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Keywords: recurring, terminating, decimals, fractions, pattern, repetend
At Right Angles | Vol. 2, No. 3, November 2013 49 in the classroom
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.
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
Exploration of
Recurring Decimals
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50 At Right Angles | Vol. 2, No. 3, November 2013
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:
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,
2 PADMAPRIYA SHIRALI
Fractions with terminating decimals. The unit fractions with terminating decimals were:
Notice the denominators:
2, 4, 5, 8, 10, 16, 25, 32, 40, 50, 64, 80.
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, 52 × 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.
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:
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, 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.
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 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.
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At Right Angles | Vol. 2, No. 3, November 2013 51
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 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 digits (i.e., period 3) were:
We noted that the denominators were multiples of 27 and 37; however, 81 was ‘missing’.
Fractions where the repetend has four digits. Fractions which resulted in a decimal with four repeating
digits (i.e., period 4) were . . . : none! A question which naturally crossed our minds was: Is this true only
for the first 100 unit fractions, or will this always be the case? And can we prove it either way?
Fractions where the repetend has five digits. Fractions which resulted in a decimal with five repeating
digits (i.e., period 5) were:
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 the repetend has six digits. Fractions which resulted in a decimal with six repeating
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 49 or 7×7 does
not appear in the list. We asked ourselves why this should be so.
Fractions where the repetend has seven digits. There were no such fractions.
Fractions where the repetend has eight digits. There was just one fraction which resulted in a decimal
with eight repeating digits (i.e., period 8):
EXPLORATION OF RECURRING DECIMALS 3
Fractions where the repetend has three digits. Fractions which resulted in a decimal with
three repeating digits (i.e., period 3) were:
We noted that the denominators were multiples of 27 and 37; however, 81 was ‘missing’.
Fractions where the repetend has four digits. Fractions which resulted in a decimal with
four repeating digits (i.e., period 4) were . . . : none! A question which naturally crossed our
minds was: Is this true only for the first 100 unit fractions, or will this always be the case?
And can we prove it either way?
Fractions where the repetend has five digits. Fractions which resulted in a decimal with
five repeating digits (i.e., period 5) were:
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 the repetend has six digits. Fractions which resulted in a decimal with
six repeating 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
49 or 7×7 does not appear in the list. We asked ourselves why this should be so.
Fractions where the repetend has seven digits. There were no such fractions.
Fractions where the repetend has eight digits. There was just one fraction which resulted
in a decimal with eight repeating digits (i.e., period 8):
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.
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):
EXPLORATION OF RECURRING DECIMALS 3
Fractions where the repetend has three digits. Fractions which resulted in a decimal with
three repeating digits (i.e., period 3) were:
We noted that the denominators were multiples of 27 and 37; however, 81 was ‘missing’.
Fractions where the repetend has four digits. Fractions which resulted in a decimal with
four repeating digits (i.e., period 4) were . . . : none! A question which naturally crossed our
minds was: Is this true only for the first 100 unit fractions, or will this always be the case?
And can we prove it either way?
Fractions where the repetend has five digits. Fractions which resulted in a decimal with
five repeating digits (i.e., period 5) were:
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 the repetend has six digits. Fractions which resulted in a decimal with
six repeating 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
49 or 7×7 does not appear in the list. We asked ourselves why this should be so.
Fractions where the repetend has seven digits. There were no such fractions.
Fractions where the repetend has eight digits. There was just one fraction which resulted
in a decimal with eight repeating digits (i.e., period 8):
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.
Fractions where the repetend has ten digits. There were no such fractions.
2 PADMAPRIYA SHIRALI
Fractions with terminating decimals. The unit fractions with terminating decimals were:
Notice the denominators:
2, 4, 5, 8, 10, 16, 25, 32, 40, 50, 64, 80.
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, 52 × 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.
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:
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, 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.
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 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. EXPLORATION OF RECURRING DECIMALS 3
Fractions where the repetend has three digits. Fractions which resulted in a decimal with
three repeating digits (i.e., period 3) were:
We noted that the denominators were multiples of 27 and 37; however, 81 was ‘missing’.
Fractions where the repetend has four digits. Fractions which resulted in a decimal with
four repeating digits (i.e., period 4) were . . . : none! A question which naturally crossed our
minds was: Is this true only for the first 100 unit fractions, or will this always be the case?
And can we prove it either way?
Fractions where the repetend has five digits. Fractions which resulted in a decimal with
five repeating digits (i.e., period 5) were:
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 the repetend has six digits. Fractions which resulted in a decimal with
six repeating 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
49 or 7×7 does not appear in the list. We asked ourselves why this should be so.
Fractions where the repetend has seven digits. There were no such fractions.
Fractions where the repetend has eight digits. There was just one fraction which resulted
in a decimal with eight repeating digits (i.e., period 8):
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.
EXPLORATION OF RECURRING DECIMALS 3
Fractions where the repetend has three digits. Fractions which resulted in a decimal with
three repeating digits (i.e., period 3) were:
We noted that the denominators were multiples of 27 and 37; however, 81 was ‘missing’.
Fractions where the repetend has four digits. Fractions which resulted in a decimal with
four repeating digits (i.e., period 4) were . . . : none! A question which naturally crossed our
minds was: Is this true only for the first 100 unit fractions, or will this always be the case?
And can we prove it either way?
Fractions where the repetend has five digits. Fractions which resulted in a decimal with
five repeating digits (i.e., period 5) were:
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 the repetend has six digits. Fractions which resulted in a decimal with
six repeating 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
49 or 7×7 does not appear in the list. We asked ourselves why this should be so.
Fractions where the repetend has seven digits. There were no such fractions.
Fractions where the repetend has eight digits. There was just one fraction which resulted
in a decimal with eight repeating digits (i.e., period 8):
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.
EXPLORATION OF RECURRING DECIMALS 3
Fractions where the repetend has three digits. Fractions which resulted in a decimal with
three repeating digits (i.e., period 3) were:
We noted that the denominators were multiples of 27 and 37; however, 81 was ‘missing’.
Fractions where the repetend has four digits. Fractions which resulted in a decimal with
four repeating digits (i.e., period 4) were . . . : none! A question which naturally crossed our
minds was: Is this true only for the first 100 unit fractions, or will this always be the case?
And can we prove it either way?
Fractions where the repetend has five digits. Fractions which resulted in a decimal with
five repeating digits (i.e., period 5) were:
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 the repetend has six digits. Fractions which resulted in a decimal with
six repeating 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
49 or 7×7 does not appear in the list. We asked ourselves why this should be so.
Fractions where the repetend has seven digits. There were no such fractions.
Fractions where the repetend has eight digits. There was just one fraction which resulted
in a decimal with eight repeating digits (i.e., period 8):
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.