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Fractions

Fractions Calculator

This page is about converting a fraction (i.e. a ratio of two numbers, also called a rational number) into a decimal fraction and the patterns that occur in such a decimal fraction. It is interactive and you can use the calculators on this page to investigate fractions for yourself to many decimal places. No special knowledge beyond decimals and division is required.

Contents

(o) Changing a Fraction into a Decimal number
(*) Notation
(*) Don't all fractions recur?
[frac to dec calc] Fraction-to-Decimal Calculator

(o) Converting a decimal fraction into a proper fraction
[dec to frac calc] Decimal-to-Fraction Calculator

(o) How can we tell if a given fraction will terminate or recurr as a decimal fraction?

(o) What about the other numbers whose decimal fractions neither terminate nor recur?
(*) Decimal fractions that look like special sequences
[frac to dec calc] Fraction-to-Decimal and Special Series Calculator

(o) Are there other types of numbers too?




Changing a Fraction into a Decimal number

Converting a fraction to a decimal is just a division operation. So the fraction 1/2 means 1 ÷ 2.
When we do the long division (or use a calculator!) we find 1 ÷ 2 = 0·5. This fraction was easy - just one digit and we are done.

Here is a table showing some more fractions converted to decimal form:
4 = 0·44444...  

9
1 = 0·33333...  

3
2 = 0· 285714 285714 28...  

7
1 = 0·25  

4
1 = 0·1 6666...  

6
3 = 0·375  

8
3 = 0·06 81 81 81 ...  

44

Some of them stop after a few digits, e.g.

1/4 = 0·25
3/8 = 0·375

Such fractions are called terminating decimal fractions.

Others get into a loop where a digit (or several digits) repeat for ever, e.g.

1/3 = 0·333333...
4/9 = 0·444444...
2/7 = 0·285714 285714 285714 ...
These fractions are called recurring decimal fractions.

Some of the fractions in the table above start off with some digits before they too eventually end up repeating a sequence of digits for ever, e.g.

1/ = 0·1 66666...
3/44 = 0·06 81818181...

These fractions are also called recurring decimal fractions. Sometimes we call those which start their recurring cycle immediately after the decimal point purely recurring decimals. Those that have some extra digits before their cycles are also called mixed recurring decimals.

The decimal fraction of every proper fraction* is either terminating or else it is recurring.
*A proper fraction is a fraction which cannot be simplified any further and is less than 1. So 6/12 is a fraction less than 1 but not a proper one since it can be simplified to 1/2, which is a proper fraction.

Notation

To make quite sure that we can distinguish 0·3666 (a terminating fraction with just 4 decimal places) from 0·36666... (a recurring fraction which goes on for ever), mathematicians use one of two common notations to indicate which (if any) of the digits are in the repeating part (also called the period or cycle).
  • EITHER a dot is put over the first and last part of the recurring sequence (if there is only a single digit in the repeating part, only one dot is used)
  • OR a line is drawn over the repeating part.
If there is no repeating part then the decimal fraction is terminating and no digit has a dot or a line above it. Here are some examples:

1÷7 = 0·142857 142857 14..=
142857
=
142857
3÷44 = 0·06 81 81 81 81...=
0681
=
0681
11÷30 = 0·3 6666666... =
36
=
36
1833÷5000 = 0·3666 is a terminating decimal fraction
183÷500 = 0·366 is a terminating decimal fraction
9÷25 = 0·36 is a terminating decimal fraction

However, when showing results computed by the Calculators on this page, we will use another method which takes less vertical space: enclose the recurring part inside

Here is a question to test your understanding of the bracket notation:

Can you... say which of these decimal fractions is not the same as 0· 123 123 123 ...?

[Press the button to check your answer.]
Wrong. This is 0. 123 123 123 123 ...
Wrong. This is another way of writing 0. 123 123 123 123 ...
Wrong. This is 0. 1 231 231 231 231 ... which is the same decimal fraction.
Correct! This is 0. 1 23123 23123 ... and is not the same as 0. 1231231231231...
Wrong. This is 0. 12 312 312 312 ... which is the same decimal fraction.


Don't all fractions recur?

People have suggested that all fractions are recurring ones because they all end with 000000... or they can end with 99999999..... . Let's examine these two special periods, [0] and [9]:-

Aren't all fractions recurring?

  • Argument 1:
    Since 1/2=0·5 is exactly the same as 0·50000000... you could say that 1/2=0·5[0]. This will also apply to every terminating decimal fraction. So can't we say that all terminating fractions are just recurring ones with a period of [0]? Yes, we can!
    But mathematicians always ignore this special period of just zeroes and just say that "the decimal terminates" because they choose to write the number as a finite collection of decimal digits rather than an infinite one when there is a choice.
  • Argument 2:
    Since 0.49999999... or 0.4[9] is indistingushable from 0.5 (because the series of 9's never ends), we have another way in which all terminating decimals may be written as recurring ones - always replace the last non-zero digit, D, of a terminating decimal fraction by (D-1) followed by a recurring period of 9's.
    for example 1/8 = 0·125 = 0·1249999999...
    Again, this reasoning is correct.
    Mathematically though, we do not use a period of [9] in our decimal fractions but again choose to write it as a finite sequence of digits wherever possible (i.e. so that it terminates).

It's really a matter of taste as both methods are correct.
Such decisions are made so that we can all conveniently talk the same mathematical language. They are called conventions.

The same is true when deciding on which side of the road to drive. It is a convention in the UK that we drive on the left, but the convention in France is to drive on the right. So long as you go with the convention when driving in Britain and go with the other convention when in France, then there is no problem. But make sure you know which convention is being used in any other country!



A Fraction to Decimal Calculator

Use the interactive calculator following these questions to help you answer them:
[This calculator can give as many decimal places as you like, unlike an ordinary hand-held calculator which often only gives you 8 or perhaps 12 decimal places.]

Can you... use the calculator below to find the fixed and recurring parts of all unit fractions up to 1/12 (i.e. 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12)?
[First find if it is terminating or recurring. If it is recurring, the calculator will tell you how many digits there are in the repeating cycle. Add this to the number of non-repeating digits and enter this in the "up to __ dps" box and press the "Show Decimal" key.]
You can select the text in the Results box to paste into another document if you want to keep your results in a file.
Can you... convert following fractions to decimals: 1/7, 2/7, 3/7, 4/7, 5/7, 6/7?
Can you... spot what all those 7th fractions have in common?
Can you... see if this is true of 8 (try all the fractions from 1/8 to 7/8)?
Can you... find another number, N, whose fractions M/N have the same property?
Can you... find Anne's birthday if she says its fractional form is 80031/156250?
Can you... find Bill's birthday if he says its fractional form is 16578/781250?
Can you... say if Pat's birthday is the same as Bill's if its fractional form is 656937/3125000?
Fraction to Decimal
    


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Converting a decimal fraction into a proper fraction

In the calculator below, you can give a decimal fraction and the calculator will convert it to a proper fraction. Give the fixed part (if it has one) followed by the recurring part (if your decimal fraction is not a terminating one).

If your decimal fraction is a terminating fraction such as 0·526, type the digits 526 into the FIXED part, leaving the RECURRING part empty.

If your decimal fraction is purely recurring as in 0·246 246 246..., type the digits that repeat (in this example it is just 246) into the RECURRING part.

If your decimal fraction starts with a fixed part and is followed by a recurring part, as in 0·0840 26 26 26 ... then put 0840 into the FIXED part (don't forget to type in all initial and trailing zeroes) and type 26 into the box for the RECURRING part.

THEN press the "=" button to see the conversion.

Can you... find a fraction which is all 1's (0.[1]=0.111111...)?
Can you... find a fraction which is all 2's (0.[2]=0.222222...)?
Can you... find a fraction which repeats your age (e.g. 0.[14]=0.14 14 14 14 14... )?
Can you... find a fraction which is your birthday (e.g. for 5 January 1985 you might try 0.05 01 1985)?
Can you... find a fraction which is 0.123456789?
Can you... find a fraction which is 0.[123456789]?


Decimal To Fraction
  FIXED partRECURRING part
  



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How can we tell if a given fraction will terminate or recurr as a decimal fraction?

The rules are not easy, but some experimenting will answer this question:

Use the calculator above to find all the fractions which have a denominator (bottom number) less than 50 and whose decimal fractions terminate.

Your list should begin with 2, 4, 5, 8, 10, 16, 20 since

1/2 = 0·51/4 = 0·25 1/5 = 0·2 1/8 = 0·125 1/10 = 0·1 1/16 = 0·0625 1/20 = 0·05

All these fractions terminate after a limited number of digits.

Can you... spot the pattern that is common to the numbers 2, 4, 5, 8, 10, 16, 20, ...?
[Hint: try looking at their prime factors.]
Check it out by finding some more numbers, N, whose unit fractions, 1/N, terminate.

Can you... find some more English words that begin with ir- where the ir- means not? E.g. an irregular polygon is a polygon that is not regular (a regular polygon has all sides and all angles equal). [Hint: if this isn't irrelevant, you may find it an irresistible challenge :) ]


Find some other words which begin with ir- which do not have this meaning. [Hint: Irish people like to iron out such irritating questions :) ]
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What about the other numbers whose decimal fractions neither terminate nor recurr?

Numbers whose decimal fraction terminate or end up recurring are proper fractions.
No other numbers have these properties.
So any number which does not stop and does not end with a recurring pattern are non-fractional numbers; they are not the ratio of two whole numbers; they are called ir-ratio-nal numbers. An example is the square root of 2 whose decimal fraction is 1.41421 35623 73095 .... .
No matter how far we go on expanding this number as an ever more precise decimal fraction, its decimal digits will never get into any repeating pattern. There are lots of irrational numbers too! Other examples are:
  • the square root of any non-square number. The list of these non-square numbers begins 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, ...
  • the cube roots of any non-cube number. The list of these non-cube numbers begins 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, ...
  • the fourth roots of any non-fourth-power number.
  • and so on for all the nth-roots of non-nth-powers...
  • pi = 3.14159 26535 ...
  • e = 2.71828 18284 ...
  • almost all sines, cosines and tangents of angles




Decimal fractions that look like special sequences

Can you... find 1/9801 as a decimal fraction to 100 decimal places?
Fraction to Decimal
    


 up to  dps


It looks like the decimal fraction is just the numbers 0,1,2,..., with two digits per number, and indeed it is! But we said earlier that all proper fractions when put into decimal form either terminate or recurr, so the number series cannot go on for ever in the decimal digits!

What's gone wrong?

The answer lies in the fact that the numbers appear with two digits each. When we get to 100, there will be an "overflow" into the 2-digit "number" before it. In fact, what happens is that we do include everynumber from 0 upwards but the overflows eventually cause the decimal to get into a recurring sequence.

Can you... find how long the repeating part is in 1/9801?
Can you... use the calculator to show the complete period of 1/9801?
Can you... find the pattern of fractions that would give the integers as a series, each with 5 digits?

[The calculator will fill in 1/9801 for 2-digit numbers and 1/998001 for the all-integers series 3 digits at a time, so the question asks you to try and continue the pattern. There are lots of other series such as powers of 2 (0,1,2,4,8,16,...) or the Fibonacci numbers (0,1,1,2,3,5,8,13,21,...) which appear as the recurring part of some special fractions.

A list of some of them is given in the Selection Menu in the above calculator. Pick one and the fraction with an initial segment of this series as its periodic decimal expansion will be filled in for you in the calculator's boxes. You can then use the other buttons to investigate the fraction for yourself. Can you... find the pattern of fractions that would give the powers of two to 4 digits, 5 digits and so on?
Can you... find the pattern for some of the other series in the selection list in the calculator?
Can you... find some new series of numbers not in the selection list that occur in a decimal fraction? If you do find some more, please email the author of this page, email Dr Knott, so that they can be included in a new version of this calculator together with your name as the finder.
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Are there other types of numbers too?

You might now ask if rational and irrational covers all the numbers.
Yes, it does - at least the numbers that could (ideally) measure the length of any straight line to any and all degrees of accuracy. Such numbers (lengths) are called real numbers. Beyond these there are other numbers, called complex numbers, combining both reals and a new type of number known as imaginary numbers which depend upon the square-root of -1. But that's another story!
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References

A. McD. Mercer, A Note on Some Irrational Decimal Fractions American Mathematical Monthly Vol. 101 (1994), pages 567-8.

© Dr R Knott (send me an email) , 14 August 2000