Simple continued fraction

A simple or regular continued fraction is a continued fraction with numerators all equal to one, and denominators built from a sequence { a i } {\displaystyle \{a_{i}\}} of integer numbers. The sequence can be finite or infinite, resulting in a finite (or terminated) continued fraction like a 0 + 1 a 1 + 1 a 2 + 1 ⋱ + 1 a n {\displaystyle a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\ddots +{\cfrac {1}{a_{n}}}}}}}}}} or an infinite continued fraction like a 0 + 1 a 1 + 1 a 2 + 1 ⋱ . {\displaystyle a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\ddots }}}}}}.} Typically, such a continued fraction is obtained through a recursive process which starts by representing a number as the sum of its integer part and its fractional part. The integer is recorded and the reciprocal of the fractional part is then recursively represented by another continued fraction. In the finite case, the recursion is stopped after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers a i {\displaystyle a_{i}} are called the coefficients or terms of the continued fraction. Simple continued fractions have a number of remarkable properties related to the Euclidean algorithm for integers or real numbers. Every rational number ⁠ p {\displaystyle p} / q {\displaystyle q} ⁠ has two closely related expressions as a finite continued fraction, whose coefficients ai can be determined by applying the Euclidean algorithm to ( p , q ) {\displaystyle (p,q)} . The numerical value of an infinite continued fraction is irrational; it is defined from its infinite sequence of integers as the limit of a sequence of values for finite continued fractions. Each...