E, base of the natural logarithm

, the base of the natural logarithm (e.g. ) can be expressed as the following equation:

The Euler’s formula involves , which is:

The Euler’s Identity is the special case when ,

The constant coincides in many interesting problems. For example, the average number to add random numbers () in order to obtain a sum larger than one can be obtained by the following Python code by brute-force over a few iterations.

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

#!/usr/bin/env python
from random import random, seed
from math import e
 
maxn = 100
seed()
s = 0
for i in range(maxn):
    ss = 0
    j = 0
    while ss < 1:
        j += 1
        ss += random()
    s += j
 
ee = s * 1.0 / maxn
print "%.6f" % ee
print e, abs(e - ee)

#!/usr/bin/env python
from random import random, seed
from math import e

maxn = 100
seed()
s = 0
for i in range(maxn):
    ss = 0
    j = 0
    while ss < 1:
        j += 1
        ss += random()
    s += j

ee = s * 1.0 / maxn
print "%.6f" % ee
print e, abs(e - ee)

The 100 iterations give us an approximate value of , which is 2.75 (with error = 0.0317).

2.750000
2.71828182846 0.031718171541

If we increase the iterations to 1000000,

Surprisingly, the output is the following.

2.718216
2.71828182846 6.58284590451e-05

We can see that if we keep increasing the iterations, the absolute error to will be improved. And if maxn reaches , the will be equal to . Anybody can explain this in mathematics?

See also: Simple and Efficient C Program to Compute the Mathematic Constant E (Euler’s number)

–EOF (The Ultimate Computing & Technology Blog) —