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12,720
entries
Tue Oct 23
2007

Number
Theory > Constants > e
Recreational
Mathematics > Mathematical
Records
History
and Terminology > Mnemonics
MathWorld
Contributors > Abbott,
Paul
MathWorld
Contributors > Sondow

e

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,

(1)

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

(2)

(Sloane's A001113).

can be defined by the limit

(3)

(illustrated above), or by the infinite series

(4)

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

is given by the unusual limit

(5)

(Brothers and Knox 1998).

Euler 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

(6)

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

(7)

as well as

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(Brothers 2004).

The special case of the Euler
formula

(15)

with gives the beautiful identity

(16)

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

The simple
continued fraction representations of is

(17)

(Sloane's A003417),
giving the first few convergents as 2, 3, 8/3, 11/4, 19/7, 87/32,
106/39, 193/71, ... (Sloane's A007676
and A007677),
which are good to 0, 0, 1, 1, 2, 3, 3, 4, 5, 5, ... (Sloane's A114539)
decimal digits, respectively.

A plot of the first 256 terms of the continued
fraction represented as a sequence of binary bits is shown above.

A beautiful non-simple continued
fraction for is given by

(18)

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

(19)

(20)

(21)

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

Other continued fraction representations are

(22)

(23)

(24)

(Olds 1963, pp. 135-136). Amazingly, not only the
continued fractions , but those of rational powers of show regularity, for example

(25)

(26)

(27)

(28)

Let the continued fraction of be denoted and let the denominators of the convergents be
denoted , , ..., . Then plots above show successive values of , , ..., (left figure) and (right figure). As can be seen from the plots,
the regularity in the continued fraction of means that is one of a set of numbers of measure 0 whose
continued fraction sequences do not converge to Khinchin's
constant or the Khinchin-Lévy
constant.

has a very regular Engel
expansion, namely 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
... (Sloane's A000027).

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

(29)

(Sloane's A084148
and A084149;
Pippenger 1980). Another product for given by

(30)

due to Guillera (Sondow 2005). This is analogous to
the products

(31)

and

(32)

(Guillera and Sondow 2005, Sondow 2005).

Using the recurrence
relation

(33)

with , compute

(34)

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

(35)

(36)

(Sloane's 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 (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

(37)

(38)

(39)

(Sloane's A068996).

Stirling's
formula gives

(40)

(41)

(Sloane's 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.

Scanning the decimal expansion of until all -digit numbers have occurred, the last appearing is
6, 12, 548, 1769, 92994, 513311, ... (Sloane's A036900).
These end at positions 21, 372, 8092, 102128, 1061613, 12108841, ...
(Sloane's A036904).

was computed to digits by P. Demichel, and the first have been verified by X. Gourdon on Nov. 21, 1999
(Plouffe).

SEE ALSO: e
Approximations, 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. [Pages Linking Here]

RELATED WOLFRAM
SITES: http://functions.wolfram.com/Constants/E/

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

REFERENCES:

Bailey, D. H. "Numerical Results on the
Transcendence of Constants Involving , e, and Euler's Constant." Math.
Comput. 50, 275-281, 1988.

Barel, Z. "A Mnemonic for e."
Math. Mag. 68, 253, 1995.

Baruvelle, H. V. "The Number e: The
Base of Natural Logarithms." Math. Teacher 38,
350-355, 1945.

Borwein, J. and Bailey, D. Mathematics
by Experiment: Plausible Reasoning in the 21st Century.
Wellesley, MA: A K Peters, 2003.

Borwein, J. M.; Borwein, P. B.; and Bailey, D. H.
"Ramanujan, Modular Equations, and Approximations to Pi or How to
Compute One Billion Digits of Pi." Amer. Math. Monthly
96, 201-219, 1989.

Brothers, H. J. and Knox, J. A. "New Closed-Form
Approximations to the Logarithmic Constant e."
Math. Intell. 20, 25-29, 1998.

Brothers, H. J. "Improving the Convergence of
Newton's Series Approximation for ." College Math. J. 35, 34-39, 2004.

Caldwell, C. K. and Dubner, H. "Primes in Pi."
J. Recr. Math. 29, 282-289, 1998.

Castellanos, D. "The Ubiquitous Pi. Part I."
Math. Mag. 61, 67-98, 1988.

Conway, J. H. and Guy, R. K. The
Book of Numbers. New York: Springer-Verlag, pp. 201 and
250-254, 1996.

Finch, S. R. "The Natural Logarithm Base." §1.3
in Mathematical
Constants. Cambridge, England: Cambridge University Press,
pp. 12-17, 2003.

Friedman, E. "Problem of the Month (August
2004)." http://www.stetson.edu/~efriedma/mathmagic/0804.html.

Gardner, M. "Memorizing Numbers." Ch. 11 in The
Scientific American Book of Mathematical Puzzles and
Diversions. New York: Simon and Schuster, pp. 103 and 109,
1959.

Gardner, M. "The Transcendental Number
e." Ch. 3 in The
Unexpected Hanging and Other Mathematical Diversions.
Chicago, IL: Chicago University Press, pp. 34-42, 1991.

Gourdon, X. and Sebah, P. "The Constant
e." http://numbers.computation.free.fr/Constants/E/e.html.

Hatzipolakis, A. P. "PiPhilology." http://www.cilea.it/~bottoni/www-cilea/F90/piphil.htm.

Hermite, C. "Sur la fonction exponentielle."
C. R. Acad. Sci. Paris 77, 18-24, 74-79, and 226-233,
1873.

Knox, J. A. and Brothers, H. J. "Novel
Series-Based Approximations to e." College Math.
J. 30, 209-215, 1999.

Le Lionnais, F. Les
nombres remarquables. Paris: Hermann, p. 47, 1983.

Maor, E. e:
The Story of a Number. Princeton, NJ: Princeton University
Press, 1994.

Minkus, J. "A Continued Fraction." Problem 10327.
Amer. Math. Monthly 103, 605-606, 1996.

Mitchell, U. G. and Strain, M. "The Number
e." Osiris 1, 476-496, 1936.

Nagell, T. "Irrationality of the numbers
e and ." §13 in Introduction
to Number Theory. New York: Wiley, pp. 38-40, 1951.

Newton, I. The
Mathematical Papers of Isaac Newton, Vol. 2.: 1667-1670 (Ed.
D. T. Whiteside). New York: Cambridge University Press, 1968.

Olds, C. D. Continued
Fractions. New York: Random House, 1963.

Olds, C. D. "The Simple Continued Fraction
Expression of e." Amer. Math. Monthly
77, 968-974, 1970.

Pippenger, N. "An Infinite Product for ." Amer. Math. Monthly 87, 391, 1980.

Plouffe, S. "Table of Current Records for the
Computation of Constants." http://pi.lacim.uqam.ca/eng/records_en.html.

Rabinowitz, S. and Wagon, S. "A Spigot Algorithm
for the Digits of ." Amer. Math. Monthly 102, 195-203,
1995.

Reid, C. In From
Zero to Infinity, 4th ed. Washington, DC: Math. Assoc.
Amer., 1992.

Sloane, N. J. A. Sequences A000029/M0563,
A001113/M1727,
A003417/M0088,
A007676/M0869,
A007677/M2343,
A036900,
A036904,
A068985,
A068996,
A084148,
A084149,
A100074,
and A114539
in "The On-Line Encyclopedia of Integer Sequences."

Sondow, J. "A Geometric Proof that Is Irrational and a New Measure of Its
Irrationality." Amer. Math. Monthly 113, 637-641,
2006.

Stoneham, R. "A General Arithmetic Construction
of Transcendental Non-Liouville Normal Numbers from Rational
Functions." Acta Arith. 16, 239-253, 1970.

Weisstein, E. W. "Books about e." http://www.ericweisstein.com/encyclopedias/books/e.html.

Wells, D. The
Penguin Dictionary of Curious and Interesting Numbers.
Middlesex, England: Penguin Books, p. 46, 1986.

LAST MODIFIED: July 9,
2007

CITE THIS
AS:

Sondow,
Jonathan and Weisstein,
Eric W. "e." From MathWorld--A Wolfram
Web Resource. http://mathworld.wolfram.com/e.html

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