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e


The number e is a famous irrational number, and is one of the most important numbers in mathematics.

The first few digits are:

2.7182818284590452353602874713527 (and more ...)

It is often called Euler's number after Leonhard Euler

e is the base of the Natural Logarithms (invented by John Napier). On the other hand Common Logarithms have 10 as their base.

 

Calculating

The value of (1 + 1/n)n approaches e as n gets bigger and bigger:

n (1 + 1/n)n
1 2.00000
2 2.25000
5 2.48832
10 2.59374
100 2.70481
1,000 2.71692
10,000 2.71815
100,000 2.71827
  graph of (1+1/n)^n

 

The value of e is also equal to 1 + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + 1/7! + ... (etc)

(Note: "!" means factorial)

The first few terms add up to: 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 = 2.718055556

Remembering

To remember the value of e (to 10 places) just remember this saying (count the letters!):

  • To
  • express
  • e
  • remember
  • to
  • memorize
  • a
  • sentence
  • to
  • simplify
  • this

Or you can remember the curious pattern that after the "2.7" the number "1828" appears TWICE:

2.7 1828 1828

And following THAT is the angles in a Right-Angled Isosceles (two equal angles) Triangle of 45°, 90°, 45°:

2.7 1828 1828 45 90 45

(An instant way to seem really smart!)

Where

Often the number e makes an appeareance where it is not expected.

For example, it gives the value for Continuous Compounding (used with loans and investments):

e^r-1

Formula for Continuous Compounding

Another Interesting Property

Cut Up Then Multiply

Let us say that we cut a number into equal parts and then multiply those parts together.

How large should each part be, so that when we multiply them together they make the biggest possible number?

The answer: make the parts "e", ... well, as close to e as possible.

Example: 10

10 cut into 3 parts is 3.3... 3.3...×3.3...×3.3... (3.3...)3 = 37.037...
10 cut into 4 equal parts is 2.5 2.5×2.5×2.5×2.5 = 2.54 = 39.0625
10 cut into 5 equal parts is 2 2×2×2×2×2 = 25 = 32

The winner is the number closest to "e", in this case 2.5.

Try it with another number yourself, say 50, ... what do you get?

Transcendental

e is also a transcendental number