A Real Exploration of Euler’s Imaginary

In elementary schools the students are taught that negative numbers do not have square roots.The appearance of a square root of a negative number in the course of a computation indicates that either the problem has no solution or an error has occurred.Subsequently students are told that negative numbers have "imaginary" square roots which can be constructed using the symbol i which represents the square root of -1.However, this reasoning appears logically inconsistent.There is nothing imaginary about the symbol i or its use.This paper treats the following interesting topics in the theory of functions of a complex variable: 1) sensible introductions to Euler's i that conform to the way engineers and technicians use the symbol in analyzing alternating current circuits and mechanically vibrating systems;2) the derivation of the algebraic and topological features of the complex plane and a comparison of these features to the properties of "real" numbers;3) the description of the isomorphism between phasors and combinations of same-frequency sinusoidal oscillations that underlies the theory of alternating current analysis promoted and successfully used by C. P. Steinmetz for the distribution of electrical power throughout the United States; 4) the derivation of the linear 2-dimensional rotational mappings represented by a system of 2×2 matrices with real number entries.These mappings can represent complex numbers and serve as an alternative definition of the meaning of the symbol i.This paper should provide a reasonable introduction to theories of alternating currents and vibrations and encourage further study of the theory of complex variables.