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 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 $n > 0$.

Simple Infinite Continued Fraction

A simple infinite continued fraction is a infinite continued fraction in $\R$ whose partial denominators 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

• Results about simple continued fractions can be found here.

Sources

• 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
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): continued fraction

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