Continued Fraction Laboratory

Inspired by a conversation with Alexander Young (UCSD)

A playground for the "best rational approximations" of real numbers. Below, you can compute with them interactively or explore the symbolic masterpieces of Srinivasa Ramanujan.

1. Define Your Fractions

Fraction A (Standard Notation $[a_0; a_1, \dots]$)

Value:

Fraction B

Value:

2. Perform Arithmetic

4. Gallery of Symbolic Beauty

The Indian mathematician Srinivasa Ramanujan (1887–1920) possessed an intuitive mastery of continued fractions that remains unmatched. While we calculate with numbers, Ramanujan saw deep relationships between variables.

References & History

Ramanujan's Notebooks: Ramanujan recorded thousands of these identities, many without proof, which mathematicians spent decades verifying.
Leonhard Euler (1737): Proved that $e$ and $\phi$ have elegant infinite continued fractions, while rational numbers always terminate.
The "Best" Approximation: $\pi \approx 22/7$ comes from the continued fraction $[3; 7]$. The next term gives $355/113$, which is accurate to 6 decimal places!

Van2025/Gemini 3 Pro

Continued Fraction Laboratory

1. Define Your Fractions

2. Perform Arithmetic

3. Computational Result

4. Gallery of Symbolic Beauty

References & History