© t madas = 0.333… = 0.1666… = 0.142857142857142857…

© 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