Angle and Length

Instead of thinking about a complex number with $x$ and $y$ coordinates, as in $x + yi$, we can think of it as an angle and length of the vector. Just like in unit circle trig (TODO), a bigger angle means counter-clockwise and zero means right.

The angle may be bigger than 180 degrees:

Alternatively, the angle can be negative:

In general, angles that differ by a full turn (360 degrees or $2\pi$ radians), such as the ones in the above two pictures, are considered to be the same.

Examples:

Generally, the length of a complex number $a + bi$ is the vector length $\sqrt{a^2+b^2}$. The angle is more difficult to calculate. In computer programming, you can use atan2 to calculate the angle. TODO: explain how to calculate angle in general

Absolute value of complex number

The absolute value of a complex number $a+bi$ is defined to be its length: $$ \abs{a+bi} = \text{length of $a+bi$} = \sqrt{a^2+b^2} $$ Because the length of a real number is its absolute value, this behaves as expected when $a+bi$ happens to be a real number (that is, $b=0$).

Argument

The notation $\arg(z)$ means an angle of a nonzero complex number $z$. Because $z$ has infinitely many different angles that differ from each other by a full turn $2\pi$, we need to somehow choose which angle is called $\arg(z)$. A common choice is $-\pi < \arg(z) \le \pi$:

So, $\arg(z)$ increases as $z$ rotates around $0$ counter-clockwise, but it "jumps" at $\pi$. It must jump somewhere, because it's periodic and so it can't increase forever.

I sometimes need to represent any angle of a complex number, and I use $\arg(z)+n2\pi$ for that, with an arbitrary integer $n$. For this purpose, the choice of $\arg(z)$ doesn't matter, because we get all the possible angles anyway.