Express complex numbers in modulus-argument and exponential form, convert between forms, and use them to multiply, divide and take powers via de Moivre's theorem

What this dot point is asking

SEAB wants you to express complex numbers in modulus-argument (polar) form and in exponential form, convert between Cartesian and these forms, and use them to multiply, divide and raise to powers, applying de Moivre's theorem.

The answer

Modulus and argument

For $z = a + bi$:

• The modulus $|z| = r = \sqrt{a^2 + b^2}$ is the distance from the origin.
• The argument $\arg z = \theta$ is the angle from the positive real axis, taken in the range $-\pi < \theta \leq \pi$ (the principal argument).

Then $z = r(\cos\theta + i\sin\theta)$, the polar form.

Exponential form

Euler's relation $e^{i\theta} = \cos\theta + i\sin\theta$ gives the compact exponential form:

This is the most efficient form for multiplication, division and powers.

Multiplication and division

In exponential form the rules are simply:

You multiply the moduli and add the arguments (subtract for division). This is far quicker than Cartesian multiplication.

De Moivre's theorem

Raising to a power:

So you raise the modulus to the power and multiply the argument by $n$. Care with the quadrant of the argument is essential when converting back.

Finding the nth roots of a complex number

De Moivre's theorem also runs in reverse to find roots. The $n$ distinct $n$th roots of $re^{i\theta}$ have modulus $r^{1/n}$ and arguments $\tfrac{\theta + 2k\pi}{n}$ for $k = 0, 1, \ldots, n-1$, because adding a full turn of $2\pi$ to the argument before dividing produces a genuinely different root. So the cube roots of $8e^{i\pi}$ have modulus $8^{1/3} = 2$ and arguments $\tfrac{\pi}{3}, \pi, \tfrac{5\pi}{3}$. Geometrically the $n$ roots are equally spaced around a circle of radius $r^{1/n}$, separated by $\tfrac{2\pi}{n}$. Remembering to add multiples of $2\pi$ to the argument before dividing is what generates all $n$ roots instead of just one.

Deriving trigonometric identities with de Moivre

Expanding $(\cos\theta + i\sin\theta)^n$ by de Moivre and comparing real and imaginary parts produces multiple-angle identities, a classic H2 application. For $n = 2$, de Moivre gives $\cos 2\theta + i\sin 2\theta = (\cos\theta + i\sin\theta)^2 = \cos^2\theta - \sin^2\theta + 2i\sin\theta\cos\theta$. Equating real parts yields $\cos 2\theta = \cos^2\theta - \sin^2\theta$ and imaginary parts yields $\sin 2\theta = 2\sin\theta\cos\theta$. Using de Moivre as a generator of trigonometric identities, by expanding and matching parts, connects the complex-number work directly to trigonometry and is a frequently examined technique.

Examples in context

Example 1. Rotating a point. Multiplying a complex number by $e^{i\theta}$ rotates it anticlockwise by $\theta$ about the origin without changing its modulus, which is why complex multiplication is the algebra of rotations in the plane.

Example 2. AC phasors. In electronics a sinusoidal signal is represented as $r e^{i\theta}$; combining signals by adding phasors and scaling by gain factors uses the multiply-moduli, add-arguments rule, making polar form the engineer's default.

Try this

Q1. Find the modulus and argument of $z = -1 + i$. [3 marks]

• Cue. $|z| = \sqrt{2}$; second quadrant, $\arg z = \dfrac{3\pi}{4}$.

Q2. Given $z = 4e^{i\pi/3}$, find $z^2$ in exponential form. [2 marks]

• Cue. $16 e^{i 2\pi/3}$.

Q3. State the rule for the argument of a product of two complex numbers. [1 mark]

• Cue. The arguments add: $\arg(z_1 z_2) = \arg z_1 + \arg z_2$.