Diff — Bernoulli numbers

The **Bernoulli numbers** are a sequence of rational numbers that surface throughout mathematics — in power sums, series expansions, and number theory. In computing they are famous for a different reason: computing them is the subject of [[Ada Lovelace]]'s 1843 algorithm for the [[Analytical Engine]], the first program published for a machine.

First studied by **Jakob Bernoulli** around 1713 in his work on sums of powers, the sequence begins:

| B₀ | B₁ | B₂ | B₃ | B₄ | B₅ | B₆ | B₇ | B₈ |

|----|----|----|----|----|----|----|----|----|

| 1 | −1/2 | 1/6 | 0 | −1/30 | 0 | 1/42 | 0 | −1/30 |

The odd-indexed numbers past B₁ are all zero; the even ones alternate sign and grow explosively. They appear wherever powers accumulate — the sum 1ᵏ + 2ᵏ + … + nᵏ, the series for the tangent, and deep results like the **Riemann zeta function** at negative integers: ζ(−n) = −Bₙ₊₁/(n+1).

When Lovelace translated Menabrea's paper on Babbage's engine, she needed a computation that would show the machine's real power — not a fixed formula but a *process* with repetition and accumulation. Bernoulli numbers fit perfectly: each one can be derived from the ones before it, so the engine would loop, feeding its own intermediate results back through the mill.

**Note G** presented the method as a step-by-step table of operations — variables in, intermediate columns, and the cascade of dependent values — essentially a trace of execution written before any machine existed to run it. In modern terms she wrote a loop with a recurrence inside it, verified the register flow by hand, and shipped the listing in a journal.

- **They needed the whole machine** — simple formulas would have wasted the demo; the recurrence demanded storage, sequencing, and reuse of results.

- **The program was traceable** — her operation table let a reader *step through* the computation, which is why historians can still execute Note G today and why it serves as evidence of what the engine could do.

- **They outlived the machine** — the engine was never built, but Bernoulli-number recurrences remain a standard test program; emulations of Note G's algorithm run correctly on modern reconstructions of Babbage's design.

- [[Ada Lovelace]]

- [[Analytical Engine]]

- [[Charles Babbage]]

- [[History of Programming]]

1. Menabrea, Luigi Federico (tr. Ada Lovelace). "Sketch of the Analytical Engine invented by Charles Babbage." *Taylor's Scientific Memoirs* 3 (1843) — includes Lovelace's Notes A–G.

2. Babbage, Charles. *Passages from the Life of a Philosopher* (Longman, 1864).

3. Swade, Doron. *The Cogwheel Brain: Charles Babbage and the Quest to Build the First Computer* (Little, Brown, 2000).

4. Boyer & Merzbach, *A History of Mathematics*, 3rd ed. (Wiley, 2011) — for the Bernoulli family context.

[[Category:History]]

[[Category:Mathematics]]