Continued fractions of square roots are infinite (they go on forever) but repeating. And so are all quadratic irrationals – numbers that arise as solutions of quadratic equations – i.e. of the form
Some irrational continued fractions, of the form , can be built with almost no calculation whatsoever.
The constructions rely totally on multiplication by conjugate surds. If that sounds unfamiliar, read the very short explanation of using conjugate surds.
The case of involves continually multiplying by to get .
(Usually conjugate surds are used to get the to the top!)
Then 1s and 2s are shifted around to make another to repeat the process:
Obviously that may go on for some time.
The continued fraction for uses the fact that