Converting decimals into fractions

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Converting fractions to decimals is easy - any calculator will tell you that 4/9 = 0.4444...

Doing things the other way around is a bit more tricky. For instance, do you know what 0.141414... would be as a fraction? Here is a method for finding out.

This algorithm could also be used as a constructive proof that all recurring decimals are rational

Method

  • x = 0.141414...

  • Notice that two digits repeat so multiply through by 10^2.
    100x = 14.141414...

  • Take one line from the other and the recurring bit disappears - magic!
    99x = 14

  • Divide by 99
    x = 14/99

Harder Examples

This method easily extends to numbers which only start repeating after several digits. e.g. 0.13666666...

In this case, you would have to multiply by both 100 and 1000 to get


1000x-100x = 136.6666... - 13.6666...


900x = 123


x = 123/900

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