2.1 Complex numbers: geometry of complex numbers

The most fundamental new idea in the study of complex numbers is the “imaginary number” $j$ . This imaginary number is defined to be the square root of $-1$ :

The imaginary number $j$ is used to build complex numbers $x$ and $y$ in the following way:

We say that the complex number $z$ has “real part” $x$ and “imaginary part” $y$ :

In MATLAB, the variable $x$ is denoted by, and the variable $y$ is denoted by. In communication theory, $x$ is called the “in-phase” component of $z$ , and $y$ is called the “quadrature” component. We call $z=$ $x+jy$ the Cartesian representation of $z$ , with real component $x$ and imaginary component $y$ . We say that the Cartesian pair $(x,y)$ codes the complex number $z$ . imag(z)

We may plot the complex number $z$ on the plane as in [link] . We call the horizontal axis the “real axis” and the vertical axis the “imaginaryaxis.” The plane is called the “complex plane.” The radius and angle of the line to the point $z=x+jy$ are

See [link] . In MATLAB, $r$ is denoted by, and $\theta$ is denoted by. angle(z)

The original Cartesian representation is obtained from the radius $r$ and angle $\theta$ as follows:

The complex number $z$ may therefore be written as

The complex number $\text{cos}\theta +j\text{sin}\theta$ is, itself, a number that may be represented on the complex plane and coded with the Cartesian pair $(cos\theta ,\text{sin}\theta )$ . This is illustrated in [link] . The radius and angle to the point $z=cos\theta +j\text{sin}\theta$ are 1 and $\theta$ . Can you see why?

The complex number $\text{cos}\theta +j\text{sin}\theta$ is of such fundamental importance to our study of complex numbers that we give it the special symbol $e^{j\theta }$ :

As illustrated in [link] , the complex number $e^{j\theta }$ has radius 1 and angle $\theta$ . With the symbol $e^{j\theta }$ , we may write the complex number $z$ as

We call $z=r{e}^{j\theta }$ a polar representation for the complex number $z$ . We say that the polar pair $r\angle \theta$ codes the complex number $z$ . In this polar representation, we define $\left|z\right|=r$ to be the magnitude of $z$ and $arg\left(z\right)=\theta$ to be the angle, or phase , of $z$ :

With these definitions of magnitude and phase, we can write the complex number $z$ as

Let's summarize our ways of writing the complex number z and record the corresponding geometric codes:

In "Roots of Quadratic Equations" we show that the definition $e^{j\theta }=\text{cos}\theta +j\text{sin}\theta$ is more than symbolic. We show, in fact, that $e^{j\theta }$ is just the familiar function $e^x$ evaluated at the imaginary argument $x=j\theta$ . We call $e^{j\theta }$ a “complex exponential,” meaning that it is an exponential with an imaginary argument.