Continued fractions

Definition

Continued fractions are expressions obtained by iteratively adding integers and reciprocals, representing real numbers in a unique way. They can provide excellent approximations of irrational numbers and are especially useful in number theory and approximation theory, revealing relationships between rational numbers and their best approximations.

5 Must Know Facts For Your Next Test

1. Continued fractions can be finite or infinite, with finite ones representing rational numbers and infinite ones representing irrational numbers.
2. The coefficients in the continued fraction can give insight into the properties of the number being represented, such as its irrationality or periodicity.
3. Continued fractions often yield the best rational approximations to real numbers, making them an essential tool for understanding approximation theory.
4. The theory of continued fractions provides algorithms for calculating approximations and establishing bounds on the errors of these approximations.
5. Padé approximants can be derived from continued fractions by expressing them in a form that relates closely to polynomial approximations of functions.

Review Questions

• How do continued fractions contribute to finding the best rational approximations of real numbers?
  • Continued fractions provide a systematic way to represent real numbers through iterative processes of integer addition and reciprocal operations. This representation yields convergents that are the best rational approximations, especially for irrational numbers. Each convergent gives a fraction that closely approximates the original number, minimizing the difference and showcasing the efficiency of continued fractions in producing optimal approximations.
• Discuss how convergents derived from continued fractions can be utilized in Diophantine approximation.
  • Convergents from continued fractions play a crucial role in Diophantine approximation by offering sequences of rational numbers that approximate real numbers closely. These convergents allow mathematicians to analyze how well real numbers can be represented by rationals, effectively addressing questions about optimality in approximation. The properties of these convergents also help establish bounds on errors associated with these approximations, enriching our understanding of number theory.
• Evaluate the significance of continued fractions in the context of Padé approximants and their applications.
  • Continued fractions are significant in the derivation of Padé approximants, which are used to approximate functions more effectively than Taylor series. By relating continued fractions to rational functions, Padé approximants utilize the convergents derived from continued fractions to create better numerical approximations. This connection highlights not only the theoretical underpinnings of approximation methods but also their practical applications in numerical analysis, leading to improved error analysis and convergence properties in function approximation tasks.

Related terms

Convergents: Convergents are the fractions derived from truncating a continued fraction at various points, which provide increasingly accurate approximations to the original number.

Diophantine approximation: Diophantine approximation is a branch of number theory that studies how closely real numbers can be approximated by rational numbers, often utilizing continued fractions.

Farey sequence: A Farey sequence is a sequence of completely reduced fractions between 0 and 1, which can be related to continued fractions as it provides an ordered list of rational approximations.