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Let a be a real positive number, the exponential function ax
(base a) is a differentiable function, that is the following limit exists
ax = Cax.
A very important case is given when the derivative of the exponential
function is equal to itself, which implies
C = 1 =
and can be written as
( 1+h) 1/h.
This limit exists and it was denoted by the letter e by Euler, first in a
letter to Goldbach in 1731, later in 1736 in his work. The origin of this
choice is maybe due to the first letter of the word exponential.
Hence the constant e is defined by the monotone increasing sequence
but the convergence is very slow
A quicker convergence is easily obtained by (see ,
It's interesting to note that the constant e holds a key position in the
simplest first order differential equation
yў = y,
while p occurs in a similar second order differential equation.
Links between the previous definition and the law of compound interest in
financial calculations are given in .
A famous sequence
Using the Newton's binomial theorem
(1+x)n = 1+nx+
for x = 1/n and after some manipulations gives the famous relation
k = 0
given by Euler in 1748 . This relation is very efficient to
compute e because the factorial of a number grows very quickly and it's
easy to show that
sn < e < sn+
Euler used this relation to find 23 digits of e.
The constant e is also known as the natural base of logarithm,
and is equal to the value of the exponential function at 1 :
e = exp(1).
The number e is irrational (Euler 1744) and transcendental
(Hermite 1873, ).
Let pk/qk be the convergent of order k of a simple continued
fraction, and let x = [a0;a1,a2,...,ak], we have the following
with initial conditions
1,p0 = a0
0,q0 = 1.
If we use the continued fraction for (e-1)/2, then ak = 4(k-1)+2 for k > 1 and it's easy to build an algorithm to compute a fast converging
sequence to e. With k = 1500, e is given to 104 decimal places,
with k = 12000, e is given to 105digits.
We are looking for a rational approximation of this sequence with numerator
of degree m and denominator of degree n such as:
ж з з
з з и
k = 0
k = 0
ц ч ч
ч ч ш
= O(tm+n+1), t® 0
When such an approximation exists, it's called a Padé approximant of degree (m,n) of the series s. There are many properties associated
with those approximants, they are used to compute series with bad
convergence properties. Here we will just give the Padé approximant of
the et function:
k = 0
k = 0
For example here are some approximant of increasing degree for et:
From this we may deduce that for small values of t, e = (et)1/twill
be well approximated by
The error being respectively O(t2),O(t4),O(t6),... The famous
formula (1+t)1/t converges with an error O(t). As an example of the
efficiency of the other formulas, the last formula for different values of t gives
This program has 117 characters (try to do better !). It can be changed to
compute more digits (change the value 9009 to more) and to be faster (change
the constant 10 to another power of 10 and the printf
command). A not so obvious question is to find the algorithm used.
John von Neumann and his team 
used the ENIAC. The result confirmed a previous computation to 808 digits
published in 1946.
D. Shanks and J.W. Wrench, Jr.
Euler's formula was used
on a IBM 7090. The computation took 2.5 hours . (See Math.
Comp. 23, 679-680 (1969))
R. Nemiroff and J. Bonnell
The computation took 182.5 hours
on a Pentium at 133 Mhz
The computation took 714 hours
of a HP 9000/778 (160Mhz).
The method was based on binary
The method was based on a
continued fraction expansion of e.
The formula used was the
exponential series e = е1/n!, the verification was made with e = 1/(е(-1)n/n!). A binary splitting technique was used. The computation took
40 hours on a PentiumII 350 with 320 Mo. 4 Go of disk space was used during
the process. (Note : a previous 1.7 billion digits computation by Patrick
Demichel was made before, but no verification was performed. It appears that
its first 1.25 billion digits are OK).
2000, Jul 10
S. Kondo and X. Gourdon
The computation was launched by Shigeru Kondo with the program
written by X. Gourdon. The same technique as in the previous record was used.
The computation took 21 hours on a PentiumIII 933 with 512 Mo. 8 Go of
disk space were used during the process.
2000, Jul 16
C. Martin and X. Gourdon
Colin Martin used the same
on an Athlon 650 with just 128 Mo. of physical memory and 17.2 GB of
The initial calculation took just over
77 hours and completed on July 13th
and the verification took 80 hours.
2000, Aug 02
S. Kondo and X. Gourdon
Shigeru Kondo used
The computation took 70 hours on a PentiumIII 800 with 1024 Mo. 24 Go of
disk space were necessary. The verification took 71 hours on the