AHL 1.13—Polar and Euler form

So far you've represented complex numbers as $z=a+bi$ , the Cartesian form. But there's a second, equally powerful representation that uses the geometry of the complex plane instead of coordinates.

Modulus-Argument (Polar) Form: A complex number $z$ written as $z=r(\cos \theta +i\sin \theta )=r\text{cis}\theta$ where $r=|z|$ is the modulus and $\theta$ is the argument of $z$.

The two key quantities are:

• Modulus: $r=|z|=\sqrt{a^2+b^2}$ , the distance from the origin to the point $z$ in the complex plane.
• Argument: $\theta =\arg (z)$ , the angle measured anticlockwise from the positive real axis to the line segment $Oz$. It is computed as $\arctan \left(\frac{b}{a}\right)$ with a quadrant adjustment (see below).

The shorthand $\text{cis}\theta$ simply means $\cos \theta +i\sin \theta$, and is commonly used in IB problems.

The formula $\theta =\arctan \left(\frac{b}{a}\right)$ only gives the correct argument when $z$ is in the first or fourth quadrant (i.e., $a>0$). You must adjust for other quadrants.

The clearest approach uses the reference angle $\alpha =\arctan \left(\frac{|b|}{|a|}\right)$, which is always positive and lies in $\left[0,\frac{\pi }{2}\right]$.