
# 1.4: The Complex Plane


## The 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.

## The 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.

Triangle inequality: $$|AB| + |BC| > |AC|$$

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

Precisely: for complex numbers $$z_1, z_2$$

$$|z_1| + |z_2| \ge |z_1 + z_2|$$

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.

Triangle inequality: $$|z_1| + |z_2| \ge |z_1 + z_2|$$

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.