Definition:Continued Fraction

Definition
Let $F$ be a field, such as the field of real numbers $\R$.

Notation
A continued fraction can be denoted using ellipsis:
 * $a_0 + \cfrac 1 {a_1 + \cfrac 1 {a_2 + \cfrac 1 {\ddots \cfrac {} {}}}}$

which suggests the definition of its value, but it should be noted that this is only a notation.

By definition, a continued fraction is its sequence of partial quotients and can thus be denoted:
 * $(a_n)_{n\geq 0}$
 * $[a_0; a_1, a_2, \ldots ]$
 * $[a_0, a_1, a_2, \ldots ]$

where the last two notations are usually reserved for its value.

Also defined as
When one is primarily concerned with simple continued fractions of real numbers, it is common to require a continued fraction to have strictly positive partial quotients, except perhaps the first.

Also known as
A continued fraction is also known as a:
 * continued fraction in canonical form
 * regular continued fraction
 * simple continued fraction

as opposed to a generalized continued fraction.

Also see

 * Definition:Value of Continued Fraction
 * Definition:Sequence of Partial Quotients
 * Definition:Sequence of Complete Quotients
 * Definition:Convergent of Continued Fraction
 * Definition:Numerators and Denominators of Continued Fraction

Other continued fraction expansions

 * Definition:Continued Fraction Expansion of Real Number
 * Definition:Continued Fraction Expansion of Laurent Series
 * Definition:p-Adic Ruban Continued Fraction

Generalizations

 * Definition:Generalized Continued Fraction