The constant e is base of the natural logarithm. e is sometimes known as Napier's constant, although its symbol (e) honors Euler.

e is the unique number with the property that the area of the region bounded by the hyperbola y=1/x, the x-axis, and the vertical lines x=1 and x=e is 1. In other words,


With the possible exception of pi, e is the most important constant in mathematics since it appears in myriad mathematical contexts involving limits and derivatives. The numerical value of e is


(OEIS A001113).


e can be defined by the limit


(illustrated above), or by the infinite series


as first published by Newton (1669; reprinted in Whiteside 1968, p. 225).

e is given by the unusual limit


(Brothers and Knox 1998).

Euler (1737; Sandifer 2006) proved that e is irrational by proving that e has an infinite simple continued fraction (e=[2,1,2,1,1,4,1,1,6,...]; Nagell 1951), and Liouville proved in 1844 that e does not satisfy any quadratic equation with integral coefficients (i.e., if it is algebraic, it must be algebraic of degree greater than 2). Hermite subsequently settled the issue, proving e to be transcendental in 1873. However, e is the "least" transcendental possible, with irrationality measure mu(e)=2.

Sondow (2006) proved that e is irrational using a construction for e as the intersection of a nested sequence of closed intervals. This method also provides a measure of irrationality in terms of the Smarandache function (denoted here as S(n) instead of the conventional mu(n) in order to avoid confusion with the irrationality measure) by showing that if p and q are any integers with q>1, then


It is not known if pi+e or pi/e is irrational. It is known that pi+e and pi/e do not satisfy any polynomial equation of degree <=8 with integer coefficients of average size 10^9 (Bailey 1988, Borwein et al. 1989), but it is not known if either of these is transcendental.

It is not known if e is normal to any base (Stoneham 1970).

e has the series representation


as well as


The special case of the Euler formula


with x=pi gives the beautiful identity


an equation connecting the fundamental numbers i, pi, e, 1, and 0 (zero) and involving the fundamental operations of equality (=), addition (+), multiplication (×), and exponentiation.

A nested series for e can be obtained by rewriting the series (2) for e as


which gives a pretty nested radical result when x is taken to the power of both sides.

An unexpected Wallis-like formula for e is given by the Pippenger product


(OEIS A084148 and A084149; Pippenger 1980). Another product for e given by


due to Guillera (Sondow 2006). This is analogous to the products




(Guillera and Sondow 2005, Sondow 2006).

Using the recurrence relation


with a_1=a^(-1), compute


The result is e^a. Gosper gives the unusual equation connecting pi and e,


(OEIS A100074).

Rabinowitz and Wagon (1995) give an algorithm for computing digits of e based on earlier digits (Borwein and Bailey 2003, p. 140), but a much simpler spigot algorithm was found by Sales in 1968. Around 1966, MIT hacker Eric Jensen wrote a very concise program (requiring less than a page of assembly language) that computed e by converting from factorial base to decimal.

Let p(n) be the probability that a random one-to-one function on the integers 1, ..., n has at least one fixed point. Then


(OEIS A068996).

Stirling's approximation gives


(OEIS A068985).

Steiner's problem asks for the largest value of the function x^(1/x), which is given by e^(1/e).

Examples of e mnemonics (Gardner 1959, 1991) include:

"By omnibus I traveled to Brooklyn" (6 digits).

"To disrupt a playroom is commonly a practice of children" (10 digits).

"It enables a numskull to memorize a quantity of numerals" (10 digits).

"I'm forming a mnemonic to remember a function in analysis" (10 digits).

"He repeats: I shouldn't be tippling, I shouldn't be toppling here!" (11 digits).

"In showing a painting to probably a critical or venomous lady, anger dominates. O take guard, or she raves and shouts" (21 digits). Here, the word "O" stands for the number 0.

A much more extensive mnemonic giving 40 digits is

"We present a mnemonic to memorize a constant so exciting that Euler exclaimed: '!' when first it was found, yes, loudly '!'. My students perhaps will compute e, use power or Taylor series, an easy summation formula, obvious, clear, elegant!"

(Barel 1995). In the latter, 0s are represented with "!". A list of e mnemonics in several languages is maintained by A. P. Hatzipolakis.

See also

e Approximations, e Continued Fraction, e Digits, Carleman's Inequality, Compound Interest, de Moivre's Identity, Euler Formula, Exponential Decay, Exponential Function, Exponential Growth, Hermite-Lindemann Theorem, Natural Logarithm, Pickover's Sequence, Steiner's Problem Explore this topic in the MathWorld classroom

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Portions of this entry contributed by Jonathan Sondow (author's link)

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Sondow, Jonathan and Weisstein, Eric W. "e." From MathWorld--A Wolfram Web Resource.

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