i

What kind of number is √−1? In a way that parallels the unexpected discovery of √2 by the Pythagoreans, when this number surfaced as a solution to a quadratic equation, mathematicians asked themselves what it could possibly mean. Not knowing what to call it, René Descartes named it an imaginary number. Like the irrationals, the discovery of i led to new ideas and discoveries. One of these was complex numbers—numbers having the form (a + bi), where a and b are real numbers and i is √−1. Incredibly, complex numbers turn out to have many applications. They are used to describe electric circuits and electromagnetic radiation and they are fundamental to quantum theory in physics. This chapter deals with imaginary numbers, which constitute another of the great ideas of mathematics that have not only changed the course of mathematics but also of human history.