Continued fractions

Definition

Continued fractions are expressions that represent numbers through a sequence of integer divisions, where each division can be expressed as a whole number plus a fraction. This format is particularly useful for approximating irrational numbers, allowing us to see their fractional parts in a structured way. By breaking down a number into continued fractions, we can gain insight into its properties and how closely it can be approximated by rational numbers.

5 Must Know Facts For Your Next Test

1. Continued fractions can represent both rational and irrational numbers, but they provide the most insight into irrational numbers.
2. The process of creating a continued fraction for a number involves repeatedly taking the integer part and inverting the remaining fractional part.
3. Each continued fraction has a unique sequence of coefficients that allows it to be approximated by simpler rational numbers called convergents.
4. Some famous irrational numbers, like the square root of 2 or the golden ratio, have simple and repeating continued fractions that reveal their mathematical beauty.
5. The efficiency of continued fractions in approximating irrational numbers is quantified by how quickly their convergents approach the actual value.

Review Questions

• How do continued fractions provide insight into the properties of irrational numbers compared to rational numbers?
  • Continued fractions offer a structured way to express irrational numbers, highlighting their non-repeating nature. Unlike rational numbers, which terminate or repeat in their decimal forms, the continued fraction representation reveals how closely an irrational number can be approximated by rational numbers through its convergents. This makes it easier to study their properties, such as approximation quality and mathematical relationships.
• What is the significance of convergents in the context of continued fractions and how do they relate to approximating irrational numbers?
  • Convergents are critical in understanding how well continued fractions approximate irrational numbers. They are obtained by truncating the continued fraction at various points, yielding the best rational approximations at each step. The closer the convergents get to the actual value of an irrational number, the more efficient the continued fraction is in representing that number. This relationship shows how powerful continued fractions are in numerical analysis and approximation techniques.
• Evaluate the role of continued fractions in mathematical research and applications, especially concerning irrational numbers.
  • Continued fractions play a pivotal role in mathematical research and have practical applications in number theory and approximation theory. They help mathematicians understand properties of irrational numbers, such as their distribution and behavior within real numbers. Continued fractions also find uses in fields like cryptography and algorithm development, where efficient approximations of real numbers are necessary. As researchers uncover deeper connections between continued fractions and other mathematical concepts, they continue to shape our understanding of numerical relationships.

Related terms

Rational Numbers: Numbers that can be expressed as the quotient of two integers, where the denominator is not zero.

Irrational Numbers: Numbers that cannot be expressed as a simple fraction; their decimal representation goes on forever without repeating.

Convergents: The best rational approximations of a continued fraction, formed by truncating the continued fraction at different lengths.