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A continued fraction is a mathematical expression that can be writen as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, the continued fraction is finite or infinite.
Continued Fractions are important in many branches of mathematics. They arise naturally in long division and in the theory of approximation to real numbers by rationals.
There are two types of continued fractions: finite continued fractions. infinite continued fractions. A finite continued fraction is a general representation of a real number x x in the form.
continued fraction is given by two sequences of numbers and One traditional way to write a continued fraction is: {bn}n≥0 {an}n≥1. is 0). Similarly, an infinite continued fraction will be the limit, if it exists, of the sequence of numbers. Q0 = b0, Q1 = b0 + a1/b1, Q2 = b0 + a1/(b1 + (a2/b2)), ... .
Unlike regular fractions, which have a single numerator and denominator, a continued fraction is expressed as the sum of an integer and a fraction, where the fraction’s denominator itself contains another sum of an integer and a fraction, continuing this process indefinitely or until it terminates.
The simple continued fraction takes for all , leaving. If is an integer and the remainder of the partial denominators for are positive integers, the continued fraction is known as a regular continued fraction.
Continued fractions are a topic in number theory which has applications to rational approximations of real numbers. We will first explain what a continued fraction is, prove some basic theorems about them, and then show how they can be used to find good rational approximations.
In this chapter, we introduce continued fractions, prove their basic properties and apply these properties to solve some problems. Being a very natural object, continued fractions appear in many areas of Mathematics, sometimes in an unexpected way.
Definition: Continued fractions. A simple continued fraction is of the form, denoted by [a0, a1, …], a0 + 1 a1 + 1 a2 + …, where a0, a1, a2, … ∈ Z. Continued fraction has been studied extensively, but we will only explore some of them in this class. Example 8.3.1:
Using the Nesting Lemma, it is easy to evaluate a continued fraction from right to left (or looking at (1.1), from bottom to top): repeatedly take reciprocals and add preceding (higher) terms. The Wallis Algorithm tells how to evaluate a continued fraction from left to right (top to bottom).