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A continued fraction is a form of fraction where the numerator or denominator contains another fraction.

## Examples

• $\sqrt 2 = 1 + \frac{1}{ 2+\frac{1}{ 2+\frac{1}{...} } }$
• $\sqrt 2 = [1;(2)]$

## Calculating from decimal number

With a decimal number, you can obtain a continued fraction with the following process.

1. Write the whole number portion to start the continued fraction. ()
2. Set the whole number portion to zero, leaving the number less than 0.
3. If 0, stop. Otherwise, take the reciprocal
4. Repeat step 1 for the following numbers in the continued fraction.

For example, with 2.285:

• 2.285 - 2 -> 0.285 ()
• 1 / 0.285 -> 3.508772
• 3.508772 - 3 -> 0.508772 ([2; 3])
• 1/0.508772 -> 1.965517
• 1.965517 - 1 -> 0.965517 ([2; 3, 1])
• 1/0.965517 -> 1.035714
• 1.035714 - 1 -> 0.035714 ([2; 3, 1, 1])
• 1/0.35714 -> 28.000224
• 28.000224 - 28 -> 0.000224 ([2; 3, 1, 1, 28])

The process can be stopped here, as the number begins to approach 0, or otherwise generate rounding errors.