Exploration of
Recurring Decimals
We first expose children to the topic of converting fractions to
decimal numbers in class 5 or 6. At that point children notice that
some fractions terminate and some do not, and they come across
terms like terminating decimals and recurring decimals. They are
also shown the usage of the bar or dot notation. Generally most
textbooks do not proceed beyond this point. Later (class 8 or 9)
they are taught how to rationalize numbers. The activity I describe
here is one which I have tried with class 8 children. It proved to be
an interesting investigation into the patterns in recurring decimals
leading to generalization and looking at the reverse process initially
through a trial and error approach followed by arriving at the
procedure for rationalization.
in the classroom
Math Club
P��������� S������
I first posed a question to the children, “Will all fractions give
rise to either terminating decimals or recurring decimals of some
periodicity”? They were not too certain. Some confidently said yes. I
asked in return “Can you prove why they should either terminate or
recur with some periodicity?” They were not yet exposed to formal
proof. So though they knew the answer intuitively, they found it
Keywords: recurring, terminating, decimals, fractions, pattern, repetend
At Right Angles | Vol. 2, No. 3, November 2013
49
difficult to articulate it. So I asked them further questions: “What remainders can you get when you divide
a number by 5?” They responded, “0, 1, 2, 3, 4”. I asked the same question for other divisors, and within a
short while they saw that if the divisor is n, then there are just n possible remainders (including zero), and
once the same remainder appears again, the quotient pattern repeats itself from that point.
I then set them the task of finding the decimal expansions for all unit fractions (i.e., fractions of the type
1/n where n is a positive integer) with denominators from 2 till 100. The class had twenty students, and
each one computed the value of five such fractions within an hour. In the case of fractions whose decimals
terminated, they had to write the complete answer. In the case of fractions whose decimals recurred,
they had to stop at the point where the digits began to recur for the second time. However I asked them
to omit fractions for which repetition had not happened by the tenth decimal place. (Later, I provided
the computer generated result.) As they did this, some began to notice some interesting patterns in their
answers.
We then collated all the fractions on a chart classifying them into the following groups. Fractions which
terminated were grouped together; then fractions with period 1 (i.e., where the repeating portion or
repetend has just one digit) were grouped together; then came fractions with period 2, period 3, period 4,
etc. Here are the summarized results.
Fractions with terminating decimals. The unit fractions with terminating decimals were:
Notice the denominators:
Children noticed that the list contains all the powers of 2 (i.e., 2, 4, 8, 16, 32, 64) and another set 5, 10, 20,
25, 40, 50, 80 which could be rewritten as 5, 5 × 2, 5 × 4, 5 × 8, 25 × 2, 5 × 16, 52 × 4. After some discussion
they generalized the result by stating that the denominators are of the form 2n, or 2 × 5n, or 5 × 2n, with n
belonging to the set of positive integers N. Each denominator listed above is of one of these forms.
Generalizing further, we may say that whenever the denominator has the form 2a × 5b where a and b are
non-negative integers, the decimal expansion terminates.
Fractions where the repetend has one digit. Here are the fractions for which the repetend has just one
digit:
hildren quickly noticed that the denominators are not consecutive and have gaps emergi
Children quickly noticed that the denominators are not consecutive and have gaps emerging after the
initial set of numbers. They factorized the denominators as 3 × 1, 3 × 2, 3 × 3, 3 × 4, 3 × 5, 3 × 6, 3 × 8,
3 × 10, 3×12, 3 × 15, 3 × 16, 3 × 20, 3 × 24, 3 × 25, 3 × 30, 3 × 32.
These numbers could now be sorted as a set consisting of 3 times powers of 2 (3 × 1, 3 × 2, 3 × 4, 3 × 8, 3
× 16, 3 × 32), another set consisting of 3 times multiples of 2 and 5 (3 × 5, 3 × 10, 3 × 15, 3 × 20, 3 × 25, 3
× 30) and a third set consisting of 32 times powers of 2 (3 × 3, 3 × 6 or 3 × 3 × 2, 3 × 12 or 32 × 22, 3 × 24
which is 32 × 23); more generally, fractions in which the denominator has the form 3 × 2n, 32 × 2n, 3 × 5n.
Generalizing,
50
At Right Angles | Vol. 2, No. 3, November 2013
we may say that all fractions where the denominator has the form 3 × 2a × 5b and 32 × 2a × 5b give rise to
decimal numbers with period 1. Each denominator listed above is of one of these forms.
Fractions where the repetend has two digits. Fractions which resulted in a decimal with two repeating
digits (i.e., period 2) were:
When we first saw the list we concluded that the denominators w
When we first saw the list we concluded that the denominators were all the multiples of 11, but then we
noticed that 77 is not in this list. This initially came as a surprise. We could see why it was not so only later
when we studied fractions with period 6.
Fractions where the repetend has three digits. Fractions which resulted in a decimal with three
repeating
(i.e., digits
period(i.e.,
3) were:
three digits
repeating
period 3) were:
minds was: Is this true only for the first 100 unit fractions, o
minds was: Is thisWe
truenoted
onlythat
for the
first 100 unit fractions,
o
the denominators
were multiples
of 27 and 37; however, 81 was ‘missing’.
We noted that the denominators were multiples of 27 and 37; however, 81 was ‘missing’.
minds
was:
this
true
for
the
100
unit
oo
mindswhere
was: Is
Isthe
thisrepetend
true only
onlyhas
for four
the first
first
100Fractions
unit fractions,
fractions,
Fractions
digits.
which
resulted in a decimal with four repeating
Fractions where the repetend has five digits.
digits
(i.e.,
period
4)
were
.
.
.
:
none!
A
question
which
naturally
crossed
our minds was: Is this true only
five repeating digits (i.e., period 5) were:
Fractions where
the
repetend
has
five
digits.
for the first 100 unit fractions, or will this always be the case? And can we prove it either way?
five repeating digits
(i.e.,was:
period
minds
Is 5)
thiswere:
true only for has
the first 100
unit fractions, o
Fractions
where
the
digits.
Fractions
where
the repetend
has five
digits.
Fractions which resulted in a decimal with five repeating
Fractions
where
the repetend
repetend
has five
five
digits.
digitsfive
(i.e.,repeating
period 5)digits
were:(i.e.,
five
repeating
digits
(i.e., period
period 5)
5) were:
were:
This was the third time we noticed a multiple of the first denom
This was the thirdFractions
time we noticed
multiple
of the
where athe
repetend
hasfirst
fivedenom
digits.
five
repeating
digits (i.e.,weperiod
5) were:
This
This was
was the
the third
third time
time we noticed
noticed aa multiple
multiple of
of the
the first
first denom
denom
This was the third time we noticed a multiple of the first denominator appearing in the list, and it
provoked us to look for an explanation for this.
Fractions
where
has six adigits.
Fractions
resulted in a decimal with six repeating
This was
the the
thirdrepetend
time we noticed
multiple
of the which
first denom
digits (i.e., period 6) were many in number:
Multiples of 7 and 13 could be seen in the denominators, but it was interesting to see that
Fractions where
the
hascould
sevenbedigits.
Multiplesrepetend
of 7 and 13
seen in the denominators, but it was interesting to see that 49 or 7×7 does
not appear
in the list.
asked
ourselves why this should be so.
Fractions where
the repetend
hasWe
seven
digits.
Fractions
where the repetend
has seven
digits. There were no such fractions.
Fractions
Fractions where
where the
the repetend
repetend has
has seven
seven digits.
digits.
Fractions where the repetend has eight
digits. There was just one fraction which resulted in a decimal
re was
with eight repeating digits (i.e., period 8):
):
Fractions
where
the
repetend
has
digits.
reseven
was
Fractions where the repetend has nine
digits.
There was just one fraction which resulted in a decimal
with nine repeating digits (i.e., period 9):
):
Fractions where the repetend has ten digits. There were no such fractions.
At Right Angles | Vol. 2, No. 3, November 2013
51
Finding the fraction matching a given recurring decimal
Following the above exercise I wanted them to look at the process in reverse. I posed the question “Given a
decimal number how do we convert it into a fraction?” Terminating decimals do not pose any difficulty as
they can be written as whole numbers divided by suitable powers of 10 and then reduced (e.g., 0.034 =
34 / 1000 = 17 / 500). So the question got narrowed down to: what does one do with recurring decimals?
We used a trial-and-error approach. We took the recurring decimal .027027027. . .. What fraction will give
this decimal? The children first looked at 1 / 2 (= .5) and 1 / 4 (= .25) and realized that the denominator had
to be bigger than 4. Then they tried with 1 / 10 (= .1) and 1 / 20 (= .05) and realized that the denominator
had to be bigger than 20 as well. They now tried with 1/30 (= .0333. . .) and 1 / 40 (= .025) and realized that
the fraction lies between 1 / 30 and 1 / 40. Then they tried 1 / 35 (= .02814. . .) and further narrowed down
the range to 1 / 35 to 1 / 40. Soon they obtained the result that 1 / 37 = .027027. . ..
Now they had to be introduced to a more systematic procedure for ‘rationalization’. We looked at the
result obtained by multiplying a decimal number with powers of 10. We followed this by looking at
various recurring decimals to determine by what power of 10 they had to be multiplied in order to move
the recurring part to the whole number position. (For example, .027027. . . needs to be multiplied by 1000
to change it to 27.027027. . ., whose repeating portion is identical to that of the original number). Then I
asked them what operation could be performed using this and the original number that would result in
the elimination of the recurring part. Soon they saw that subtraction of the original number from the new
one would yield the answer 27. Now I introduced them to the procedure of treating the original number as
x and subtracting x from 1000x to give 999x which must equal 27. From this we deduce that x = 27 / 999,
which when reduced yields 1 / 37.
This whole exercise took us three classes (some calculations were set as homework). We found many
interesting patterns; for example, 1 / 81 = 0.012345679 012345679 . . ., and we see that 8 is missing from
the repetend. It also raised many questions, not all of which we could answer. For example, why did we
not find any fractions which give rise to decimals with period 4, 7 or 10 in the unit fractions from 1 / 2 to
1 / 100? Does this continue to hold if we extend our range? And is there a way by which we can prove this
conclusively?
Maybe some day we will find answers to some of these questions! [Editor’s note. We shall take up some of
these questions in the next issue of At Right Angles.]
PAdmAPRiyA ShiRAli is part of the Community math Centre based in Sahyadri School (Pune) and
Rishi Valley (AP), where she has worked since 1983, teaching a variety of subjects – mathematics,
computer applications, geography, economics, environmental studies and Telugu. For the past few
years she has been involved in teacher outreach work. At present she is working with the SCERT (AP)
on curricular reform and primary level math textbooks. in the 1990s, she worked closely with the
late Shri P K Srinivasan, famed mathematics educator from Chennai. She was part of the team that created
the multigrade elementary learning programme of the Rishi Valley Rural Centre, known as 'School in a Box'.
Padmapriya may be contacted at padmapriya.shirali@gmail.com
52
At Right Angles | Vol. 2, No. 3, November 2013