.Variable vectors,Functions, Holomorph, Quaternions.

The centuries-old art of calculating 2D-vectors needs promotion with outspoken opposition to the higher-mathematics nonsense of the "imaginary".

Only after doing this we can go on also disentangling the knots in our thinking of variables and functions of the R2.

```  - The representation, the visualisation, you are used to - rectangular coordinates,
f(x) perpendicular to x - here it can be different, they can be parallel too.
- I restrict myself here to termdefined functions, you build in the domain a term
and the calculated values you assign to the range.
- Never forget to look at the domain, the range and the algebraic rules in them !
- It took me years to come to the following simple-seeming conclusion:
An inverse function is a holy wish, which nearly never comes true !
In computer-programs of functions you can grab a point in the domain and move it
around and see the dependent point in the range moving accordingly. So You might
think, there is may be a function the opposite way. Nearly always you fail,
either it's not a function(one point in the former range is  associated with
several points in the former domain) or it's not complete, only a part is
inverted. And if it's not that, than may  be the former range, which should
become now the domain, is not fully exhausted.There are only very few functions,
which can be inverted. The simple function f:z--> zē already delivers two "inverse"
functions: + squareroot and - squareroot.
- z^w or z to the power of w - it really behaves different in 2D, to what you are
used to in R.
```

```    TRAVEL COMPANION OVER THE COMPLEX PLANE AND ALONG THE IMAGINARY AXIS
ALGEBRAIC FAIRY TALE : C IS EUCLIDIAN