The modulus and argument of a complex number, modulus-argument form, the exponential form re^(i theta), and the multiplication and division rules in which moduli multiply or divide and arguments add or subtract.

What this dot point is asking

OCR wants you to find the modulus and argument of a complex number (choosing the correct quadrant), write it in modulus-argument form $r(\cos\theta + i\sin\theta)$ and in exponential form $re^{i\theta}$, convert between Cartesian and these forms, and use the rules that when you multiply complex numbers the moduli multiply and the arguments add, while for division the moduli divide and the arguments subtract.

Modulus and argument

The modulus measures size and the argument measures direction. The subtlety is the quadrant: a calculator's inverse tangent only returns angles in $\left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right)$, so you must adjust by sketching $z$ first.

Modulus-argument and exponential form

Once $r$ and $\theta$ are known, the two standard forms follow. Euler's relation $e^{i\theta} = \cos\theta + i\sin\theta$ links them, so exponential form is just a compact way of writing modulus-argument form.

These forms are essential for de Moivre's theorem and for finding roots of complex numbers, where powers and roots act simply on $r$ and $\theta$.

Multiplication and division

The reason these forms matter is that multiplication and division become geometric. Multiplying by $re^{i\theta}$ enlarges by factor $r$ and rotates anticlockwise by $\theta$, exactly the spiral-enlargement transformation seen with matrices.

These forms are the gateway to de Moivre's theorem, which extends the "multiply moduli, add arguments" rule to powers and roots.

Try this

Q1. Write $z = 1 - i$ in exponential form. [3 marks]

• Cue. $r = \sqrt{2}$; the point $(1, -1)$ is in the fourth quadrant, so $\arg z = -\tfrac{\pi}{4}$, giving $z = \sqrt{2}\,e^{-i\pi/4}$.

Q2. Given $|z_1| = 5$, $|z_2| = 2$, $\arg z_1 = \tfrac{\pi}{2}$ and $\arg z_2 = \tfrac{\pi}{6}$, find the modulus and argument of $\dfrac{z_1}{z_2}$. [2 marks]

• Cue. Modulus $\tfrac{5}{2}$, argument $\tfrac{\pi}{2} - \tfrac{\pi}{6} = \tfrac{\pi}{3}$.