Definition:Continued Fraction

Finite Continued Fraction

Let $a_1 \in \R$ be any real number and $a_2, a_3, \ldots, a_n$ be any positive real numbers.

Then the expression:

$a_1 + \cfrac 1 {a_2 + \cfrac 1 {a_3 + \cfrac 1 {\ddots \cfrac {} {a_{n-1} + \cfrac 1 {a_n}} }}}$

is a finite continued fraction.

A finite continued fraction is often abbreviated FCF.

Infinite Continued Fraction

Let $a_1 \in \R$ be any real number and $a_2, a_3, \ldots, a_n, \ldots$ be any infinite sequence in $\R$.

Then the expression:

$a_1 + \cfrac 1 {a_2 + \cfrac 1 {a_3 + \cfrac 1 {\ddots \cfrac {} {a_{n-1} + \cfrac 1 {a_n + \cfrac 1 {\ddots}}} }}}$

is an infinite continued fraction.

• An infinite continued fraction is often abbreviated ICF.

Partial Quotient

The numbers $a_1, a_2, a_3, \ldots$ are called the partial quotients of the continued fraction.

Simple Continued Fraction

When all the partial quotients of a continued fraction are integers, the continued fraction is described as simple.

Simple Finite Continued Fraction

A simple finite continued fraction is often abbreviated SFCF.

Simple Infinite Continued Fraction

A simple infinite continued fraction is often abbreviated SICF.

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

Continued Fraction Expansion

Any continued fraction is completely determined by its partial quotients.

Therefore, to reduce the cumbersome nature of its representation, the continued fractions in the definitions are usually written as:

• $\left[{a_1, a_2, a_3, \ldots, a_n}\right]$ for the finite case, and
• $\left[{a_1, a_2, a_3, \ldots}\right]$ for the infinite case.

For example:

$\left[{1, 2, 3}\right] = 1 + \cfrac 1 {2 + \cfrac 1 3}$

Such an expression is known as the continued fraction expansion of the continued fraction, especially in the case of the infinite version.

Value

The value of a continued fraction is the number which results from calculating out the fractions.

Note that formally the continued fraction and its value are considered to be distinct; a continued fraction is an arithmetic representation of its value.

However, this is a nicety of interpretation and may usually be ignored - we often say something like $x = \left[{a_1, a_2, a_3, \ldots, a_n}\right]$ to mean that $x$ is the value of the given continued fraction.

Convergent

Let $\left[{a_1, a_2, a_3, \ldots, a_n}\right]$ or $\left[{a_1, a_2, a_3, \ldots}\right]$ be a continued fraction (i.e. either finite or infinite).

Then the $k$th convergent $C_k$ of the given continued fraction is the finite continued fraction:

$C_k = \left[{a_1, a_2, \ldots, a_k}\right]$.

(In the finite case it is of course taken as read that $k \le n$.)

From Value of Simple Continued Fraction we have that the sequence of convergents of a SICF does in fact converge to a limit.

From Uniqueness of Simple Infinite Continued Fraction we see we can talk directly about the convergents to any irrational number $x$.

Odd Convergent

The odd convergents of $\left[{a_1, a_2, a_3, \ldots, a_n}\right]$ are the convergents $C_1, C_3, C_5, \ldots$, that is, those with an odd subscript.

Even Convergent

The even convergents of $\left[{a_1, a_2, a_3, \ldots, a_n}\right]$ are the convergents $C_2, C_4, C_6, \ldots$, that is, those with an even subscript.

Numerators and Denominators

Let $\left[{a_1, a_2, a_3, \ldots, a_n}\right]$ or $\left[{a_1, a_2, a_3, \ldots}\right]$ be a continued fraction (i.e. either finite or infinite).

Then the numerators $p_1, p_2, p_3, \ldots$ and denominators $q_1, q_2, q_3, \ldots$ of this continued fraction are defined as:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle p_1\)	\(=\)	\(\displaystyle a_1\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle q_1\)	\(=\)	\(\displaystyle 1\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle p_2\)	\(=\)	\(\displaystyle a_1 a_2 + 1\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle q_2\)	\(=\)	\(\displaystyle a_2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle p_k\)	\(=\)	\(\displaystyle a_k p_{k-1} + p_{k-2}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle q_k\)	\(=\)	\(\displaystyle a_k q_{k-1} + q_{k-2}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

From Value of Simple Continued Fraction we have that $C_k = \dfrac {p_k} {q_k}$, where $C_k$ is the $k$th convergent of the continued fraction.

Generalized Form

The form of continued fraction in which the successive numerators are all $1$ is called a regular continued fraction, or to be in canonical form.

If the numerators can be allowed to be any value, even functions in some contexts, the continued fraction is known as a generalized continued fraction:

$b_0 + \cfrac {a_1} {b_1 + \cfrac {a_2} {b_2 + \cfrac {a_3} {\ddots \cfrac {} {b_{n-1} + \cfrac {a_n} {b_n + \cfrac {a_{n+1}} {\ddots}}} }}}$

Notes

The concept of continued fractions has been around a long time, since Euclid at least, and a great deal of research has been done and terminology developed.