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What are recurring decimals?

By Jack Putt on the 11th of June, 2012

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    Recurring decimals are decimal numbers that go on forever in a repeating pattern.

    Examples1.3333333333...
    4.343434343434...
    -81.9999999999... 

    Refine By Lee Mansfield on the 14th of June, 2012

Suggested reading…

Convert recurring decimals to fractions and fractions to recurring decimals

As we discussed before there are three types of decimals:

  • Terminating decimals stop at some point such as: 1.5, 4.75 and 0.33265985
  • Recurring decimals repeat forever, but there is a pattern to their repetition such as: 0.66666666666 and 0.142857142857142857
  • Non-recurring decimals are decimals that go on forever, but there is no pattern such as π and √2

Convert Recurring Decimals to Fractions

Depending on the formatting of your calculator you might notice that sometimes your calculator might say 1/3 instead of the decimal number 0.333 ̇, in this section we are going to learn the method which your calculator used to convert a recurring decimal into a fraction. Remember just because your calculator tells you that 2/3 = 0.6 ̇ , you don't have to simply take it's word for it. Instead, there is a very simple method of turning a decimal into a fraction which will allow you to prove that 0.6 ̇= 2/3, which I will explain below!

1) Firstly, you should write down X = your recurring decimal:

x = 0.666666

You will now need to pick out your recurring digits, for our example 0.6 ̇ we can clearly see that it is 6. 

2) Secondly, we must also find a way to get our recurring digit to the left of the decimal point, we can do this by multiplying the decimal number by 10 in our example, but as you know when we multiply one side of a number we must also multiply the other side of the number.

(x = 0.666 ̇)*10  = 10x = 6.666 ̇

So now we know that:

x = 0.666 ̇ and 10x = 6.666 ̇

Usually you must check to make sure that your recurring digit is also on the right hand side of the decimal, but for our example of 6.666 ̇ we can clearly see that it is already in position, so we do not have to worry about this step.

3) Thirdly,  we must subtract the left sides of both of our equations from one another and do the same for the right sides of our equations, see below:

10x - x = 6.666 ̇ - 0.666 ̇

Now from looking at our calculation we can conclude that we are left with: 9x = 6

4) Finally, we now want to divide both sides by the number infront of x which in our example is 9 to reach our final answer:

9x = 6 becomes x = 6/9

To simplfy this fraction we can now divide both the numerator and denominator by 3 which will give us our fraction: 2/3!

So now when our calculator gives us a fraction and that we are not to sure about, we can easily check by following the steps which we have just worked through! Just to make sure that we understand the method, let's do one more example.

Step 1: x = 0.888 ̇

Step 2: 10 x = 8.888 ̇

Step 3: 10x - x = 8.888 ̇ - 0.888 ̇ = 9x = 8

Step 4: x = 8/9

Final Answer = 0.888 ̇ = 8/9

Convert Fractions to Recurring Decimals

Now that we know how to convert recurring decimals to fractions (which was the tricky bit) we are now going to learn the method for converting fractions to recurring decimals... which is quite simple really! Look at the following examples:

All you have to do is divide the numerator by the denominator!

2/3 = 0.666 ̇

1/3 = 0.333 ̇

8/9 = 0.888 ̇

2/9 = 0.222 ̇

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