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

(1) |

(2) |

(3) |

Euler proved that is Irrational, and Liouville proved in 1844 that
does not satisfy any Quadratic Equation with integral Coefficients. Hermite proved
to be Transcendental in 1873. It is not known if or is
Irrational. However, 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).

The special case of the Euler Formula

(4) |

(5) |

Some Continued Fraction representations of include

(6) | |||

(7) |

(Sloane's A003417) and

(8) | |||

(9) | |||

(10) | |||

(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

(12) |

(13) |

(14) |

Rabinowitz and Wagon (1995) give an Algorithm for computing digits of 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 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

(15) |

(16) |

Castellanos (1988) gives several curious approximations to ,

(17) | |||

(18) | |||

(19) | |||

(20) | |||

(21) | |||

(22) |

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

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!''

Scanning the decimal expansion of until all -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, ....

**References**

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

Barel, Z. ``A Mnemonic for .'' *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 .'' *Osiris* **1**, 476-496, 1936.

Olds, C. D. ``The Simple Continued Fraction Expression of .'' *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 .'' *Amer. Math. Monthly* **102**, 195-203, 1995.

Sales, A. H. J. ``The Calculation of 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.

© 1996-9

1999-05-25