Recurring Decimals = 0.333… = 0.1666… = 0.142857142857142857… © T Madas

A recurring decimal can always be written as a fraction 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… 1 3 7 6 5 31 12 913 9990 2 A recurring decimal can always be written as a fraction © T Madas

These recurring decimals are worth remembering A recurring decimal is a never ending decimal, whose decimal part repeats with a pattern = 0.333… 0.666… 0.1666… 0.8333… 1 3 2 6 5 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 1.4755859375 1 4 2 5 13 25 31 80 919 400 1511 1024 © T Madas

An Explanation for 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

0.3 0.3 0.1 0.3 Break the whole into tenths Break the tenth into hundredths 0.3 0.3 0.1 0.3 © T Madas

0.3 0.3 0.3 Break the whole into tenths Break the tenth into hundredths 0.3 0.3 0.3 © T Madas

0.31 0.3 0.3 Break the whole into tenths Break the tenth into hundredths 0.31 0.3 0.3 © T Madas

0.31 0.31 0.3 Break the whole into tenths Break the tenth into hundredths 0.31 0.31 0.3 © T Madas

0.31 0.31 0.31 Break the whole into tenths Break the tenth into hundredths 0.31 0.31 0.31 © T Madas

0.32 0.31 0.31 Break the whole into tenths Break the tenth into hundredths 0.32 0.31 0.31 © T Madas

0.32 0.32 0.31 Break the whole into tenths Break the tenth into hundredths 0.32 0.32 0.31 © T Madas

0.32 0.32 0.32 Break the whole into tenths Break the tenth into hundredths 0.32 0.32 0.32 © T Madas

0.33 0.32 0.32 Break the whole into tenths Break the tenth into hundredths 0.33 0.32 0.32 © T Madas

0.33 0.33 0.32 Break the whole into tenths Break the tenth into hundredths 0.33 0.33 0.32 © T Madas

0.33 0.33 0.01 0.33 Break the whole into tenths Break the tenth into hundredths Break the hundredth into thousandths 0.33 0.33 0.01 0.33 © T Madas

0.33 0.33 0.33 Break the whole into tenths Break the tenth into hundredths Break the hundredth into thousandths 0.33 0.33 0.33 © T Madas

0.331 0.33 0.33 Break the whole into tenths Break the tenth into hundredths Break the hundredth into thousandths 0.331 0.33 0.33 © T Madas

0.331 0.331 0.33 Break the whole into tenths Break the tenth into hundredths Break the hundredth into thousandths 0.331 0.331 0.33 © T Madas

0.331 0.331 0.331 Break the whole into tenths Break the tenth into hundredths Break the hundredth into thousandths 0.331 0.331 0.331 © T Madas

0.332 0.331 0.331 Break the whole into tenths Break the tenth into hundredths Break the hundredth into thousandths 0.332 0.331 0.331 © T Madas

0.332 0.332 0.331 Break the whole into tenths Break the tenth into hundredths Break the hundredth into thousandths 0.332 0.332 0.331 © T Madas

0.332 0.332 0.332 Break the whole into tenths Break the tenth into hundredths Break the hundredth into thousandths 0.332 0.332 0.332 © T Madas

0.333 0.332 0.332 Break the whole into tenths Break the tenth into hundredths Break the hundredth into thousandths 0.333 0.332 0.332 © T Madas

0.333 0.333 0.332 Break the whole into tenths Break the tenth into hundredths Break the hundredth into thousandths 0.333 0.333 0.332 © T Madas

0.333 0.333 0.001 0.333 Break the whole into tenths Break the tenth into hundredths Break the hundredth into thousandths Break the thousandth into tenthousandths 0.333 0.333 0.001 0.333 © T Madas

0.333 0.333 0.333 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

0.3331 0.333 0.333 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

0.3331 0.3331 0.333 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

0.3331 0.3331 0.3331 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

0.3332 0.3331 0.3331 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

0.3332 0.3332 0.3331 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

0.3332 0.3332 0.3332 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

0.3333 0.3332 0.3332 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

0.3333 0.3333 0.3332 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

0.3333 0.3333 0.0001 0.3333 Break the whole into tenths Break the tenth into hundredths Break the hundredth into thousandths Break the thousandth into tenthousandths 0.3333 0.3333 0.0001 0.3333 © T Madas

When does a Fraction produce 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 © 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… © T Madas

Does the number to be divided (i.e. the numerator) affect the outcome? 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 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 © T Madas

© T Madas

THIS ONLY HOLDS TRUE PROVIDED THE FRACTION IS IN ITS SIMPLEST FORM 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

It is in its simplest form It is in its simplest form F o r E x a m p l e : It is in its simplest form It is in its simplest form 12 = 2 x 2 x 3 25 = 5 x 5 It will recur It will terminate © T Madas

It is now in its simplest form It is now in its simplest form F o r E x a m p l e : It is now in its simplest form It is now in its simplest form 24 = 2 x 2 x 2 x 3 25 = 5 x 5 It will terminate It will recur © T Madas

It is in its simplest form F o r E x a m p l e : It is in its simplest form 28 = 2 x 2 x 7 It will recur © T Madas

It is now in its simplest form It is now in its simplest form F o r E x a m p l e : It is now in its simplest form It is now in its simplest form 15 = 3 x 5 8 = 2 x 2 x 2 It will terminate It will recur © T Madas

It is in its simplest form F o r E x a m p l e : It is in its simplest form 52 = 2 x 2 x 13 It will recur © T Madas

© T Madas