Continued fractions

Definition

Continued fractions are expressions of the form $$a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \cdots}}$$ where each $$a_i$$ is an integer. They serve as a powerful tool for approximating real numbers, particularly in the context of rational and irrational numbers, and have significant applications in Diophantine approximation and transcendence theory.

5 Must Know Facts For Your Next Test

1. Continued fractions can provide the best rational approximations of irrational numbers, making them useful in Diophantine approximation.
2. The convergents of a continued fraction are obtained by truncating the infinite series, and they form a sequence of fractions that converge to the original number.
3. Any real number can be represented as a continued fraction, and this representation is unique for most numbers, except for some rational numbers.
4. Continued fractions can be finite or infinite; finite continued fractions correspond to rational numbers, while infinite ones represent irrational numbers.
5. The theory of continued fractions is closely related to transcendence theory, as it can help identify whether certain numbers are algebraic or transcendental.

Review Questions

• How do continued fractions provide insights into the approximation of irrational numbers by rational numbers?
  • Continued fractions offer a systematic way to find the best rational approximations to irrational numbers. Each step in a continued fraction represents an increasingly accurate approximation, with the convergents providing the closest rational fractions. This method allows us to determine how well a particular rational number approximates an irrational one and to analyze the rate at which these approximations improve.
• Discuss the relationship between continued fractions and Diophantine approximation, especially regarding their applications in finding integer solutions.
  • Continued fractions play a crucial role in Diophantine approximation by enabling mathematicians to find the best possible rational approximations for irrational numbers. This is significant when seeking integer solutions to equations involving real numbers. By analyzing the convergents of a continued fraction, one can determine how closely these solutions can come to representing real values, thus providing insights into the nature of integer relationships within mathematical problems.
• Evaluate how continued fractions can aid in distinguishing between algebraic and transcendental numbers within transcendence theory.
  • Continued fractions are instrumental in transcendence theory as they help identify whether a number is algebraic or transcendental. For instance, if a number has a periodic continued fraction expansion, it is guaranteed to be quadratic algebraic. On the other hand, certain types of non-repeating or complex continued fractions can indicate that a number is transcendental. This distinction is vital for understanding the properties and classifications of different types of numbers in mathematics.

Related terms

Diophantine approximation: The study of how closely real numbers can be approximated by rational numbers.

Convergents: The best rational approximations derived from the continued fraction representation of a number.

Transcendental numbers: Numbers that are not the root of any non-zero polynomial equation with integer coefficients.