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- af**i** + bd**i** - 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.

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