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

**Original set:**Consider the image of the set of real numbers

under the transform

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

to the complemented set of images.-
**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?