There is a certain historical sequence of difficulties and their solution which arose as the use of numbers was extended to ever more complicated calculations. Fractions were needed to divide entities up into parts and rearrange the pieces. Negative numbers as the equivalent of debts represent another level of abstraction, requiring some explanations about the negatives of negatives, and expecially about how the product of two negative numbers ought to be positive. That was decided upon long before concepts like associative laws or distributive laws were formalized, but the ideas behind them were appreciated and lent a certain order to rules which could have been hard to understand.

The solution of equations containing products now and then leads to having to multiply two equal numbers together to obtain a negative result The product of positive numbers can't be negative, but trying to square negative numbers to get a negative result goes against the conclusion that such a product of negative numbers should also be positive. Sometimes this contradiction can be avoided by reformulating the problem, or the terms of its solution.

Still, the basic contradiction persists. In order not to remain a permanent obstacle, some adjustment in the idea of what constitutes a number is needed. One approach is to postulate a quantity whose square is minus one, afer having decided that it can be scaled to get square roots of other negative quantities. But there aren't any such ``numbers,'' which seems to be why they got to be called *imaginary*. Even so, it is not clear that such an invention solves all problems; one of the first reactions of persons who hear about *i* and understand (but maybe not too deeply) the reasons for its use is to ask: ``What about the square root of -*i*?''

A recent book of Paul J. Nahin [23] considers the historical evolution of the concept of imaginary numbers and their symbolism at some length. In turn, the book has been reviewed by Brian E. Blank [3], who finds that mathematicians and electrical engineers think somewhat differently.

- complex numbers in the real plane via the Argand diagram
- polar and cartesian coordinates
- absolute value, phase, modulus
- stereographic projection
- Smith Chart
- graphical representation of complex functions