e (Euler's 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.
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:
|n||(1 + 1/n)n|
Try it! Put "(1 + 1/100000)^100000" into the calculator:
(1 + 1/100000)100000
What do you get?
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.
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.
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:
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):
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!
e is also a transcendental number.