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© T Madas
13
= 0.333…
16
= 0.1666…
17
= 0.142857142857142857…
© T Madas
A recurring decimal is a never ending decimal, whose decimal part repeats with a pattern
0.333…
1.1666…
0.8333…
2.58333…
0.0913913913…
0.285714285714285714
…
=
=
=
=
=
=
13
76
56
3112
9139990
27
A recurring decimal can always be written as a fraction
© T Madas
A recurring decimal is a never ending decimal, whose decimal part repeats with a pattern
0.333…
0.666…
0.1666…
0.8333…
=
=
=
=
13
23
16
56
These recurring decimals are worth remembering
© T Madas
A terminating decimal comes to an end after a number of decimal places
0.25
0.4
0.52
0.3875
2.2975
1.475585937
5
=
=
=
=
=
=
14
25
13 25
3180
919400
15111024
© T Madas
An Explanationfor
Recurring Decimals
© T Madas
We have a whole and we want to divide it into 3
© T Madas
Break the whole into tenths
© T Madas
Break the whole into tenths
© T Madas
Break the whole into tenths
0.1
© T Madas
Break the whole into tenths
0.1
0.1
© T Madas
Break the whole into tenths
0.1
0.1
0.1
© T Madas
Break the whole into tenths
0.2
0.1
0.1
© T Madas
Break the whole into tenths
0.2
0.2
0.1
© T Madas
Break the whole into tenths
0.2
0.2
0.2
© T Madas
Break the whole into tenths
0.3
0.2
0.2
© T Madas
Break the whole into tenths
0.3
0.3
0.2
© T Madas
Break the whole into tenths
Break the tenth into hundredths
0.3
0.3
0.3
0.1
© T Madas
Break the whole into tenths
Break the tenth into hundredths
0.3
0.3
0.3
© T Madas
Break the whole into tenths
Break the tenth into hundredths
0.31
0.3
0.3
© T Madas
Break the whole into tenths
Break the tenth into hundredths
0.31
0.31
0.3
© T Madas
Break the whole into tenths
Break the tenth into hundredths
0.31
0.31
0.31
© T Madas
Break the whole into tenths
Break the tenth into hundredths
0.32
0.31
0.31
© T Madas
Break the whole into tenths
Break the tenth into hundredths
0.32
0.32
0.31
© T Madas
Break the whole into tenths
Break the tenth into hundredths
0.32
0.32
0.32
© T Madas
Break the whole into tenths
Break the tenth into hundredths
0.33
0.32
0.32
© T Madas
Break the whole into tenths
Break the tenth into hundredths
0.33
0.33
0.32
© T Madas
Break the whole into tenths
Break the tenth into hundredths
Break the hundredth into thousandths
0.33
0.33
0.330.01
© T Madas
Break the whole into tenths
Break the tenth into hundredths
Break the hundredth into thousandths
0.33
0.33
0.33
© T Madas
Break the whole into tenths
Break the tenth into hundredths
Break the hundredth into thousandths
0.331
0.33
0.33
© T Madas
Break the whole into tenths
Break the tenth into hundredths
Break the hundredth into thousandths
0.331
0.331
0.33
© T Madas
Break the whole into tenths
Break the tenth into hundredths
Break the hundredth into thousandths
0.331
0.331
0.331
© T Madas
Break the whole into tenths
Break the tenth into hundredths
Break the hundredth into thousandths
0.332
0.331
0.331
© T Madas
Break the whole into tenths
Break the tenth into hundredths
Break the hundredth into thousandths
0.332
0.332
0.331
© T Madas
Break the whole into tenths
Break the tenth into hundredths
Break the hundredth into thousandths
0.332
0.332
0.332
© T Madas
Break the whole into tenths
Break the tenth into hundredths
Break the hundredth into thousandths
0.333
0.332
0.332
© T Madas
Break the whole into tenths
Break the tenth into hundredths
Break the hundredth into thousandths
0.333
0.333
0.332
© T Madas
Break the whole into tenths
Break the tenth into hundredths
Break the hundredth into thousandths
0.333
0.333
0.333
Break the thousandth into tenthousandths
0.001
© T Madas
Break the whole into tenths
Break the tenth into hundredths
Break the hundredth into thousandths
Break the thousandth into tenthousandths
0.333
0.333
0.333
© T Madas
Break the whole into tenths
Break the tenth into hundredths
Break the hundredth into thousandths
Break the thousandth into tenthousandths
0.3331
0.333
0.333
© T Madas
Break the whole into tenths
Break the tenth into hundredths
Break the hundredth into thousandths
Break the thousandth into tenthousandths
0.3331
0.3331
0.333
© T Madas
Break the whole into tenths
Break the tenth into hundredths
Break the hundredth into thousandths
Break the thousandth into tenthousandths
0.3331
0.3331
0.3331
© T Madas
Break the whole into tenths
Break the tenth into hundredths
Break the hundredth into thousandths
Break the thousandth into tenthousandths
0.3332
0.3331
0.3331
© T Madas
Break the whole into tenths
Break the tenth into hundredths
Break the hundredth into thousandths
Break the thousandth into tenthousandths
0.3332
0.3332
0.3331
© T Madas
Break the whole into tenths
Break the tenth into hundredths
Break the hundredth into thousandths
Break the thousandth into tenthousandths
0.3332
0.3332
0.3332
© T Madas
Break the whole into tenths
Break the tenth into hundredths
Break the hundredth into thousandths
Break the thousandth into tenthousandths
0.3333
0.3332
0.3332
© T Madas
Break the whole into tenths
Break the tenth into hundredths
Break the hundredth into thousandths
Break the thousandth into tenthousandths
0.3333
0.3333
0.3332
© T Madas
Break the whole into tenths
Break the tenth into hundredths
Break the hundredth into thousandths
Break the thousandth into tenthousandths
0.3333
0.3333
0.33330.0001
© T Madas
When
doesa
Fraction
producea
Recurring
Decimal?
© T Madas
0.5
0.333…
0.25
0.2
0.1666…
0.142857142857142857…
0.125
0.111…
0.1
0.090909…
0.08333…
0.076923076923076923…
0.07148577148577148577…
0.0666…
0.0625
¾¾®12
¾¾®13
¾¾®14
¾¾®15
¾¾®16
¾¾®17
¾¾®18
¾¾®19
¾¾®110
¾¾®111
¾¾®112
¾¾®113
¾¾®114
¾¾®115
¾¾®116
© T Madas
0.058823529411764705882352941176470588235294117647…
0.0555…
0.052631578947368421052631578947368421052631578947…
0.05
0.047619047619047619047619047619047619…
0.0454545…
0.043478260869565217391304347826086956521739130434…
0.41666…
0.04
0.0384615384615384615384615384615384615…
0.037037037…
0.03571428571428571428571428571428…
0.03448275862068965517241379310344827586206896551…
0.0333…
0.03225806451612903225806451612903225806451612903…
¾¾®117
¾¾®118
¾¾®119
¾¾®120
¾¾®121
¾¾®122
¾¾®123
¾¾®124
¾¾®125
¾¾®126
¾¾®127
¾¾®128
¾¾®129
¾¾®130
¾¾®131
© T Madas
What do we observe?
Dividing by most numbers produces recurring decimals
Does the number we are dividing by (i.e. the denominator) affect the outcome?
Does the number to be divided (i.e. the numerator) affect the outcome?
© T Madas
0.5
0.25
0.2
0.1
0.0625
0.04
0.03125
0.025
0.02
0.015625
0.0125
0.01
0.008
0.0078125
0.00625
¾¾®12
¾¾®14
¾¾®15
¾¾®110
¾¾®116
¾¾®125
¾¾®132
¾¾®140
¾¾®150
¾¾®164
¾¾®180
¾¾®1100
¾¾®1125
¾¾®1128
¾¾®1160
0.005
0.004
0.00390125
0.003125
0.0025
0.002
0.001953125
0.0016
0.0015625
0.00125
0.001
0.0009765625
0.00078125
0.00625
0.0005
¾¾®1200
¾¾®1250
¾¾®1256
¾¾®1320
¾¾®1400
¾¾®1500
¾¾®1512
¾¾®1625
¾¾®1640
¾¾®1800
¾¾®11000
¾¾®11024
¾¾®11280
¾¾®11600
¾¾®12000
© T Madas
2
= ´4 2 2
5
= ´10 2 5
= ´ ´ ´16 2 2 2 2
= ´25 5 5
= ´ ´ ´ ´32 2 2 2 2 2
= ´ ´ ´40 2 2 2 5
= ´ ´50 2 5 5
= ´ ´ ´ ´ ´64 2 2 2 2 2 2
= ´ ´ ´ ´80 5 2 2 2 2
= ´ ´ ´100 5 5 2 2
= ´ ´125 5 5 5
= ´ ´ ´ ´ ´ ´128 2 2 2 2 2 2 2
= ´ ´ ´ ´ ´160 5 2 2 2 2 2
= ´ ´ ´ ´200 5 5 2 2 2
= ´ ´ ´250 5 5 5 2
= ´ ´ ´ ´ ´ ´ ´256 2 2 2 2 2 2 2 2
= ´ ´ ´ ´ ´ ´320 5 2 2 2 2 2 2
= ´ ´ ´ ´ ´ ´400 5 5 2 2 2 2 2
= ´ ´ ´ ´500 5 5 5 2 2
= ´ ´ ´ ´ ´ ´ ´ ´512 2 2 2 2 2 2 2 2 2
= ´ ´ ´625 5 5 5 5
= ´ ´ ´ ´ ´ ´ ´640 5 2 2 2 2 2 2 2
= ´ ´ ´ ´ ´ ´800 5 5 2 2 2 2 2
= ´ ´ ´ ´ ´1000 5 5 5 2 2 2
= ´ ´ ´ ´ ´ ´ ´ ´ ´1024 2 2 2 2 2 2 2 2 2 2
= ´ ´ ´ ´ ´ ´ ´ ´1280 5 2 2 2 2 2 2 2 2
= ´ ´ ´ ´ ´ ´ ´1600 5 5 2 2 2 2 2 2
= ´ ´ ´ ´ ´ ´2000 5 5 5 2 2 2 2
© T Madas
It turns out that:
The number we are dividing (i.e. the numerator) plays no role into whether the decimal will be terminating or recurring
The divisor (i.e. the denominator) is important:• If the denominator can be broken into prime
factors of 2 and/or 5 only then the decimal will be terminating
• If the denominator contains any other prime factors then the decimal will be recurring
• THIS ONLY HOLDS TRUE PROVIDED THE FRACTION IS IN ITS SIMPLEST FORM
© T Madas
5
12
F o r E x a m p l e :
12 = 2x 2x 3
It is in its simplest form
It will recur
. ...= 0 41666 7
25
25 = 5x 5
It is in its simplest form
It will terminate
.= 0 28
© T Madas
12
75
F o r E x a m p l e :
25 = 5x 5
It is now in its simplest form
It will terminate
.= 0 16 15
72
24 = 2x 2
It is now in its simplest form
It will recur
. ...= 0 208333=4
25=
5
24
x 2 x 3
© T Madas
11
28
F o r E x a m p l e :
28 = 2x 2x 7
It is in its simplest form
It will recur
. ...= 285285714 2857714 140 39
© T Madas
=5
8
F o r E x a m p l e :
35
56
8 = 2x 2
It is now in its simplest form
It will terminate
.= 0 625
x 2
22
30
15 = 3x 5
It is now in its simplest form
It will recur
. ...= 0 7333=11
15
© T Madas
15
52
F o r E x a m p l e :
52 = 2x 2x 13
It is in its simplest form
It will recur
. ...= 846846153 8461153 530 28
© T Madas