\begin{definition}[Definition:Continued Fraction/Finite]
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.
\end{definition}