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Assignment 3
Types of Numbers – Operations with Rational numbers
Fractions and Decimals
By Messrs Angus Begg & Timothy Williamson (with contributions from other class
members’ wikis and webpages)
Focus Question: All fractions can be expressed as terminating or repeating decimals.
This is because all rational numbers can be expressed as a/b where b ≠ 0, a is the
numerator and b is the denominator. Therefore, all fractions fit into the descriptor of
rational numbers which are always terminating or repeating decimals. However, note that
if the denominator is 1, then the fraction is expressed as an integer, which is arguably
neither a terminating nor repeating decimal.
From the two spreadsheets investigated, we find that the denominator determines the
nature of the fraction.
The results follow:
Denominator Decimal for 1/denominator Decimal Nature
2
.5
1 terminating figure
3
.333…
1 repeating figure
4
.25
2 terminating figures
5
.2
1 terminating figure
6
.1666…
1 terminating then 1 repeating figure
7
.142857142857…
6 repeating figures
8
.125
3 terminating figures
9
.111…
1 repeating figure
10
.1
1 terminating figure
11
.0909
2 repeating figures
12
.08333…
2 terminating then 1 repeating figures
13
.076923076923
6 repeating figures
14
.0714285714285
1 terminating 6 repeating figures
15
.0666…
1 terminating 1 repeating figure
41
.0243902439
5 repeating figures
Note: some fractions do not obviously follow the rule, for example 2/4 has 1 instead of 2
terminating decimals as predicted. This is because 2/4 is a ‘disguised fraction’ and
actually equals ½, so remember to cancel fully to avoid confusion.
The above table gives a large enough sample to demonstrate that all fractions follow a
pattern depending on their denominator.
Investigation: We have seen that all fractions follow sets of rules, but is it possible to
predict what a fraction’s rule will be without having to calculate it?
To do this, first take the denominator and calculate its prime factors. For example, 28 has
prime factors of 22x7. The rule for 22 (4) is 2 terminating decimals, and the rule for 7 is 6
repeaters. If the rules are combined, you get 2 terminating and 6 repeating figures, this is
the rule for 28. When calculated, 1/28 = 0.03571428571428…… which matches the
prediction.
Examples are as follows:
Denominator
32
10
20
14
Prime Factors
25
2x5
22x5
2x7
1155000
2^3x5^4x7x3x11
Rule
5 non-repeating digits(0.03125)
1 non-repeating* digit(0.1)
2 non-repeating digits(.05)
1 non-repeating 6 repeating
digits (0.0714285714285…)
4 non-repeating digits, 6 cyclic
repeating digits (0.0000008658)
* Note that if there are two prime factors with the same rule, example 2 and 5, you only
take the rule once, so you have one terminating instead of 2 terminating. However if there
are higher powers of the same number, eg 22, you use the rule twice, so you have 2
terminating figures.
Because of the results of the investigation above, we can say that a decimal will only
terminate, if the fraction’s denominator has only prime factors of 2x and/or 5x. This
works because we use a base 10 (or decimal) system and 2 and 5 are the only factors of
10, meaning that these and their multiples are the only numbers that will produce a
terminating result.
It is also important to note that if there is a 2 and a 5 as the prime factor that there will
only be one terminating figure instead of two as you might expect. Only one number will
influence the number of non-repeating decimal places and that is determined by the
highest power of the factors 2 and 5. The highest power dictates the number of decimal
places.
As shown above, all odd denominators, excluding 5’s, have some repeating component.
As stated in the proposed rule, numbers that are multiples of prime numbers repeat the
same way as the prime factor.
3 is a single repeating decimal. Therefore, 6 and 9 are also single repeaters eg 5/9 equals
0.555… and 1/6 equals 0.1666…
There are single repeaters, double repeating, triple repeaters and cyclic repeaters.
An example of a double repeater is 1/11, which equals 0.0909…
An example of a triple repeater is 1/37, which equals 0.027027…
Mixed decimals have denominators like 6, 15, 18 and 36. By breaking these numbers
down we find that they have one prime factor of either 2 or 5, or powers of 2 or 5, and
one other prime factor, which dictates the type of repeater. For example, 18 has factors of
9 and 2 which, in terms of primes, is 32 x 2. This will give a decimal with a one digit nonrepeater (because the 2 is to a power of 1) and a single digit repeater (because the 3 gives
a single digit repeater) such as 0.KTTTTTTTTTTTTTT…… using pronumerals to
represent the repeating figures. When calculated, 1/18 = 0.05555…. as predicted.
The equivalent decimal of a fraction is a cyclic repeater when the denominator is a
prime number >5 (with the exception of 11) when the fraction is expressed in its simplest
form. The multiples of all these numbers are also cyclic repeaters (1/7 is a cyclic repeater
and so is 1/14). The number of repeating digits is always less than the denominator (1/7
is a 6 digit repeater and 1/17 is a 16 digit repeater).
Cyclic repeaters ‘shuffle’ as shown below ie the numbers stay the same but appear in a
different order. For example, 1/7=0.142857142857 and 2/7=0. 285714285714, these two
numbers use the same digits but just in a different order as shown by the colour coding.
Example:
Numerator Denominator Decimal
1
7
0.142857143
2
7
0.285714286
1
13
0.076923077
3
13
0.230769230
It is observed that occasionally there is a number where the prime factors of the
denominator “conflict”. A number like this is 33, the prime factors are 3 and 11, the rule
for 3 is one repeating digit and the rule for 11 is 2 repeating digits. In a situation such as
this, the number with the most repeating digits will take priority, so 33 has 2 repeating
decimals. When calculated, 1/33 = 0.030303…. as predicted.
A list of the prime numbers to 30:
Denominator
3
5
7
11
13
17
19
23
29
Nature of decimal
1 repeating
1 terminating
6 cyclic repeating
2 repeating
6 cyclic repeating
16 cyclic repeating
18 cyclic repeating
22 cyclic repeating
28 cyclic repeating
From this, we can conclude that most fractions with a denominator of a prime over 11
will produce a cyclic repeating decimal. A cyclic repeating decimal is any decimal where
the numbers involved in the repeating pattern appear the same for the different
numerators with the same denominator but in a different order.
For example 1/13 is 0.076923… and 3/13 is 0.230769…, these are the same numbers
appearing but in a different order.
It is also apparent that the number of cyclic repeaters is only ever equal to the
denominator -1. For example 1/29 has 29-1=28 cyclic repeaters.
Interesting Patterns:
1/11=0.090909090909
2/11=0.1818181818
7/11=0.6363636363
If you take the number over 11 (the numerator) and multiply it by 9 you get the two
repeating numbers (this means that the two numbers will also equal 9 when added).
1/49=0.02040816… (2,4,8,16)
3/49=0.061224… (6,12,24)
8/49=0.1632…(16,32)
This works until the numbers exceed 60 then it adds 1 to the expected pattern then stops
working for the figures after this (2/49=0.0204163265[+1 to 32x2, pattern stops after
this]).
Powers of prime numbers produce unusual patterns which do not follow the pattern of the
original prime number. For example, 1/27 should follow a ‘3’ pattern because 27=33,
however 1/27 = 0.037037… which is three repeaters instead of the predicted one. Other
powers of three have different patterns.
Another example, as shown above, is 49 which is 72, and yet it does not follow the same
pattern as 7 does. When it comes to these numbers there is no discernable pattern
although interesting number sequences do come through as shown in x/49.
Conclusion:
 All fractions are rational numbers and, hence, repeating or terminating decimals.
 The nature of the decimal produced by each fraction follows a rule that depends
on the denominator and its prime factors.
 The rules for each prime factor can be used to predict the nature of fractions with
larger denominators.
 You can tell whether a number terminates or not by examining it’s prime factors.
If they have prime factors which only involve 2 and 5, or their powers, the
decimal will terminate. The number of places equals the highest power of the
factors.
 Denominators which have some prime factors other than 2 and/or 5, will have a
repeating component.
 There is, however, one set of numbers that is harder to classify and no rule has
been found for these. They involve powers of prime numbers.
And a final example from Joe Kozera:
If you understand everything we have said so far then you're a very cool person but here
comes the true test. The fraction, one over 1155000 contains many of these patterns. If
you took the liberty to get the prime factors yourself before reading this you would know
that they are: 2^3x5^4x7x3x11.
Now predict what this number will look like when 1 is divided by it. If you can make a
prediction very good. I predict it will look something like this; 0.(terminate) x4 (6
number cyclic repeat).
0.wxyz (abcdef)
And the answer is: 0.0000008658008658
This answer is correct because 0.0000008658008658 in orange we have four 0’s which
represent our four terminates, then in blue we have our 6 number cyclic repeat.
One last explanation for people who still don’t get the complex relationship.
The prime factors were 2^3x5^4x7x3x11. So first we find out how many terminates there
will be before the repeat. We see 2^3 and 5^4. Remember that the highest number of
powers above a terminating prime factor will equal the number of them in the decimal.
So 2 has three but 5 has four and since the latter is larger we now know what our number
is starting to look like. 0. (terminate) x4.
Then we need to find what our repeater is going to look like. Remember that the highest
number of repeating digits is the repeater that will be used in our decimal. So we have 3,
which gives a 1 number repeat, 11, which gives a 2 number repeat, and 7, which gives a 6
number repeat. So 7 obviously has the highest amount of repeats so 7 goes into our mix.
We now have 0.(terminate) x4 (6 number cyclic repeat).
And so that is the correct “look” of the decimal only using the denominator. Now we all
have the knowledge of how to predict how a decimal will “look”.