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

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

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

(OEIS A001113).

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).

is given by the unusual limit

(Brothers and Knox 1998).

Euler (1737; Sandifer 2006) proved that is irrational by proving that has an infinite simple continued fraction (; Nagell 1951), and Liouville proved in 1844 that 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 to be transcendental in 1873. However, is the “least” transcendental possible, with irrationality measure .

Sondow (2006) proved that is irrational using a construction for 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 instead of the conventional in order to avoid confusion with the irrationality measure) by showing that if and are any integers with , then

It is not known if or is irrational. It is known that and do not satisfy any polynomial equation of degree with integer coefficients of average size (Bailey 1988, Borwein et al. 1989), but it is not known if either of these is transcendental.

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

has the series representation

as well as

The special case of the Euler formula

with gives the beautiful identity

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

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

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

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

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

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

and

(Guillera and Sondow 2005, Sondow 2006).

Using the recurrence relation

with , compute

The result is . Gosper gives the unusual equation connecting and ,

(OEIS A100074).

Rabinowitz and Wagon (1995) give an algorithm for computing digits of 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 by converting from factorial base to decimal.

Let be the probability that a random one-to-one function on the integers 1, …, 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 , which is given by .

Examples of 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 , use power or Taylor series, an easy summation formula, obvious, clear, elegant!”

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