19/05/2021

In Folha de S.Paulo: mathematics that seems like magic

Curso do projeto Mentalidades Matemáticas. Foto: Divulgação

Reproduction of Marcelo Viana's column in Folha de S.Paulo.

It's one of the most mysterious mathematical discoveries, one of those that seems like magic. It's very useful, and nobody really knows why it works.

The story began in 1881, when the Canadian-American astronomer and mathematician Simon Newcomb (1835-1909) noticed that his table of logarithms was handled much more on the first pages than on the last.

Before the invention of practical calculators, the logarithm table was a fundamental tool for performing complex calculations in any field of science and engineering. Newcomb's observation meant, for example, that he had dealt with far more astronomical data with a leading digit (the leftmost one) of 1 than with a leading digit of 9.

Even more incredible, all the observatory's logarithm tables looked the same: astronomical data "prefer" to start with small digits rather than large ones! Newcomb proposed Table 1 (see below) for the frequency of the first digit.

Although he proposed an explanation, this observation must have seemed bizarre and was forgotten. Until 1937, when it was rediscovered by the American physicist Frank Benford (1883–1948), who ended up giving his name to the law.

But this is not entirely unfair, as Benford went further, pointing out that this behavior emerges in almost all the data we encounter: air distances, molecular weights, mortality rates, number of passes in a football match, apartment prices, gross domestic product of countries, mathematical constants, newspaper circulation, etc. — they all follow the table above!

Of course, I had to see it to believe it. I downloaded the population figures for Brazilian municipalities from the IBGE website, calculated the frequency of the first digit, and it was exactly as described (see table 2).

To read the full text, visit the newspaper's website.