e (Euler's Number)

e (eulers number)

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 (pronounced "Oiler").

e is the base of the Natural Logarithms (invented by John Napier).

e is found in many interesting areas, so it is worth learning about.

Calculating

There are many ways of calculating the value of e, but none of them ever give a totally exact answer, because e is irrational (not the ratio of two integers).

But it is known to over 1 trillion digits of accuracy!

For example, the value of (1 + 1/n)n approaches e as n gets bigger and bigger:

graph of (1+1/n)^n

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

 

Try it! Put "(1 + 1/100000)^100000" into the calculator:

(1 + 1/100000)100000

What do you get?

Another Calculation

The value of e is also equal to 10! + 11! + 12! + 13! + 14! + 15! + 16! + 17! + ... (etc)

(Note: "!" means factorial)

The first few terms add up to: 1 + 1 + 12 + 16 + 124 + 1120 = 2.718055556

You can try it yourself at the Sigma Calculator.

Remembering

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

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

2.7 1828 1828

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

2.7 1828 1828 45 90 45

(An instant way to seem really smart!)

An Interesting Property

Just for fun, try "Cut Up Then Multiply"

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

Example: Cut 10 into 2 pieces and multiply them:

Each "piece" is 10/2 = 5 in size

5×5 = 25

Now, ... how could we get the answer to be as big as possible, what size should each piece be?

The answer: make the parts as close as possible to "e" in size.

Example: 10

10 cut into 2 equal parts is 5:5×5 = 25 = 25
10 cut into 3 equal parts is 313:(313)×(313)×(313) = (313)3 = 37.0...
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 100, ... what do you get?

100 Decimal Digits

Here is e to 100 decimal digits:

2.71828182845904523536028747135266249775724709369995957
49669676277240766303535475945713821785251664274...

Advanced: Use of e in Compound Interest

Often the number e appears in unexpected places.

For example, e is used in Continuous Compounding (for loans and investments):

e^r-1

Formula for Continuous Compounding

Why does that happen?

Well, the formula for Periodic Compounding is:

FV = PV (1+r/n)n

where FV = Future Value
PV = Present Value
r = annual interest rate (as a decimal)
n = number of periods within the year

But what happens when the number of periods heads to infinity?

The answer lies in the similarity between:

(1+r/n)n and (1 + 1/n)n
Compounding Formula   e (as n approaches infinity)

The Compounding Formula is very like the formula for e (as n approaches infinity), just with an extra r (the interest rate). When the number of periods n approaches infinity we can substitiute e into the fomula.

And that is why e makes an appearance in interest calculations!

Transcendental

e is also a transcendental number.