# Definition:Continued Fraction/Simple/Finite

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## Definition

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

Let $n \ge 0$ 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 $a: \closedint 0 n \to \Z$ with $a_n > 0$ for $n > 0$.

## Also known as

A **simple finite continued fraction** can be abbreviated **SFCF**.

It is also known as a **regular finite continued fraction**.

The order of the words can be varied, that is **finite simple continued fraction** for example, but $\mathsf{Pr} \infty \mathsf{fWiki}$ strives for consistency and does not use that form.

## Also see

- Definition:Value of Finite Continued Fraction
- Definition:Simple Infinite Continued Fraction
- Correspondence between Rational Numbers and Simple Finite Continued Fractions

## Sources

- Weisstein, Eric W. "Simple Continued Fraction." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/SimpleContinuedFraction.html - 1963: C.D. Olds:
*Continued fractions*: $\S$ $1.2$: Definitions and Notation