e

The base of the Natural Logarithm, named in honor of Euler. It appears in many mathematical contexts involving Limits and Derivatives, and can be defined by

$$e\equiv \lim_{x\to \infty} \left({1+{1\over x}}\right)^x,$$ (1)

or by the infinite sum

$$e=\sum_{k=0}^\infty {1\over k!}.$$ (2)

The numerical value of $e$ is

$$e = 2.718281828459045235360287471352662497757\ldots$$ (3)

(Sloane's A001113).

Euler proved that $e$ is Irrational, and Liouville proved in 1844 that $e$ does not satisfy any Quadratic Equation with integral Coefficients. Hermite proved $e$ to be Transcendental in 1873. It is not known if $\pi+e$ or $\pi/e$ is Irrational. However, it is known that $\pi+e$ and $\pi/e$ do not satisfy any Polynomial equation of degree $\leq 8$ with Integer Coefficients of average size $10^9$ (Bailey 1988, Borwein et al. 1989).

The special case of the Euler Formula

$$e^{ix}=\cos x+i\sin x$$ (4)

with $x=\pi$ gives the beautiful identity

$$e^{i\pi}+1=0,$$ (5)

an equation connecting the fundamental numbers i, Pi, $e$, 1, and 0 (Zero).

Some Continued Fraction representations of $e$ include

$$e = {2+{1\over {\strut\displaystyle 1+{\strut\displaystyle 1\over\str... ...ut\displaystyle 3+{\strut\displaystyle 3 \over \strut\displaystyle \ddots}}}}}}$$ (6)

$$= [2, 1, 2, 1, 1, 4, 1, 1, 6, \ldots]$$ (7)

(Sloane's A003417) and

$${e-1\over e+1} = [2, 6, 10, 14, \ldots]$$ (8)

$$e-1 = [1, 1, 2, 1, 1, 4, 1, 1, 6, \ldots]$$ (9)

$${\textstyle{1\over 2}}(e-1) = [0, 1, 6, 10, 14, \ldots]$$ (10)

$$\sqrt{e} = [1, 1, 1, 1, 5, 1, 1, 9, 1, 1, 13, \ldots].$$ (11)

The first few convergents of the Continued Fraction are 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, ... (Sloane's A007676 and A007677).

Using the Recurrence Relation

$$a_n=n(a_{n-1}+1)$$ (12)

with $a_1=a^{-1}$, compute

$$\prod_{n=1}^\infty (1+{a_n}^{-1}).$$ (13)

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

$$\sum_{n=1}^\infty {1\over n^2}\cos\left({9\over n\pi+\sqrt{n^2\pi^2-9}}\right)= -{\pi^2\over 12e^3} = -0.040948222\ldots.$$ (14)

Rabinowitz and Wagon (1995) give an Algorithm for computing digits of $e$ based on earlier Digits, 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 $e$ 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

$$\lim_{n\to\infty} p(n)=\sum_{k=1}^\infty {(-1)^{k+1}\over k!} =1-{1\over e}=0.6321205588\ldots.$$ (15)

Stirling's Formula gives

$$\lim_{n\to\infty} {(n!)^{1/n}\over n}={1\over e}.$$ (16)

Castellanos (1988) gives several curious approximations to $e$,

$$e \approx 2+{54^2+41^2\over 80^2}$$ (17)

$$\approx (\pi^4+\pi^5)^{1/6}$$ (18)

$$\approx {271801\over 99990}$$ (19)

$$\approx \left({150-{87^3+12^5\over 83^3}}\right)^{1/5}$$ (20)

$$\approx 4-{300^4-100^4-1291^2+9^2\over 91^5}$$ (21)

$$\approx \left({1097-{55^5+311^3-11^3\over 68^5}}\right)^{1/7},$$ (22)

which are good to 6, 7, 9, 10, 12, and 15 digits respectively.

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 A032511). These end at positions 21, 372, 8092, 102128, 1061613, 12108841, ....

See also Carleman's Inequality, Compound Interest, de Moivre's Identity, Euler Formula, Exponential Function, Hermite-Lindemann Theorem, Natural Logarithm

References

e

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.

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.

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. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/e/e.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. Ch. 3 in The Unexpected Hanging and Other Mathematical Diversions. Chicago, IL: Chicago University Press, p. 40, 1991.

Hatzipolakis, A. P. ``PiPhilology.'' http://users.hol.gr/~xpolakis/piphil.html.

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

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.

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

Plouffe, S. ``Plouffe's Inverter: Table of Current Records for the Computation of Constants.'' http://www.lacim.uqam.ca/pi/records.html.

Rabinowitz, S. and Wagon, S. ``A Spigot Algorithm for the Digits of $\pi$.'' Amer. Math. Monthly 102, 195-203, 1995.

Sales, A. H. J. ``The Calculation of $e$ to Many Significant Digits.'' Computer J. 11, 229-230, 1968.

Sloane, N. J. A. Sequences A032511, A001113/M1727, A003417/M0088, A007676/M0869, A007677/M2343 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.