acemuzzy wrote:

Right, before the proof, somebody asked for a noddy explanation of all this shizzle (i, e, pi, euler's identity, complex numbers).  So, feeling brave, I'm giong to try.  Indulge me.  And ask questions...

Let's start with i.  i is kinda useless by itself.  i only becomes useful if you square it.  In fact, that's how it's defined - the thing that if you square it you get -1.

So, what is i?  Well, doesn't really mean much till you square it.

But, if you then start doing maths with it, fun stuff happens.

Like, what's (2i) squared?  Well that's (2i)^2 = (2*i)^2, and you can multiply out, so = (2^2) x (i^2) = 4 * i^2, and i^2 = -1, so that's -4.  Whoop.

But then, more fun things happen.  Like, what are the solution to x^2-2x+2=0?  (i.e. the values of x for which that formula holds true; it doesn't for x=1 say, as that's 1^2-2x1+2 = 1-2+2 = 1, not 0).

You may remember the formula, x=(-b +/- sqrt(b^2-4ac)/2a.  Let's try it.  

 x = (-(-2) +/- sqrt((-2)^2-4x1*2))/2x1
   = (2 +/- sqrt(4 - 8 ))/2
   = (2 +/- sqrt (-4))/2

oh wait, we just decided that sqrt(-4) = 2i.  

So

 x = (2 +/- 2i) / 2 = (1 +/- i)

Nice.

Let's test it (with (1 + i)).

x^2-2x+2 = (1+i)^2 - 2x(1+i) + 2
              = (1^2 + 2i + i^2) - 2(1+i) + 2
              = (1 + 2i - 1) - 2 - 2i + 2
              = 0 as everything cancels.

Whoopitydoop.

More to follow....