Continued Fraction Calculator
Calculate the continued fraction representation of any irrational number. Explore approximations and understand the concept with our easy-to-use tool.
Enter Irrational Number
Input an irrational number or a mathematical expression (e.g., sqrt(2), pi, (1+sqrt(5))/2) to find its continued fraction representation.
Continued Fraction Representation:
Approximations
Continued fractions provide a sequence of rational approximations that get progressively closer to the irrational number.
Approximation
Value:
What are Continued Fractions?
A continued fraction is a way to represent any real number as a sequence of integers. For an irrational number, this sequence is infinite. Each term in the sequence, called a convergent, provides a rational approximation of the number. For example, the continued fraction of π starts as [3; 7, 15, 1, 292, ...]. The approximations are obtained by truncating this sequence: 3, 3 + 1/7 = 22/7, 3 + 1/(7 + 1/15) = 333/106, and so on, each getting closer to π. Continued fractions are useful in number theory, approximation theory, and computer science.
- Representation: Expressing numbers as nested fractions.
- Approximations: Generating rational values close to irrational numbers.
- Applications: Used in various fields like number theory and computing.