Bernoulli Number Bₙ Calculator
Enter a degree n to get the Bernoulli number Bₙ as an exact fraction and a decimal, plus a table of B0 through Bn.
Input
Enter an integer n that is 0 or greater to compute the Bernoulli number Bₙ as an exact fraction and a decimal.
Enter an integer from 0 to 100.
Result
Bernoulli number B(12)
-691 / 2730
Decimal: -0.253113553113553113
Degree n
12
Parity of n
Even
Value is zero
No
Table of B0 through Bₙ
| Degree | Fraction | Decimal |
|---|---|---|
| B(0) | 1 | 1 |
| B(1) | -1 / 2 | -0.5 |
| B(2) | 1 / 6 | 0.166666666666666666 |
| B(3) | 0 | 0 |
| B(4) | -1 / 30 | -0.033333333333333333 |
| B(5) | 0 | 0 |
| B(6) | 1 / 42 | 0.023809523809523809 |
| B(7) | 0 | 0 |
| B(8) | -1 / 30 | -0.033333333333333333 |
| B(9) | 0 | 0 |
| B(10) | 5 / 66 | 0.075757575757575757 |
| B(11) | 0 | 0 |
| B(12) | -691 / 2730 | -0.253113553113553113 |
How it works
- The Bernoulli numbers Bₙ are found one by one by solving the recurrence Σ C(n+1,k) Bₖ = 0 for Bₙ.
- This tool uses the convention B1 = -1/2.
- Computation uses BigInt rational arithmetic (numerator and denominator), so the fraction is exact with no rounding error.
- For odd degrees n with n ≥ 3, Bₙ is exactly 0.
- The decimal value is a high precision approximation obtained by long division of the fraction.
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Bernoulli Number Bₙ Calculator