# Complex Number Regions

Regions defined by complex numbers $\displaystyle z = x + yi$ where $x$ and $y$ are real numbers that can be drawn using inequalities for complex number regions.

### Worked Example of Complex Number Regions

Find the inequality of complex region $\displaystyle \Bigg|1+\frac{1}{z}\Bigg| \le 1$.

\begin{aligned} \displaystyle \require{color} \frac{|z+1|}{|z|} &\le 1 \\ |z+1| &\le |z| \\ |x+yi+1| &\le |x+yi| \\ |(x+1) + yi| &\le |x+yi| \\ \sqrt{(x+1)^2+y^2} &\le \sqrt{x^2+y^2} \\ (x+1)^2+y^2 &\le x^2+y^2 \\ x^2 + 2x + 1 +y^2 &\le x^2+y^2 \\ 2x + 1 &\le 0 \\ 2x &\le -1 \\ \therefore x &\le -\frac{1}{2} \end{aligned}