Complex analysis can roughly be thought of as that subject which applies the ideas of calculus to imaginary numbers. But what exactly are imaginary numbers? Usually, students learn about them in high school with introductory remarks from their teachers along the following lines: ``We can't take the square root of a negative number. But, let's pretend we can---and since these numbers are really imaginary, it will be convenient notationally to set i = sqrt(-1) .'' Rules are then learned for doing arithmetic with these numbers. The rules make sense. If i = sqrt(-1) , it stands to reason that i^2 = -1 . On the other hand, it is not uncommon for students to wonder all along whether they are really doing magic rather than mathematics.