Definition:Continued Fraction/Simple/Finite

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 denominators 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

• Results about simple continued fractions can be found here.

Sources

• 1963: C.D. Olds: Continued Fractions: $\S \ 1.2$: Definitions and Notation

• Weisstein, Eric W. "Simple Continued Fraction." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SimpleContinuedFraction.html