The absolute value of a complex number is its distance from zero.

We can compute this distance using the Pythagorean Theorem.

Let's see one more.

Instead of the formula, here we try factorization.

And now let’s see what else we can do with these complex numbers.

Let’s try to graph on the complex plane those complex numbers where:

We use the algebraic form,

that is

Next come some coordinate geometry horror stories.

The equation of

is the equation of a circle centered at the origin, with a radius of r.

Based on this, is also a circle centered at the origin, with a radius of r=2.

And means the circle and its inside.

Coordinate geometry horror stories:

The equation of a line:

The equation of a circle:

Let’s find on the complex plane complex numbers such that:

We use the algebraic form, so we replace z

with everywhere.

The inequality means one side of the line.

Let's see which side.

It is always a good idea to try a=0 and b=0.

This seems to fit, so we need this side of the line.

Next, let's see what is going on with this one:

The inequality means one side of the circle.

Either the inside or the outside of the circle.

Again, it is a good idea to try a=0 and b=0.

It seems we need the outside.

And because equality is not allowed,

the circle itself is not part of the region.

Finally, let’s see what this is about:

We will need to complete the square.