Editing Simple continued fraction (section)

{{Short description|Number represented as a0+1/(a1+1/...)}}
{{redirect-distinguish|Recurring fraction|Repeating decimal}}

A '''simple''' or '''regular continued fraction''' is a [[continued fraction]] with numerators all equal one, and denominators built from a sequence <math>\{a_i\}</math> of integer numbers. The sequence can be finite or infinite, resulting in a '''finite''' (or '''terminated''') continued fraction like
:<math>a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{ \ddots + \cfrac{1}{a_n} }}}</math>
or an '''infinite''' continued fraction like
:<math>a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{ \ddots }}}</math>

Typically, such a continued fraction is obtained through an [[iterative]] process of representing a number as the sum of its [[integer part]] and the [[multiplicative inverse|reciprocal]] of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In the ''finite'' case, the iteration/[[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 (mathematics)|infinite expression]]. In either case, all integers in the sequence, other than the first, must be [[positive number|positive]]. The integers <math>a_i</math> are called the [[coefficient]]s or terms of the continued fraction.{{sfn|Pettofrezzo|Byrkit|1970|p=150}}

Simple continued fractions have a number of remarkable properties related to the [[Euclidean algorithm]] for integers or [[real number]]s. Every [[rational number]] {{sfrac|<math>p</math>|<math>q</math>}} has two closely related expressions as a finite continued fraction, whose coefficients {{mvar|a<sub>i</sub>}} can be determined by applying the Euclidean algorithm to <math>(p,q)</math>. The numerical value of an infinite continued fraction is [[irrational number|irrational]]; it is defined from its infinite sequence of integers as the [[limit (mathematics)|limit]] of a sequence of values for finite continued fractions. Each finite continued fraction of the sequence is obtained by using a finite [[Prefix (computer science)|prefix]] of the infinite continued fraction's defining sequence of integers. Moreover, every irrational number <math>\alpha</math> is the value of a ''unique'' infinite regular continued fraction, whose coefficients can be found using the non-terminating version of the Euclidean algorithm applied to the [[Commensurability (mathematics)|incommensurable]] values <math>\alpha</math> and 1. This way of expressing real numbers (rational and irrational) is called their ''continued fraction representation''.