When math looks like magic – 05/19/2021 – Marcelo Viana

It’s one of the most mysterious mathematical discoveries that looks like magic. It’s very useful and no one really knows why it works.

The story began in 1881 when the Canadian-American astronomer and mathematician Simon Newcomb (1835-1909) noticed that his log table was treated much better on the first pages than the last.

Before the invention of practical pocket calculators, the log table was a fundamental tool for performing complex calculations in all areas of science and technology. For example, Newcomb’s observation meant that he had handled far more astronomical data with a leading digit (the left) 1 than with a leading digit 9.

Most amazingly, all of the observatory’s log tables looked the same: astronomical data “prefers” to start with small digits rather than large ones! Newcomb suggested Table 1 (see below) for the frequency of the first digit.

Although he suggested an explanation, the observation must have been bizarre and overlooked. Until 1937, when it was rediscovered by the American physicist Frank Benford (1883 “” 1948), who gave the law its name.

This isn’t entirely unfair, however, as Benford went further, pointing out that this behavior appears in almost all of the data we face: air gaps, molecular weights, death rates, number of passes in a football game, house prices, countries gross domestic product, mathematical constants , Newspaper runs, etc. – all follow the table above!

Of course I had to see it to believe it. I downloaded the population of the Brazilian municipalities from the IBGE website, calculated the frequency of the first digit and didn’t give another one (see Table 2).

Cool, the dear reader will say, but what is it for? The point is, it is difficult to make data that conforms to Benford Law. Hence, this law can be used to distinguish between real data and false or fraudulent data. It will be the topic of next week.

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