Continued Fraction Calculator

Continued Fraction Calculator: Core Features and Principles

When you need to represent a complex decimal, irrational number, or rational fraction as an elegantly structured continued fraction, expanding it manually is both time-consuming and prone to errors. This tool automatically converts any real number input (decimals, fractions, or radical expressions) into a standard continued fraction. It can also work in reverse, converting a sequence of integers back into a simplified fraction, while providing the convergents at each step to help you visualize the best rational approximations.

Why Choose Our Continued Fraction Calculator?

• Supports bidirectional conversion: Expand numerical values into continued fraction sequences, or convert continued fraction sequences back into exact fractions or decimal approximations.
• Customizable expansion depth: Allows you to set the maximum number of terms for the continued fraction expansion, controlling calculation precision and output length.
• Automatically generates convergent tables: Each expansion step provides the current truncated fraction value and its error margin, making it easy to analyze the best approximation.

How to Use the Tool

1. Select the conversion direction: "Number to Continued Fraction" or "Continued Fraction to Number".
2. If you select "Number to Continued Fraction", enter a number (e.g., 3.245, 1.41421356, or the expression 22/7) and set the maximum number of expansion terms (default is 20). If you select "Continued Fraction to Number", enter a comma-separated sequence of integers, such as 3,7,15,1,292.
3. Click the "Calculate" button to view the continued fraction expression (e.g., [3; 7, 15, 1, 292]), the corresponding list of convergents, and the restored simplified fraction or decimal approximation in the results area. Example: Inputting 3.1415926 → Continued fraction [3; 7, 15, 1, 292, ...], with convergents 3/1, 22/7, 333/106, 355/113, etc.

Frequently Asked Questions (FAQ)

• Is the continued fraction representation unique?
  A simple continued fraction (where all numerators are 1) has exactly one finite representation for a rational number. For irrational numbers, the infinite simple continued fraction representation is unique. However, a finite regular continued fraction can be written in two ways (the last coefficient can be split into itself minus 1, and a final 1). This tool defaults to the standard unique form.
• Why are continued fractions better for approximations than standard decimals?
  Continued fractions provide the rational approximation with the smallest error for a given denominator size (the best rational approximation). For example, the continued fraction expansion of Pi (π) provides a highly accurate approximation with a relatively small denominator at 355/113, which standard decimal rounding cannot achieve.

Important Notes

When entering a continued fraction sequence, ensure all values are integers separated by standard commas. Irrational numbers or infinite decimals can only be expanded to a finite number of terms, so the result is always an approximation of the original number; more terms yield higher precision. This tool operates entirely locally in your browser and does not collect any input data, ensuring your privacy. If the input value is too large or the number of expansion terms is too high, it may cause calculation delays or increased memory usage. We recommend keeping the number of terms under 500.

Technical Notes & Usage Tips

Continued fractions are widely used in number theory, computer science, cryptography, and engineering approximations. Classic examples: The golden ratio φ = (1+√5)/2 expands to [1; 1, 1, 1, ...], which is the simplest infinite periodic continued fraction. Pi π ≈ [3; 7, 15, 1, 292, 1, 1, 1, 2, ...], and the base of the natural logarithm e = [2; 1, 2, 1, 1, 4, 1, 1, 6, ...]. When dealing with rational function integration, Diophantine approximations, or constructing musical temperaments, using a continued fraction tool can quickly generate a rapidly converging sequence of fractions. We recommend using the convergent table to select the approximation that best balances precision and denominator size.