measure for measure e is everywhere From determining the compound interest on borrowed money to gauging chances at the roulette wheel in Monte Carlo, Stefanie Reichert explains that there’s no way around Euler’s number. ven outside school or university, Euler’s constant, also called the Euler– we cannot escape Euler’s number. Mascheroni constant, γ ≈ 0.57721, defined EJacob Bernoulli is credited with as the limit of the difference between the discovering e while thinking about matters harmonic series and the natural logarithm. of continuous compound interest in 1683. The Euler–Mascheroni constant appears, He realized that when the compounding for example, in the Bessel function of the period became smaller and smaller and second kind, and has not been proven to more and more periods were considered, be irrational or transcendental. Another the amount of money would converge tricky case are Euler numbers (also known towards a limit that was later found to be as zig or secant numbers), referring to the one of the representations of e. Since then, number of odd alternating permutations in use of Euler’s number has become more expressions for the secant and hyperbolic widespread and now it appears in many secant (https://go.nature.com/2N0G3tc). branches of science and in everyday life. To complicate things further, at least three For example, Euler’s number shows Credit: GL Archive / Alamy Stock Photo other mathematical terminologies are up in probability theory. Imagine you are in use denoting the Euler number of a in Monte Carlo enjoying a few games of finite complex, Euler primes or the Euler roulette, which is a Bernoulli trial process. It took a while before scholars connected characteristics, a topological invariant. If you place a bet on a single number, your the dots and realized that the base of the And in fluid dynamics, the Euler number chances are 1/37 to win that game. For logarithm introduced by Napier and the characterizes the energy loss in a flow. 37 games, the probability that you will limit discovered by Bernoulli were closely We have all encountered Euler’s lose every single time is — maybe related and settled on a common notation. number in more ways than one — from surprisingly — close to 1/e. Or, pretend Gottfried Leibniz referred to what is now natural logarithms to the definition of the you are at the theatre, where you — along known as Euler’s number as b in discussions exponential function, which relies on the with everybody else — leave your coat with Christiaan Huygens, whereas others series expansion of e discovered by Euler in the cloak room, which has one hook such as Jean-Baptiste le Rond d’Alembert himself in 1748. The constant e appears per guest, and receive a number. However, preferred to use the notation c instead. practically everywhere in science: popping your coat is placed on a random hook. This dispute was eventually settled when up in the definition of the standard normal The probability that none of the coats are the Swiss mathematician Leonhard Euler distribution; allowing us to decompose a on the correct hook for a large number of (pictured) used the letter e in an early essay time-dependent signal into its frequencies guests approaches, again, 1/e. The number on the firing of cannons — and his choice via Fourier transformation; telling us how of practical examples is endless. became increasingly popular. to calculate the half-life of radioactive The history of e reads like the Who’s Similar to π, Euler’s number e ≈ 2.71828 elements; playing a crucial role in the growth Who of mathematics and physics. It all is irrational and also transcendental — of bacteria; and governing temperature- started with the discovery of the logarithm meaning it doesn’t form a solution of activated chemical reactions. by John Napier: Euler’s number is hidden a non-zero polynomial equation with Not for nothing, e counts among the deep in the many pages of the appendix integer coefficients. Whether e (or π) most important constants in mathematics tabulating natural logarithms in his 1614 is a normal number remains to be and physics, along with 0, 1, i and π that all work Mirifici Logarithmorum Canonis determined. A normal number consists of show up in Euler’s identity eiπ + 1 = 0. It is Descriptio. Later, when Bernoulli studied a sequence of digits in which single digits truly a constant in everyone’s life. ❐ the case of continuous compound interest, between 0 and 9 occur with a frequency he concluded that the limit must converge of 10%, whereas each pair of digits between Stefanie Reichert to a number between 2 and 3. As it turned 00 and 99 occurs with a frequency of 1%, Associate Editor, Nature Physics. out, this limit equals Euler’s number (less and so on. e-mail: [email protected] commonly known as Napier’s constant), Euler is credited with a whole bunch and Bernoulli came up with its first of constants besides e, so one should be Published online: 3 September 2019 approximation. careful not to mix Euler’s number up with https://doi.org/10.1038/s41567-019-0655-9 982 NATURE PHYSICS | VOL 15 | SEPTEMBER 2019 | 982 | www.nature.com/naturephysics.