Virtuosity of Complex Numbers

Virtuosity of Complex Numbers

    Virtuosity of complex numbers is demonstrated by conversion into geometrical form of the formula that relates e (the natural base of logarithms), pi and the square root of negative one. The equation can be expressed as the sum of a series of vectors. When these are added and plotted on a complex plane, they form a spiral that strangles the point equal to negative one.

   - Philip J. Davis, Number, Scientific American, September 1964.

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The square root of negative one was a useful but self-contradictory nuisance to mathematicians until they discovered that it was the portal into a whole new dimension of numbers. 

In 1797 a geometric interpretation of complex numbers was developed by the Norwegian surveyor, Caspar Wessel (1745-1818). He showed that complex numbers involving the root of negative one are equivalent to real number pairs, and thus to points on a two dimensional plane: one dimension above the flat old number line where the root of negative one could find no home.

 Paul Halmos in his article "Innovation in Mathematics" from Scientific American Magazine,  September 1958...