2 Complex numbers

Up to now, we have focused on the real numbers, but we will now consider an extension of the real number system, the set of complex numbers. One way of looking at complex numbers is to view them as elements in \(\mathbb{R}^2\) which can be multiplied. In this section we will introduce complex numbers and use them as an extended development of some of the properties of vectors that we have seen so far.

The basic idea underlying the introduction of complex numbers is to extend the set of real numbers in a way that polynomial equations have solutions. The standard example is the equation \[x^2=-1\] which has no solution in \(\mathbb{R}\). We introduce then in a formal way a new number \(\mathrm{i}\) with the property \(\mathrm{i}^2=-1\) which is a solution to this equation.

Definition 2.1: (Complex numbers)

The set of complex numbers is the set of linear combinations of multiples of \(\mathrm{i}\) and real numbers, that is \[\mathbb{C}:=\{ x+\mathrm{i}y : x,y\in \mathbb{R}\},\] where \(\mathrm{i}^2=-1\).

We will denote complex numbers by \(z=x+\mathrm{i}y\) and call \(x=\operatorname{Re}z\) the real part of \(z\) and \(y=\operatorname{Im}z\) the imaginary part of \(z\).

What is amazing is that having added this one new number, all polynomial equations with real or complex coefficients have a solution in the set of complex numbers!