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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, 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.

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, 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 fraction

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.

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.

- Examples of finite simple continued fractions Formula Numeric Remarks a 0 {\displaystyle \ a_{0}} 2 {\displaystyle \ 2} All integers are a degenerate...

- a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial...

- In number theory, the continued fraction factorization method (CFRAC) is an integer factorization algorithm. It is a general-purpose algorithm, meaning...

- A fraction (from Latin fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English,...

- In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form x = a 0 + 1 a 1 + 1 a 2 + ⋱ ⋱ a k + 1 a...

- irrational. Therefore, π cannot have a periodic continued fraction. Although the simple continued fraction for π (shown above) also does not exhibit any...

- The Rogers–Ramanujan continued fraction is a continued fraction discovered by Rogers (1894) and independently by Srinivasa Ramanujan, and closely related...

- a rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since...

- {\displaystyle b\neq k} . It is also disjunctive sequence. The simple continued fraction expansion of Champernowne's constant has been studied as well. Kurt...

- continued fractions, Euler's continued fraction formula is an identity connecting a certain very general infinite series with an infinite continued fraction...

- a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial...

- In number theory, the continued fraction factorization method (CFRAC) is an integer factorization algorithm. It is a general-purpose algorithm, meaning...

- A fraction (from Latin fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English,...

- In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form x = a 0 + 1 a 1 + 1 a 2 + ⋱ ⋱ a k + 1 a...

- irrational. Therefore, π cannot have a periodic continued fraction. Although the simple continued fraction for π (shown above) also does not exhibit any...

- The Rogers–Ramanujan continued fraction is a continued fraction discovered by Rogers (1894) and independently by Srinivasa Ramanujan, and closely related...

- a rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since...

- {\displaystyle b\neq k} . It is also disjunctive sequence. The simple continued fraction expansion of Champernowne's constant has been studied as well. Kurt...

- continued fractions, Euler's continued fraction formula is an identity connecting a certain very general infinite series with an infinite continued fraction...

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