Bernoulli numbers: B_0=1, B_1=−1/2, B_2=1/6, B_3=0, B_4=−1/30, B_6=1/42... All B_{2k+1}=0 for k≥1. They connect to ζ(2k): ζ(2)=π²/6 uses B_2=1/6. Discovered by Jacob Bernoulli, computed by Ada Lovelace's famous algorithm.
How to Use
Enter index n:
- B_n: The Bernoulli number
- Fraction: Exact rational form
- ζ connection: For even n
Zeta Values
ζ(2k) = (−1)^{k+1} · B_{2k} · (2π)^{2k} / (2·(2k)!). So ζ(2)=π²/6 (B_2=1/6), ζ(4)=π⁴/90 (B_4=−1/30), ζ(6)=π⁶/945 (B_6=1/42). Bernoulli numbers encode ALL even zeta values!
History
Jacob Bernoulli discovered them studying sums of powers: Σk^m. Ada Lovelace wrote the first algorithm to compute them (1843) — often called the first computer program. They appear in the Euler-Maclaurin formula, Todd genus, and number theory.
Step-by-Step Instructions
- 1Enter index n.
- 2Compute B_n.
- 3See fraction.
- 4Check zeta.
- 5View table.