What Are Bernoulli Numbers?

Bernoulli numbers are a sequence of rational numbers deeply connected to number theory, combinatorics, and analysis. They appear in the Taylor series expansion of $\frac{x}{e{x}^{}-1}$ and are used in formulas for sums of powers, the Euler–Maclaurin formula, and values of the Riemann zeta function at negative integers. The first few Bernoulli numbers are $1$, $\mathrm{-1/2}$, $\mathrm{1/6}$, and so on. Notably, odd-indexed Bernoulli numbers beyond $\mathrm{B\_1}$ are zero.

Historical Background

Named after the Swiss mathematician Jacob Bernoulli, these numbers were studied in the late seventeenth century. Bernoulli used them to derive general formulas for the sums of integer powers, a task that previously required tedious manual computations. Later mathematicians, including Euler, expanded their applications in series expansions and special functions. Today, Bernoulli numbers remain fundamental in analytic number theory and the study of modular forms.

The Akiyama-Tanigawa Algorithm

One efficient way to compute Bernoulli numbers is via the Akiyama-Tanigawa algorithm. This iterative method builds a triangular array where each row refines the previous one. For $\mathrm{n}$ up to several hundred, it is straightforward to implement in a browser. The idea is to initialize an array $A$ with $A_{\mathrm{m}}^0=\frac{1}{m+1}$ for $0\le m\le n$. Then, each subsequent column subtracts weighted values from the column before. The last element in each row is the Bernoulli number $B{n}_{}$.

Applications

Bernoulli numbers surface in many surprising places. They help evaluate the Riemann zeta function $\mathrm{\backslash zeta}\left(s\right)$ at negative integers via $\mathrm{\backslash zeta}\left(-,n\right)=-\frac{B}{\mathrm{n+1}}/\mathrm{n+1}$. They also appear in formulas for higher derivatives of trigonometric functions and in the computation of certain definite integrals. In numerical analysis, Bernoulli numbers are key components of the Euler–Maclaurin formula, which connects discrete sums and continuous integrals with boundary corrections.

Using the Calculator

Enter a nonnegative integer $n$. The script applies the Akiyama-Tanigawa algorithm to generate all Bernoulli numbers up to $\mathrm{B\_n}$ and returns the result with six decimal places. Because the numbers grow rapidly in complexity, very large $n$ may produce fractions with large numerators and denominators. However, for moderate $n$ values, this method is both fast and accurate.