# Symmetry of the complex plane

#### lolgarithms

How does one express mathematically the fact that:
if we complex-conjugated everything (switch i to -i (j to -j etc. in hypercomplex numbers) in all the definitions, theorems, functions, variables, exercises, jokes ;-)) in the mathematical literature the statements would still be true?

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#### rsq_a

Hmmm. Well, that's a very open-ended question.

It's not always true that you can simply switch $$z$$ with $$\overline{z}$$ without consequence. For example, given an analytic function $$df/d\overline{z}=0$$...a statement which is true of all functions! Another fact is that $$f(z) = \overline{z}$$ is a nowhere analytic map, which means that you can't simply conjugate things without worrying about consequences!

However, in the largest picture, I suppose your question is about symmetry in the complex plane. In that case, I suppose the key lies in the fact that analytic functions (and thus satisfying the Cauchy-Riemann equations) impose a restriction on how the function behaves as x and y vary...that is, the (x,y) coordinates are necessarily coupled. This results in a great deal of symmetry.

#### Landau

If you switch i to -i in the definition of i (i.e. in the definition of complex numbers themselves), then you're just renaming/relabelling a symbol.

#### elibj123

a. Choosing i as the "basic imaginary unit" that satisfies $$x^{2}=-1$$ is arbitrary, you could have chosen to work with -i instead.

b. (Equivalent) Measuring the argument counter-clockwise is also arbitrary, and you except that the sames conclusions and theorems will remain unchanged when measuring the argument clockwise (which is equivalent to conjugating)

"Symmetry of the complex plane"

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