# Definition:Continued Fraction/Simple

## Definition

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

### Simple Finite Continued Fraction

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 $n > 0$.

### 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 $a: \N_{\ge 0} \to \Z$ 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.

## Also see

- Definition:Continued Fraction Expansion of Real Number
- Correspondence between Rational Numbers and Simple Finite Continued Fractions
- Correspondence between Irrational Numbers and Simple Infinite 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 - 1992: A.M. Rockett and P. Szüsz:
*Continued Fractions*: $I$: Introduction: $\S2$: Regular continued fractions