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### Continued fraction for the square root

The continued fraction for the square root is

##### Explanation

The identity for the square root is

In the right-hand section you can substitute √*x* and this will immediately result in the continued fraction

##### Example 1

The successive convergences of √2 give

and that's pretty close to the value 1.414213562… your calculator shows.

##### Additional information

In 1579, the Italian mathematician Rafael Bombelli used continued fractions to calculate square roots.