1.4: The Complex Plane

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Geometry of Complex Numbers

Because it takes two numbers $x$ and $y$ to describe the complex number $z = x + iy$ we can visualize complex numbers as points in the $xy$-plane. When we do this we call it the complex plane. Since $x$ is the real part of $z$ we call the $x$-axis the real axis. Likewise, the $y$-axis is the imaginary axis.

$屏幕快照 2020-08-05 下午10.20.43.png$

Triangle Inequality

The triangle inequality says that for a triangle the sum of the lengths of any two legs is greater than the length of the third leg.

$屏幕快照 2020-08-05 下午10.22.04.png$
Triangle inequality: $|AB| + |BC| > |AC|$

For complex numbers the triangle inequality translates to a statement about complex magnitudes.

Precisely: for complex numbers $z_1, z_2$

\[|z_1| + |z_2| \ge |z_1 + z_2| \nonumber \]

with equality only if one of them is 0 or if $\text{arg}(z_1) = \text{arg}(z_2)$. This is illustrated in the following figure.

$屏幕快照 2020-08-05 下午10.27.11.png$
Triangle inequality: \[|z_1| + |z_2| \ge |z_1 + z_2| \nonumber \]

We get equality only if $z_1$ and $z_2$ are on the same ray from the origin, i.e. they have the same argument.