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z 2D-variable
i is no longer imaginary


1 , 2, 3, ... start with counting; counting backwards and subtracting leads to an opposite direction: MINUS.

Bombelli taught in 1572 not only reckoning with the new - at least for europe - positive and negative numbers and gave their sign-rules, he also extended this to the reverse of reckoning with powers (like 2tothepowerof3 = 2*2*2), to the square-, cubic- and other roots by introducing only two new signs: +root- and -root-. For him "necessarissima" and nothing "impossibile".

iHis successors Cotes, de Moivre,..., Euler connected these signs to the unit +1 and with angels (with difference of direction as looked at from today) and generalized into a smooth, continuous "sign":

cos @ + root-1 * sin @ = e-tothepowerofroot-1*a. A number was thought of as quantity with direction of plus or minus, and that was not, what Wurzel-1 was. So Euler designatedroot-1with the letter i. He could now express root-9 as i* root9 or write it as i*3 and he called this - like Descartes : imaginary number. And a linear combination of numbers and imaginary numbers is called complex number.

i 1797 , in the time of the first french revolution, Caspar Wessel represented directed lengths of a plane algebraically as complex numbers, similiar to geographic longitude and latitude, and so he unveiled the "imaginary numbers" as second coordinate. So he founded the geometric vector-calculus.

38 years laterJahre Hamilton generalized further on: a complex number is not one number, it is an ordered pair and therefore the "imaginary numbers" are pairs, like (0,3) = root-9 and (0,-7) = minusroot-49.Combined with the rules for calculating these, with the (R2,+,*) thus, he put the first analytical vector-space into math (and in this case its equal to a commutative field). (And Grassmann, .. i don't know, what he contributed).

By Gauss such ideas became more known (and Gauss never claimed to be an author), but then in Germany some authoritarian unnecessarily started to use words like "Gauss-Ebene"( "Gauss-plane") and "imaginary axis". And in other coutries it's spoken of "Argand diagramm" or more neutral "complex plane". And the progress of Wessel and Hamilton is ignored or displaced of far away into the most darkest corner. And by this it developed, that mathematicians were walking around with an "imaginary veil" in front of their face, and when they didn't die, ...