# Geometrical Interpretation Of Complex Equations

This section will give you more experience in dealing with complex numbers from a geometrical perspective. We will use the knowledge gained up to this point to interpret equations and inequations involving complex numbers geometrically. In particular, we will draw *regions *corresponding to equations and inequations on the complex plane; what this means will become quite clear in the following examples.

**Example - 16**

Interpret the equation \(\left| z \right| = 1\) geometrically.

**Solution:*** \(z\)* is a (variable) complex number whose modulus is 1. This means that no matter what the direction in which *\(z\)* lies (i.e. no matter what its argument), the distance of *\(z\)* from the origin is always 1. Therefore, what path can *\(z\)* possibly trace out on the complex plane or in other words, what is the locus of \(z?\) Obviously, a circle of radius 1 centered at the origin.

**Example - 17 **

Plot the regions that *\(z\)* represents if:

**(a)** \(\left| z \right| < 1\) ** (b) ** \(\left| z \right| > 2\) ** (c) **\(1 < \left| z \right| < 2\)

**Solution: (a)** \(\left| z \right| < 1\) means that the distance of *\(z\) *from the origin must be less than 1. Therefore, *\(z\)* must lie (anywhere) inside a circle of radius 1 centered at the origin.

**(b)** \(\left| z \right| > 2\) means that the distance of *\(z\)* from the origin must be greater than 2. Therefore, *\(z\)* must lie (anywhere) outside a circle of radius 2 centered at the origin.

**(c)** \(1 < \left| z \right| < 2\) means geometrically that *\(z\)* must lie outside a circle of radius 1, but inside a circle of radius 2, both the circles being centered at the origin.

**Example - 18**

Plot the region represented by *\(z\)* if *\(z\)* satisfies

\[\begin{align}{}\left| {\arg (z)} \right| < \frac{\pi }{6}\\\left| z \right| < 3\end{align}\]

**Solution:** Since \(\begin{align}\left| {\arg (z)} \right| < \frac{\pi }{6}\end{align}\) this implies that\(\begin{align}\frac{{ - \pi }}{6} < \arg (z) < \frac{\pi }{6}\end{align}\) . This means that *\(z\)* must lie in a triangular region with one edge making an angle \(\begin{align}\frac{\pi }{6}\end{align}\) and the other edge making an angle \(\begin{align}\frac{{ - \pi }}{6}\end{align}\) with the *\(x\)*-axis. Also, the distance of *\(z\)* from the origin must be less than 3. Thus, *\(z\)* lies in a region that is in the shape of a sector of a circle: