What is "continued fraction"

Fraction \Frac"tion\, n. [F. fraction, L. fractio a breaking, fr. frangere, fractum, to break. See Break.]

1. The act of breaking, or state of being broken, especially by violence. [Obs.]

   Neither can the natural body of Christ be subject to any fraction or breaking up.
   --Foxe.

2. A portion; a fragment.

   Some niggard fractions of an hour.
   --Tennyson.

3. (Arith. or Alg.) One or more aliquot parts of a unit or whole number; an expression for a definite portion of a unit or magnitude.

   Common fraction, or Vulgar fraction, a fraction in which the number of equal parts into which the integer is supposed to be divided is indicated by figures or letters, called the denominator, written below a line, over which is the numerator, indicating the number of these parts included in the fraction; as 1/2, one half, 2/5, two fifths.

   Complex fraction, a fraction having a fraction or mixed number in the numerator or denominator, or in both.
   --Davies & Peck.

   Compound fraction, a fraction of a fraction; two or more fractions connected by of.

   Continued fraction, Decimal fraction, Partial fraction, etc. See under Continued, Decimal, Partial, etc.

   Improper fraction, a fraction in which the numerator is greater than the denominator.

   Proper fraction, a fraction in which the numerator is less than the denominator.

Continued \Con*tin"ued\, p. p. & a. Having extension of time, space, order of events, exertion of energy, etc.; extended; protracted; uninterrupted; also, resumed after interruption; extending through a succession of issues, session, etc.; as, a continued story. ``Continued woe.''
--Jenyns. ``Continued succession.''
--Locke.

Continued bass (Mus.), a bass continued through an entire piece of music, while the other parts of the harmony are indicated by figures beneath the bass; the same as thorough bass or figured bass; basso continuo. [It.]

Continued fever (Med.), a fever which presents no interruption in its course.

Continued fraction (Math.), a fraction whose numerator is 1, and whose denominator is a whole number plus a fraction whose numerator is 1 and whose denominator is a whole number, plus a fraction, and so on.

Continued proportion (Math.), a proportion composed of two or more equal ratios, in which the consequent of each preceding ratio is the same with the antecedent of the following one; as, 4 : 8 : 8 : 16 :: 16 : 32.

n. (context mathematics English) A fraction whose numerator is an integer and whose denominator is an integer plus a fraction whose denominator is an integer plus a fraction - and so on to an infinite number of terms.

n. a fraction whose numerator is an integer and whose denominator is an integer plus a fraction whose numerator is an integer and whose denominator is an integer plus a fraction and so on

} |caption=A finite continued fraction, where n is a non-negative integer, a is an integer, and a is a positive integer, for i = 1, …, n. }}

In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In a finite continued fraction (or terminated continued fraction), the iteration/ recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers a are called the coefficients or terms of the continued fraction.

Continued fractions have a number of remarkable properties related to the Euclidean algorithm for integers or real numbers. Every rational number has two closely related expressions as a finite continued fraction, whose coefficients can be determined by applying the Euclidean algorithm to (p, q). The numerical value of an infinite continued fraction is irrational; it is defined from its infinite sequence of integers as the limit of a sequence of values for finite continued fractions. Each finite continued fraction of the sequence is obtained by using a finite prefix of the infinite continued fraction's defining sequence of integers. Moreover, every irrational number α is the value of a unique infinite continued fraction, whose coefficients can be found using the non-terminating version of the Euclidean algorithm applied to the incommensurable values α and 1. This way of expressing real numbers (rational and irrational) is called their continued fraction representation.

It is generally assumed that the numerator of all of the fractions is 1. If arbitrary values and/or functions are used in place of one or more of the numerators or the integers in the denominators, the resulting expression is a generalized continued fraction. When it is necessary to distinguish the first form from generalized continued fractions, the former may be called a simple or regular continued fraction, or said to be in canonical form.

The term continued fraction may also refer to representations of rational functions, arising in their analytic theory. For this use of the term, see Padé approximation and Chebyshev rational functions.