Complex Numbers in Algebra

Abstract

The next three chapters revisit the topics of algebra, curves, and functions, observing how they are simplified by the introduction of complex numbers. That’s right: the so-called “complex” numbers actually make things simpler. In the present chapter we see where complex numbers came from (not from quadratic equations, as you might expect, but from cubic equations) and observe how they simplify the study of polynomial equations. Equations become simpler because they always have solutions in the complex numbers, and it follows that they have the “right” number of solutions. One of the reasons for the simplifying power of complex numbers is their two-dimensional nature. The extra dimension gives more room for solutions of equations to exist. For example, the equation x n = 1, which has only one or two solutions in the real numbers, has n different solutions in the complex numbers, equally spaced around the unit circle. More generally, complex numbers give a way to divide any angle into n equal parts. This comes about because multiplication of complex numbers involves addition of angles, and is related to the famous de Moivre formula in trigonometry. The equation x n = 1 is not the only one with the “right” number of solutions in the complex numbers. In fact, any equation of degree n has n complex solutions, when solutions are properly counted. This is the fundamental theorem of algebra, and it follows from intuitively simple properties of the plane and continuous functions.