Complex Number Regions


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

Worked Example of Complex Number Regions

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

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

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