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Why do we need yet another article about complex numbers? This is a valid question and I have asked it myself. I could mention that I wanted to gather the many different views that can be found elsewhere - Euler's and Gauß's perspectives, i.e. various historical views in the light of the traditionally parallel development of mathematics and physics, e.g. the use ofcomplexcoordinates in kinematics, the analytical or topological views, e.g.the Radishor the mysteriousLiouville's theoremabout bounded entire functions that are already constant, or the algebraic view that led to the many non-algebraic proofs of the fundamental theorem of algebra. The complex numbers have so many faces and appear in so many contexts that I could as well have written a list of bookmarks. All of that is true to some extent. The real reason is, that I want to break the automatism of the association of complex numbers with, and the factual reduction to points in the Gaußian plane

$$

\mathbb{C}=\{a+i b\,|\,(a,b)\in \mathbb{R}^2\}\neq \mathbb{R}^2.

$$

We need two dimensions to visualize complex numbers but that doesn't make them two-dimensional. They area one-dimensional fieldin the first place, i.e. a single set of certain elements that obey the same axiomatic arithmetic rules as the rational numbers do. They are one set that is not just a plane! The reason they exist and bar us from visual access is finally a tiny positive distance wecansee.

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