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Last updated: Thu Dec 31 2009

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e
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EAreaPlot

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,

int_1^e(dx)/x=lne=1.
(1)

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

e=2.718281828459045235360287471352662497757...
(2)

(Sloane's A001113).

ELimit

e can be defined by the limit

e=lim_(x->infty)(1+1/x)^x
(3)

(illustrated above), or by the infinite series

e=sum_(k=0)^infty1/(k!)
(4)

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

e is given by the unusual limit

lim_(n->infty)[((n+1)^(n+1))/(n^n)-(n^n)/((n-1)^(n-1))]=e
(5)

(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

|e-p/q|>1/((S(q)+1)!).
(6)

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

e=[sum_(k=0)^infty((-1)^k)/(k!)]^(-1),
(7)

as well as

e=[sum_(k=0)^(infty)(1-2k)/((2k)!)]^(-1)
(8)
=sum_(k=0)^(infty)(2k+1)/((2k)!)
(9)
=1/2sum_(k=0)^(infty)(k+1)/(k!)
(10)
=2sum_(k=0)^(infty)(k+1)/((2k+1)!)
(11)
=sum_(k=0)^(infty)(3-4k^2)/((2k+1)!)
(12)
=sum_(k=0)^(infty)((3k)^2+1)/((3k)!)
(13)
=[sum_(k=0)^(infty)(4k+3)/(2^(2k+1)(2k+1)!)]^2.
(14)

The special case of the Euler formula

e^(ix)=cosx+isinx
(15)

with x=pi gives the beautiful identity

e^(ipi)+1=0,
(16)

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

The simple continued fraction representations of e is

e=[2;1,2,1,1,4,1,1,6,...].
(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.

EContinuedFractionBinaryPlot

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 e is given by

e=2+1/(1+1/(2+2/(3+3/...)))
(18)

(Wall 1948, p. 348).

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

e=1+1+1/(2!)+1/(3!)+1/(4!)+...
(19)
=1+1+1/2(1+1/3+1/(4·3)+...)
(20)
=1+1+1/2(1+1/3(1+1/4(1+1/5(1+...)))),
(21)

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

Other continued fraction representations are

(e-1)/(e+1)=[0;2,6,10,14,...]
(22)
e-1=[1;1,2,1,1,4,1,1,6,...]
(23)
1/2(e-1)=[0;1,6,10,14,...]
(24)

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

e^(1/2)=[1,1,1,1,5,1,1,9,1,1,13,...]
(25)
e^(1/3)=[1,2,1,1,8,1,1,14,1,1,20,...]
(26)
e^(1/4)=[1,3,1,1,11,1,1,19,1,1,27,...]
(27)
e^(1/5)=[1,4,1,1,14,1,1,24,1,1,34,...].
(28)
EKhinchinLevy

Let the continued fraction of e be denoted [a_0,a_1,a_2,...] and let the denominators of the convergents be denoted q_1, q_2, ..., q_n. Then plots above show successive values of a_1^(1/1), (a_1a_2)^(1/2), ..., (a_1a_2...a_n)^(1/n) (left figure) and q_n^(1/n) (right figure). As can be seen from the plots, the regularity in the continued fraction of e means that e 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.

e 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 e is given by the Pippenger product

e/2=(2/1)^(1/2)(2/34/3)^(1/4)(4/56/56/78/7)^(1/8)...
(29)

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

e=(2/1)^(1/1)((2^2)/(1·3))^(1/2)((2^3·4)/(1·3^3))^(1/3)((2^4·4^4)/(1·3^6·5))^(1/4)...
(30)

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

e^gamma=(2/1)^(1/2)((2^2)/(1·3))^(1/3)((2^3·4)/(1·3^3))^(1/4)((2^4·4^4)/(1·3^6·5))^(1/5)...,
(31)

and

pi/2=(2/1)^(1/2)((2^2)/(1·3))^(1/4)((2^3·4)/(1·3^3))^(1/8)((2^4·4^4)/(1·3^6·5))^(1/16)...
(32)

(Guillera and Sondow 2005, Sondow 2006).

Using the recurrence relation

a_n=n(a_(n-1)+1)
(33)

with a_1=a^(-1), compute

product_(n=1)^infty(1+a_n^(-1)).
(34)

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

sum_(n=1)^(infty)1/(n^2)cos(9/(npi+sqrt(n^2pi^2-9)))=-(pi^2)/(12e^3)
(35)
=-0.040948222...
(36)

(Sloane's 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

lim_(n->infty)p(n)=sum_(k=1)^(infty)((-1)^(k+1))/(k!)
(37)
=1-1/e
(38)
=0.6321205588...
(39)

(Sloane's A068996).

Stirling's formula gives

lim_(n->infty)((n!)^(1/n))/n=1/e
(40)
=0.367879441...
(41)

(Sloane's 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.

Scanning the decimal expansion of e until all n-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).

e was computed to 1.7×10^9 digits by P. Demichel, and the first 1.25×10^9 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

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

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

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.

Euler, L. "De fractionibus continuis dissertation." Comm. Acad. Sci. Petr. 9, 98-137, (1737) 1744. Reprinted in Leonhardi Euleri Opera Omnia, Ser. I, Vol. 14. Leipzig, Germany: Teubner, pp. 187-215, 1924.

Euler, L. "An Essay on Continued Fractions." Trans. M. F. Wyman and B. F. Wyman. Math. Systems Th. 18, 295-328, 1985.

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.

Guillera, J. and Sondow, J. "Double Integrals and Infinite Products for Some Classical Constants Via Analytic Continuations of Lerch's Transcendent." 16 June 2005. http://arxiv.org/abs/math.NT/0506319.

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 pi." §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 e." 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 pi." Amer. Math. Monthly 102, 195-203, 1995.

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

Sandifer, E. "How Euler Did It: Who proved e is Irrational?" Feb. 2006. http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2028%20e%20is%20irrational.pdf.

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 e 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.

Wall, H. S. Analytic Theory of Continued Fractions. New York: Chelsea, 1948.

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.




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