ex+iy: The Imaginary Becomes Real
That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill-adapted notation. If, for instance, +1, −1, √−1 had been called direct, inverse, and lateral units, instead of positive, negative, and imaginary (or even impossible), such an obscurity would have been out of the question.—CARL FRIEDRICH GAUSS (1777–1855)1
The introduction of expressions like eix into mathematics raises the question: What, exactly, do we mean by such an expression? Since the exponent is imaginary, we cannot calculate the values of eix in the same sense that we can find the value of, say, e3.52—unless, of course, we clarify what we mean by “calculate” in the case ...