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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.

 


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