Continued fraction

A continued fraction is a fraction of the form \[ a_1 + \cfrac{b_1}{ a_2 + \cfrac{b_2}{ a_3 + \cfrac{b_3}{a_4 + \dotsb}}} \] A simple continued fraction is one in which every \(b_i\) is equal to \(1\), and every \(a_i\) is a positive integer (except that \(a_1\) may be any integer).

Every rational number can be written as a terminating simple continued fraction.

Some numbers can be written as recurring simple continued fractions, for example \[\sqrt{5} = 2 + \cfrac{1}{ 4 + \cfrac{1}{ 4 + \cfrac{1}{ 4 + \dotsb}}}\]