math fun...

Euler's identity:

e^iπ+1=0

Perhaps the coolest equation in mathematics.
It includes exactly once each of three basic mathematical operations:
- addition, multiplication, and exponentiation.
... and by way of these three basic operations links 5 fundamental mathematical constants: 0, 1, i, e, & π.
Finally, this linkage is represented by the fundamental equivalence relationship, "=".

For you math neophytes: I'm betting you're comfortable with "0", "1", "+", and "=".
... Most of us learned about π - the ratio of a circle's circumference to its diameter - somewhere along the way.
... and you've likely encountered 'i' = √(-1)... you know, those unreal 'imaginary' numbers!
That leaves only 'e' - the base of the natural logarithm.

The way I learned 'e'?
Plot the function f(x) = 1/x.
The area under the curve between 1 and e = 1.


... oh yeah: e = 2.718281828...
'e' is an irrational, transcendental number -
- irrational: e cannot be expressed as the ratio of two integers;
- transcendental: e is not the solution of any polynomial equation with integer coefficients.

(... in both these respects, e is just like π, which is also irrational & transcendental.)

p.s. thanks to loyal reader PM for teaching me how to get Excel graphs saved as pictures!

p.p.s. ... no, i've not a clue why i'm thinking about Euler's identity in the early morning hours of Wednesday, 12 Jan 2011.