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.

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