Question

Describe geometrically the set of points in the complex plane satisfying the following equations. .

Short answer

The equation is the lower part of the xy-plane.

Step-by-step solution

Given Information

The equation is $Imz<0.$

Definition of the Complex number

A complex number can be expressed as:

$\mathit{z}\mathbf{=}\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$

Where z is the complex number, x and y are real numbers, and i is known as iota, whose value is $\sqrt{\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}}$ .

The modulus of a complex number can be calculated as:

$\mathbf{\left|}\mathit{z}\mathbf{\right|}\mathbf{=}\sqrt{\mathbf{(}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{)}}$

Find the value

The equation is $Imz<0.$.

The complex number is $z=x+iy$.

$$\begin{gathered} Imz<0 \\ Y<0 \end{gathered}$$

Hence, the equation is the lower part of the xy-plane.