Can any non-repeating decimal be written as a fraction?

For example, 0.21456456456456456

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- Jul 18th 2007, 08:38 PMBlueStarnon-repeating decimal
Can any non-repeating decimal be written as a fraction?

For example, 0.21456456456456456 - Jul 18th 2007, 08:49 PMJhevon
- Jul 18th 2007, 08:53 PMearboth
Hello,

provided that the ...456... is repeated endlessly then the decimal can be transformed into a fraction.

Let x = 0.21456456456456...

1. The non-repeating part contains 2 digits. Now multiply x by 10². You get:

100x = 21.456456456456...

2. The repeating part contains 3 digits. Now multipy the result of 1. by 10³. You get:

100000x = 21456.456456456456...

3. Subtract both results:

99900x = 21435. Solve for x: