continued fraction

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continued fraction

A student writes a continued fraction on the chalkboard.

Definition
  1. Noun:
    • A mathematical expression: A continued fraction is a representation of a number as a sum of its integer part and a fraction. In this fraction, the numerator is an integer, and the denominator is itself an integer plus another fraction, which continues in this pattern indefinitely or until it terminates.
    • A specific type of fraction: It is an expression of the form (a0 + \cfrac{1}{a1 + \cfrac{1}{a2 + \cfrac{1}{a3 + \ddots}}}), where (a0) is an integer and (a1, a2, a3, \ldots) are positive integers.
Usage Examples
  • Noun:
    • The golden ratio has a beautifully simple representation as a continued fraction.
    • To approximate the square root of 2, one can use a simple continued fraction.
    • The mathematician studied the properties of the infinite continued fraction for pi.
Advanced Usage
  • "Simple continued fraction": A continued fraction where all the numerators are 1.
    • Every rational number can be expressed as a finite simple continued fraction.
  • "Convergent of a continued fraction": A rational number obtained by truncating the infinite continued fraction at a certain point.
    • Each convergent provides a successively better approximation to the original number.
Variants and Related Words
  • Simple continued fraction (n): A continued fraction of the standard form where all partial numerators are equal to 1.
  • Generalized continued fraction (n): A continued fraction where the numerators can be integers other than 1.
  • Convergent (n): A fraction that approximates the value of an infinite continued fraction by terminating it at a finite depth.
Synonyms
  • Chain fraction: An older, less common term for a continued fraction.
Related Phrases and Concepts
  • "Finite continued fraction": A continued fraction that terminates after a finite number of steps, representing a rational number.
  • "Infinite continued fraction": A continued fraction that continues indefinitely, representing an irrational number.
  • "Periodic continued fraction": An infinite continued fraction that eventually repeats a block of integers, representing a quadratic irrational number.
continued fraction

A student writes a continued fraction on the chalkboard.

Noun
  1. 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