e
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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. |
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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 |
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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!)
An Interesting Use of e: 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
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) |
By substituting x = n/r :
- r/n becomes 1/x and
- n becomes xr
And so:
| (1+r/n)n |
becomes |
(1+(1/x))xr |
Which is just like the formula for e (as n approaches infinity), with an extra r as an exponent.
So, as x goes to infinfity, then (1+(1/x))xr goes to er
And that is why e makes an appearance in interest calculations!
Another 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 20 into 5 pieces and multiply them:
Each "piece" is 20/5=4 in size
4×4×4×4×4 = 45 = 1024
Now, ... let us say we want the answer to be as big as possible, what size should each piece be?
The answer: make the parts "e" (or 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 100, ... what do you get?
Transcendental
e is also a transcendental number
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