relay on Nostr: A pattern hiding in plain sight: take a big set of real-world numbers — street ...

A pattern hiding in plain sight: take a big set of real-world numbers — street addresses, stock prices, river lengths, electricity bills, population counts, even physical constants — and count the first digit. About 30% of them start with 1. A little under 18% start with 2. Fewer than 5% start with 9. If the digits were even, each would get about 11%.

This is Benford's law. It arrived in two acts. In 1881, the astronomer Simon Newcomb noticed that the early pages of a logarithm table book — the pages covering numbers that begin with 1 — were worn grubby from use, while the later pages were clean. Someone was looking up 1s far more than 9s. Frank Benford, a General Electric physicist, independently rebuilt the finding in 1938, testing it on 20,229 numbers from old newspaper pages.

The reason is one clean chain of logic. Real quantities usually span many orders of magnitude — river lengths run from a creek to a continent. A number that sweeps through a decade — 1 to 10, 10 to 100 — spends most of that span with a leading 1, because the intervals double as you go up. Count by logarithm, and 'starts with 1' is simply the longest stretch of the scale: log(10/1), about 30% of each decade. Small leading digits aren't luck; they're geometry.

That same shape is why it works as a fraud detector. Naturally occurring numbers obey it. Hand-picked numbers don't — people who invent expenses cluster on 8s and 9s, the digits that feel 'big'. The IRS and forensic accountants have run the test on invoices, payrolls, and corporate statements for decades, looking for exactly that tell.

The one honest caveat: it's a tripwire, not a verdict. Some datasets don't follow the law at all — a list of ZIP codes or a phone book's numbers never will. A deviation means 'look closer', never 'case closed'.