e (Euler's Number)
The number e 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 an irrational number (it cannot be written as a simple fraction).
e is the base of the Natural Logarithms (invented by John Napier).
e is found in many interesting areas, so 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 and its digits go on forever without repeating.
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} |
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 |
Another Calculation
The value of e is also equal to \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + \frac{1}{6!} + \frac{1}{7!} + ... (etc)
(Note: "!" means factorial)
The first few terms add up to: 1 + 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \frac{1}{120} = 2.71666...
In fact Euler himself used this method to calculate e to 18 decimal places.
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!):
- To
- express
- e
- remember
- to
- memorize
- a
- sentence
- to
- memorize
- 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 are the digits of the angles 45°, 90°, 45° in a Right-Angled Isosceles Triangle (no real reason, just how it is):
2.7 1828 1828 45 90 45
(An instant way to seem really smart!)
Growth
e is used in the "Natural" Exponential Function:
Graph of f(x) = e^{x}
It has this wonderful property: "its slope is its value"
At any point the slope of e^{x} equals the value of e^{x} :
when x=0, the value e^{x} = 1, and the slope = 1
when x=1, the value e^{x} = e, and the slope = e
etc...
This is true anywhere for e^{x}, and makes some things in Calculus (where we need to find slopes) a whole lot easier.
Area
The area up to any x-value is also equal to e^{x} :
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
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. Such as in finance.
Imagine a wonderful bank that pays 100% interest.
In one year you could turn $1000 into $2000.
Now imagine the bank pays twice a year, that is 50% and 50%
Half-way through the year you have $1500,
you reinvest for the rest of the year and your $1500 grows to $2250
You got more money, because you reinvested half way through.
That is called compound interest.
Could we get even more if we broke the year up into months?
We can use this formula:
(1+r/n)^{n}
r = annual interest rate (as a decimal, so 1 not 100%)
n = number of periods within the year
Our half yearly example is:
(1+1/2)^{2} = 2.25
Let's try it monthly:
(1+1/12)^{12} = 2.613...
Let's try it 10,000 times a year:
(1+1/10,000)^{10,000} = 2.718...
Yes, it is heading towards e (and is how Jacob Bernoulli first discovered it).
Why does that happen?
The answer lies in the similarity between:
Compounding Formula: | (1 + r/n)^{n} | |
and | ||
e (as n approaches infinity): | (1 + 1/n)^{n} |
The Compounding Formula is very like the formula for e (as n approaches infinity), just with an extra r (the interest rate).
When we chose an interest rate of 100% (= 1 as a decimal), the formulas became the same.
Read Continuous Compounding for more.
Euler's Formula for Complex Numbers
e also appears in this most amazing equation:
e^{iπ} + 1 = 0
Transcendental
e is also a transcendental number.