This one blows my mind.
e ^ (pi times i) = -1

Interesting, I guess. But, how is this at all practical?

If I remember my electronics college courses well "i" is used quite a bit in solving electrical problems, like sinusoidal waveform phases and all that. Sorry, it's been 30 years but I do remember that it factored in quite a bit with electrical theory.

>> ^schlub:

Interesting, I guess. But, how is this at all practical?

Good question, but I would phrase it a bit differently: Why is it needed?

It is needed because without it, mathematics would be incomplete. (It actually turns out later, thanks to Gödel, that mathematics in a sense is inherently incomplete, but let's not worry about that now.)

What I mean by that is that there are certain mathematical problems that require i to have a solution. The imaginary numbers come out naturally when you try to solve certain equations, just as for instance negative and rational numbers come out of equations.

If you start out with just the positive integers, which are the most intuitive numbers for us to contemplate, you run into a barrier if you try to solve the equation x + 2 = 0. To find a solution for x, you have to introduce negative numbers.

If you want to find a solution to the equation 3/x = 1, you need to expand your numbers to include rational numbers such as 1/3, which is the solution to this equation.

Further on, you get the irrational numbers by solving equations such as
x^2 = 2.

Finally, if you want to solve the equation x^2 = -1, you have to introduce i to your set of numbers. There is no other way to solve it.


We could try to go on in the same fashion, but it has been proven by mathematicians that these numbers are all you need to solve every mathematical problem you come across. As a physicist, I come across many equations that include complex numbers, especially in quantum mechanics. Other times you don't really need the complex numbers, but certain calculations become easier to solve if you use them. That's when they are practical, but they are also in a deeper sense a natural extension of the more well known real numbers.

>> ^schlub:

Interesting, I guess. But, how is this at all practical?

>> ^Ornthoron:

Good question, but I would phrase it a bit differently: Why is it needed?
It is needed because without it, mathematics would be incomplete. (It actually turns out later, thanks to Gödel, that mathematics in a sense is inherently incomplete, but let's not worry about that now.)
What I mean by that is that there are certain mathematical problems that require i to have a solution. The imaginary numbers come out naturally when you try to solve certain equations, just as for instance negative and rational numbers come out of equations.
If you start out with just the positive integers, which are the most intuitive numbers for us to contemplate, you run into a barrier if you try to solve the equation x + 2 = 0. To find a solution for x, you have to introduce negative numbers.
If you want to find a solution to the equation 3/x = 1, you need to expand your numbers to include rational numbers such as 1/3, which is the solution to this equation.
Further on, you get the irrational numbers by solving equations such as
x^2 = 2.
Finally, if you want to solve the equation x^2 = -1, you have to introduce i to your set of numbers. There is no other way to solve it.

We could try to go on in the same fashion, but it has been proven by mathematicians that these numbers are all you need to solve every mathematical problem you come across. As a physicist, I come across many equations that include complex numbers, especially in quantum mechanics. Other times you don't really need the complex numbers, but certain calculations become easier to solve if you use them. That's when they are practical, but they are also in a deeper sense a natural extension of the more well known real numbers.
>> ^schlub:
Interesting, I guess. But, how is this at all practical?



Do not believe his witchcraft. i stands for the Word "I" and is the way to introduce yourself into your equations if you are really high on acid and such.

the mathematical equations used to explain alternating current (AC) electricity -- the kind that makes your computer turn on -- requires use of the number i.

>> ^schlub:

Interesting, I guess. But, how is this at all practical?