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Creation of complex numbers

The pair of transforms

\begin{displaymath}f(x)=x^2 \qquad f^{-1}(x) = \sqrt{x}\end{displaymath}

applied to real numbers, yields to the complex numbers by the same scheme:

  1. Original set: Consider the image of the set of real numbers

    \begin{displaymath}1, 2, 3, 4, \ldots

    under the transform

    \begin{displaymath}f(x)=x^2. \end{displaymath}

    Set of images is positive numbers.
  2. Complement: The set of images is naturally complemented to the set $Z$ of all real numbers.
  3. Inverse Transform: Now apply the inverse transform

    \begin{displaymath}f^{-1}(x)=\sqrt{x} \end{displaymath}

    to the complemented set $Z$ of images.

  4. Imaginary Numbers!

    We arrive at new numbers: These are imaginary numbers: The square roots of negative numbers.

Semantic question: Why are the names ``negative,'' ``complex,'' and ``imaginary'' refer to something unpleasant, overcomplicated, or not real?

Andre Cherkaev