terminating decimals are fractions with denominators such as 10, 100, 1000, and so on. For example: $\frac{25}{100}=0.25$ because 100 is a power of ten.
To determine whether a fraction will terminate, we can ask a simple question: Can the denominator be turned into a power of the base by multiplying by a whole number?
Take $\frac{1}{4}$ in base 10. Because $100 \div 4 = 25$ we can write $\frac{1}{4}=\frac{25}{100}$. Since $\frac{1}{4}$ can be rewritten with a power of ten on the denominator, its decimal representation terminates.
Now consider $\frac{1}{6}$. Neither 10, nor 100, nor 1000 is divisible by six, so there is no way to rewrite $\frac{1}{6}$ with a power of ten on the denominator. As a result, its decimal representation cannot terminate and must recur.
The same idea works in any base. For example, in binary the denominators of terminating representations are powers of two. A fraction such as $\frac{1}{8}$ terminates in binary because its denominator is already a power of two.
A fraction such as $\frac{1}{10}$ does not terminate in binary because no power of two is divisible by ten.
This is where prime factors become important. Every power of ten is made entirely from the prime factors two and five: $10=2\times5$, $100=2^2\times5^2$, $1000=2^3\times5^3$, and so on.
A denominator can divide a power of ten only if it is also built entirely from twos and fives.
Similarly, every power of two is built entirely from the prime factor two, so a fraction terminates in binary only when its denominator is also built entirely from twos.
This leads to the general rule: A simplified fraction has a terminating representation in a base when every prime factor of its denominator is also a prime factor of the base.
Return to the question: Is $\frac{1}{3}$ a terminating or recurring decimal?
Students should now recognise that the question cannot be answered until the base is known. In base 10, $\frac{1}{3}$ has a recurring representation. In base 3, $\frac{1}{3}$ has a terminating representation. The fraction has not changed. The representation has.