E (mathematical constant)

The constant e is the unique positive real number for which the derivative of the exponential function e^x equals the function itself. ¹ It can also be defined by the limit e = lim_n→∞(1 + 1/n)ⁿ, which describes the idealized result of compounding a unit investment at an annual rate of 100% with increasingly frequent compounding. An equivalent infinite series is e = 1 + 1/1! + 1/2! + 1/3! + ⋯.

The natural logarithm, written ln(x), is the inverse of e^x; consequently, ln(e) = 1 and e^{ln x} = x for positive x. The number is irrational, so its decimal expansion neither terminates nor repeats. It is also transcendental: no nonzero polynomial with integer coefficients has e as a root. ²

The constant e is the natural base for continuous change because exponential growth at a rate proportional to the amount present is modeled by y(t) = Ce^kt. ³ In this expression, C is the initial amount and k is the continuous growth or decay rate. Differentiation leaves e^x unchanged, while integration also returns e^x up to an additive constant, making the function especially convenient in differential equations.

The same structure occurs in compound interest, radioactive decay, cooling models, population dynamics, and the probability distribution known as the exponential distribution. The Poisson distribution uses e^−λ as the probability of observing zero events when the mean event count is λ. Natural logarithms also convert multiplication into addition, which makes e central to likelihoods, information theory, and statistical modeling. ⁴

The constant emerged from the study of logarithms and compound interest rather than from a single original formula. John Napier developed logarithms in the early seventeenth century, and Jacob Bernoulli encountered the limit defining e while analyzing compound interest. Leonhard Euler later standardized the letter e and developed many of the constant’s analytic properties. ⁵

Euler’s formula, e^ix = cos(x) + i sin(x), unifies exponential and trigonometric functions over the complex numbers. At x = π it gives the celebrated identity e^iπ + 1 = 0, joining e, π, the imaginary unit i, 1, and 0. In complex analysis, e^z is periodic with period 2πi, unlike its real counterpart, which is strictly increasing and never periodic. ¹

The constant e has several equivalent definitions that emphasize different areas of mathematics. Besides its limit and series, it is the unique positive base a for which the function a^x has derivative 1 at x = 0; other exponential bases introduce a factor ln(a) when differentiated. This characterization explains why natural logarithms simplify calculus.

Although e is often introduced through finance, its less visible roles include the normal distribution’s density, entropy formulas, generating functions, and the asymptotic behavior of factorials. Stirling’s approximation contains e through n! ≈ √(2πn)(n/e)ⁿ. The digits of e show no known simple repeating pattern, but claims that they are statistically random are not established merely by observing many digits. Efficient computation commonly uses rapidly convergent series, argument reduction, and high-precision arithmetic rather than direct evaluation of the defining limit.

Natural logarithm

The logarithm to base e, written ln(x), and the inverse function of e^x for positive x.

Exponential function

The function e^x, whose derivative and integral are both e^x up to a constant of integration.

Irrational number

A real number that cannot be expressed as a ratio of two integers.

Transcendental number

A number that is not a root of any nonzero polynomial with integer coefficients.

Euler’s formula

The identity e^(ix) = cos(x) + i sin(x), relating complex exponentials to trigonometric functions.