# Definition:Continued Fraction/Finite

## Definition

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

Let $n \geq 0$ be a natural number.

Informally, a finite continued fraction of length $n$ in $F$ is an expression of the form:

$a_0 + \cfrac 1 {a_1 + \cfrac 1 {a_2 + \cfrac 1 {\ddots \cfrac {} {a_{n-1} + \cfrac 1 {a_n}} }}}$

where $a_0, a_1, a_2, \ldots, a_n \in F$.

Formally, a finite continued fraction of length $n$ in $F$ is a finite sequence, called sequence of partial quotients, whose domain is the integer interval $\left[0 \,.\,.\, n\right]$.

A finite continued fraction should not be confused with its value, when it exists.

## Also known as

A finite continued fraction is often abbreviated FCF, and is also known as a terminated of terminating continued fraction.