# Definition:Continued Fraction/Simple

## Definition

Let $\R$ be the field of real numbers.

### Simple Finite Continued Fraction

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

A simple finite continued fraction of length $n$ is a finite continued fraction in $\R$ of length $n$ whose partial quotients are integers that are strictly positive, except perhaps the first.

That is, it is a finite sequence $a : \left[0 \,.\,.\, n\right] \to \Z$ with $a_n > 0$ for $n >0$.

### Simple Infinite Continued Fraction

A simple infinite continued fraction is a infinite continued fraction in $\R$ whose partial quotients are integers that are strictly positive, except perhaps the first.

That is, it is a sequence $a : \N_{\geq 0} \to \Z$ with $a_n > 0$ for $n >0$.

## Also known as

A simple continued fraction is also known as a regular continued fraction.

When the context is such that it is immaterial whether a simple continued fraction is finite or infinite, the abbreviation SCF can be used.