Recurring decimals

Introduction to numbers with never ending decimals

Many of the decimals we use have a limited number of digits - for example 0.75 or 0.125 or 1.2562. Because they do not continue past a certain number of digits, they are described as being terminating decimals.

There are however two types of numbers which can be written as decimals which go on forever:

irrational numbers;
numbers where a pattern is repeated continually - the recurring decimals.

Irrational numbers are distinguished by them not being able to be written as a fraction no matter what we do.
Some of the most famous irrational numbers we have are π and e. Their digits go on forever without repeating any pattern.

For example, π begins with 3.1415926535 8979323846 2643383279 5028841971 6939937510 (to 50 decimal places).
No repetition of a pattern or sequence of the same numbers is ever found.

In contrast, recurring decimals do have recurring patterns of 1 or 2 or 3 or more digits.
They look, for example, like 2.111111111... or like 5.1742424242424242.....

The irrationals cannot be simplified or "written nicely". The recurring decimals like those above can however be manipulated and written as a fraction.
With that property, they are described as being rational numbers.

How to write recurring decimals easily

The recurring decimals are not commonly written with as many digits as in the examples above. We of course cannot write them out in full because there is an infinite number of digits.

A short-hand approach is better.

So instead of writing out the same repeating pattern of digits, we use dots above particular digits to signify the string of digits being repeated.

For example:

if a single digit repeats - as in 2.11111... - we write with a dot above the 1 to indicate just the 1 is repeating.
When two digits repeat, we write dots over the first and second digits in the repeating pattern.
For example 4.1742424242424242..... has the 4 and 2 consecutively in a repeating pattern.
So we write - having a dot above each of the 4 and the 2. The 42 is the repeating pattern.
Similarly, if we have three (or more) digits repeating as in 1.217217217... we use a dot over the first and last digits just as we did previously.
So with a dot above the 1 to start the pattern and a dot above the 2 to finish the repeating pattern.
The number 3.142857 142857 142857 142857 ... can be written as .

Pretty easy and straightforward .

How to manipulate the recurring decimals.

So how do we do this manipulation to turn recurring decimals into rational numbers in the form of a fraction?

There are two approaches:

basic algebra (which we discuss here);
more sophisticated techniques involving series (which are described elsewhere).

The algebra approach.

When ONLY 1 digit is recurring - for example 2.11111... or :

1. Write x = the number	x = 2.11111...
2. Multiply both sides by 10	10x = 21.1111...
NOTE: the pattern repeats above itself.
3. Subtract the 2nd equation from the first. All digits in corresponding positions cancel out.	9x = 19
4. Divide to obtain x.

So the recurring decimal 2.11111... or can be written as the fraction .

To change from a fraction to a decimal, we just use our calculator.

Similar approaches are used for the 2 digit, 3 digit, etc patterns. The difference is that then we multiply by a multiple of 10 depending on how may numbers are in the repeating pattern.

For one digit repeating, multiply by 10.

For two consecutive digits repeating, multiply by 100.

For three digits repeating, multiply by 1000.

and so on.

When two consecutive digits are repeating - for example 4.1742424242424242..... or :

1. Write x = the number	x = 4.1742...
2. Multiply both sides by 100 (because the two digits next to one another are repeated).	100x = 417.4242...
NOTE: we only write the first occurrence of the "42" pattern in the x equation and the second occurrence in the 100 x equation. The patterns are then lined up.
3. Subtract the 2nd equation from the first. All digits after the 2nd number after the decimal point cancel out.	99x = 413.25
4. Divide to obtain x (and move any decimal in the numerator to the right) .

You should use your calculator to do the subtraction and division. Enter the larger number up to the end of the second repetition
and then enter the original number to the end of the first repetition - "to be sure, to be sure" as the Irish say.

The number of decimal places in each number will then be the same.

So the recurring decimal 4.1742424242424242..... or can be written as the fraction .

When three digits are repeating - for example the "217" pattern in 1.2172172172... or

1. Write x = the number	x = 1.217217217...
2. Multiply both sides by 1000 (because the three digits in the string are being repeated).	1000x = 1217.217...
NOTE: we only write the first occurrence of the "217" pattern in the x equation and the second occurrence in the 1000 x equation. The patterns are then lined up.
3. Subtract the 2nd equation from the first. All digits after the after the decimal point cancel out.	999x = 1216.
4. Divide to obtain x (and move any decimal in the numerator to the right) .

So the recurring decimal 1.2172172172... or can be written as the fraction .