What goes up must come down, or vice versa. Done, with no maths at all.

So what's the difference between a proof and an axiom, and is not Goober's example an axiom?

An axiom is something you state/assume/define.

A proof is something you conclude based off those axioms.

Without any axioms you can't really get going. Though there's interesting philosophical / logical stuff about what the "right" axioms are, though that's probably not even a meaningful thing to ask.

Eg. Wiki the axiom of choice for the interesting one in that regard.

It's pretty much the same as mine too - if the statement is incorrect you are left either with a function which isn't continuous, or a function which gets lower/higher over every half rotation of the circle (thus meaning you can't get back to your original point). You've got to be able to get back to where you started.

I have done a 3d model of my proof. The maths is very complicated so I thought it best to not show my workings.

Here comes a new problem*:

We all know (and love) that 3^2 + 4^2 = 5^2

Some even know 10^2 + 11^2 + 12^2 = 13^2 + 14^2

You may even be aware that 21^2 + 22^2 + 23^2 + 24^2 = 25^2 + 26^2 + 27^2

What is the next equation in this sequence, in which each equation involves an odd number of consecutive squares?

*Saturday Guardian readers need not apply!

Is the next question "why does that happen"? How many marks is that?

I find this kind of stuff seriously distracting. Curse you all!

It does as well

As I might stereotypically say:
Why i?

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....

At that point I already find it kinda beautiful.  Why should an exponential function be expressable as the sum of lots of expoents, all over factorials?  

No idea.  It's kinda magical.

More so if you know how to differentiate...

x^n differentiates to (x^(n-1)) x n.

So, differentiate e^x, and you get...

 1           -> 0
 x           -> 1                  =  1
 x^2/2!   -> x^1 * 2/2!      +  x
 x^3/3!   -> x^2 x 3/3!      + x^2/2!
 ...
 x^n/n!    -> x^(n-1)xn/n! + x^(n-1)/(n-1)!
                                      + x^n/n!

                                      = e^x


i.e. it differentiates (and hence integrates...) to itself.

Maigcal stuff.

Right, next up, pi .

Back to wikipedia:




OK, random formulae.  But, back to my spreadsheet please...

 - Put (x=0) in, and you get sin(x)=0, cos(x) = 1.
 - Put x=3.14159 in, add up all those powers of x (as per my spreadsheet), you get sin(pi)=0, cos(pi)=-1.
 - And other values work too.

So, pi is very kinda of natural, in that it's the first number that gives you zero if you bung it into the sin expansion above.  You can't _choose_ a different pi, because it wouldn't have 0 as it's sin.

So that's kinda where radians come from, as opposed to degrees.   They're "natural".

Right, now the proper magic happens...

What if in stead of x, we put ix into the e^x formula?



So 

e^ix = 1 + (ix)  + (ix)^2/2! + (ix)^3/3! + (ix)^4/4! + ...

Now let's get rid of the all the i^2 = -1, i^4=(i^2)^2 = (-1)^2=1 stuff...

       = 1 + ix -x^2/2! - ix^3/3! + x^4/4! + ...

Now let's gather terms with i in:

        = (1 - x^2/2! + x^4/4! + ...) + i (x - x^3/3! + x^5/5! + ....)

But hang on a sec... that means ...
   
 e^ix = cos(x) + i sin(x)

That some crazy shizzle.

Finally, then, you can see that if you put r = 1, theta = 180 degrees, i.e. pi, then you back at -1 on the x axis, 0 on the y axis.

i.e. -1 + 0i = -1.

And, hence -1 = r e ^ (i theta) = 1 x e ^ ( i pi)

i.e.





Simples.  

Right that's me done for the night, it's the effing weekend.

That would have been my proof.

I like a nice visual proof.

Like the pythagoras proofs - I think I could write down at least 5 but there are many more - but the pictorial one is the best.