# Benford’s law

In 1881, the American mathematician *Simon Newcomb* noticed that in the logarithm tables used by his students, the pages with logarithms beginning with the digit “1” had more *“dog-ears”* than the following pages on which the logarithms with *“2”, “3”, “4”*, etc. came first. However, his mathematical description of this phenomenon in the *American Journal of Mathematics* was quickly forgotten.

It was not until 1937 that this regularity was rediscovered by the physicist *Frank Benford* (1883–1948), studied on the basis of over 20,000 data and systematically analyzed. The law subsequently named after him states — formulated mathematically — that when a number is *randomly* selected from a table of physical constants or statistical data, the *probability* that the first digit is a “1” is about 0.301. Thus, it is far greater than 0.1, the value one might expect if all digits were equally likely to occur. In general, Benford’s law says about the probability that the first digit is equal to ““:

Here denotes the decadic logarithm and can take the values 1, 2, …, 9. In detail thus results:

Consequently, *Benford’s Law* implies that a number with a *“smaller”* first digit is *more likely* to occur in a table of statistical data than a number with a *“larger”* first digit. Benford’s law is used in detecting fraud in the preparation of financial statements, falsification in settlements, generally for the rapid detection of glaring irregularities in accounting. In 2001, Benford’s Law was used to uncover the remarkably *“creative”* accounting system at ENRON, the largest American energy company at the time (with 22,000 employees), through which management had defrauded investors of their deposits in the amount of approximately xx\langle x\rangle\coloneqq x-\lfloor x\rfloor\langle 3.1415\rangle=3.1415-\lfloor 3.1415\rfloor=3.1415-3=0.1415

\mathbb R_+E_i

PXE_i$, we get:

#### Literature

[1] Benford, F.: *The Law of Anomalous Numbers*, Proceedings of the American Philosophical Society 78, S. 551–572, 1938.

[2] Glück, M.:* Die Benford-Verteilung — Anwendung auf reale Daten der Marktforschung*, Diplomarbeit TU Dresden, Betreuer: V. Nollau und H.-O. Müller, 2007.

[3] Newcomb, S.:* Note on the Frequency of the Use of different Digits in Natural Numbers*, American Journal of Mathematics 4, S. 39–40, 1881.