At first: Read this post.

I fantasized about all the small and big real numbers and aside from that it is an interesting read.

A complex number, regarding mean human stupidity, is defined to be a linear combination of two real numbers.

Complex, conclusively, has to have some kind of caveat to it.

Let's call this one i and define it to be the square root of -1. Since nobody can even know, what i actually is, we take it as a variable.

This means, we don't know more shit about it, but at least we got a letter for it now.

So with years of existential dread and years of depressingly false calculations, one human being (namely Gauss ) can define complex numbers to be a linear combination of a real and an imaginary part. Theoretically speaking

Like this:

c = a + b* i

and

g = d + f * i.

With a, b, d, f being real numbers. We actually can add, subtract and multiply those numbers, just like their real friends. Division is, where the "fun" starts:

c / g = ( a + b * i) / (d + f* i) = ( a + b * i) / (d + f* i)

= ( ( a + b * i) * (d -f* i) ) / ( (d + f* i) * (d - f* i) )

(Why do I torture myself again with this, I got wage-slaves doing this kind of second-grade math)

= ( ad- afi + bdi - bf) / / (d² + f²), since i² = -1 per definition.

(because this gets funny sooner than later)

= { (ad -bf) + i (af + bd) } / (d² +f²) (x)

The trick we used here, is that a complex number multiplied with its conjugate is real.

So let's do these estimations from the last post once more:

If we now imagine any real number in (x) to be very small, what actually does happen ?

Naturally, we want to look at g getting "small" hence d and f getting small.

Quite obviously shines, right into our eyes, that the denominator can't get negative, no matter how small any real number gets here.

Because squares are always positive. Don't believe me ?

Try it yourself and get your fields-medal for disproving this fundamental fact.

So far into it, since we learned so much so far, how can (x) ever get negative, assuming basically nothing about the real numbers put into this ?

It is a case distinction on the nominator. Seemingly, you and me aren't smarter than Gauss was.

You tell me.

End of proof and as always: Thanks for all the fish. :)

EDIT: Anyone doubting this kind of Algebra: I also can pull Riemann and Möbius directly out of my ass with their classical projection proofs on what H² actually looks like in a geometrical sense.

In fact: It is a sphere.