Polar Form of Complex Numbers

A complex number can be represented in two primary ways: the Cartesian form ($z = a + bi$) and the polar form (also known as modulus-argument form). While Cartesian form is useful for addition and subtraction, polar form is significantly more efficient for multiplication, division, and finding powers of complex numbers.

1. Definition of Polar Form

A complex number $z = a + bi$ can be plotted on an Argand diagram at point $P(a, b)$. By considering the distance $r$ from the origin and the angle $\theta$ from the positive real axis, we derive the polar form:

$$z = r(\cos \theta + i \sin \theta)$$

This is abbreviated using the cis notation:
$$z = r \operatorname{cis} \theta$$

Key Components

• Modulus ($r$ or $|z|$): The distance of the point from the origin.
  $$r = |z| = \sqrt{a^2 + b^2}$$
• Argument ($\theta$ or $\arg z$): The angle the line segment $OP$ makes with the positive $Re(z)$ axis, measured anticlockwise.
  $$\tan \theta = \frac{b}{a}$$

Principal Argument ($\operatorname{Arg} z$)

Because $\cos$ and $\sin$ are periodic functions, a complex number has infinitely many arguments ($\theta + 2n\pi$). To ensure uniqueness, we define the Principal Argument, denoted as $\operatorname{Arg} z$ (with a capital ‘A’), which is restricted to the range:
$$-\pi < \operatorname{Arg} z \le \pi$$

EXAM TIP: VCAA questions usually require the argument to be expressed in principal form. Always check if your calculated $\theta$ falls within $(-\pi, \pi]$. If $\theta = \frac{5\pi}{4}$, the principal argument is $\frac{5\pi}{4} - 2\pi = -\frac{3\pi}{4}$.

2. Conversion Between Forms

Cartesian to Polar ($a + bi \to r \operatorname{cis} \theta$)

1. Calculate the modulus: $r = \sqrt{a^2 + b^2}$.
2. Determine the quadrant of the complex number by looking at the signs of $a$ and $b$.
3. Calculate the reference angle $\alpha = \arctan\left(\frac{|b|}{|a|}\right)$.
4. Adjust $\alpha$ to find $\theta$ based on the quadrant:
   • Quadrant 1: $\theta = \alpha$
   • Quadrant 2: $\theta = \pi - \alpha$
   • Quadrant 3: $\theta = -(\pi - \alpha)$ or $\alpha - \pi$
   • Quadrant 4: $\theta = -\alpha$

Polar to Cartesian ($r \operatorname{cis} \theta \to a + bi$)

Use the following trigonometric relations:
* $a = r \cos \theta$
* $b = r \sin \theta$

Example: Convert $z = 2 \operatorname{cis}\left(\frac{2\pi}{3}\right)$ to Cartesian form.
$$a = 2 \cos\left(\frac{2\pi}{3}\right) = 2\left(-\frac{1}{2}\right) = -1$$
$$b = 2 \sin\left(\frac{2\pi}{3}\right) = 2\left(\frac{\sqrt{3}}{2}\right) = \sqrt{3}$$
$$z = -1 + \sqrt{3}i$$

COMMON MISTAKE: Relying solely on $\theta = \arctan(\frac{b}{a})$ on a calculator. Calculators only return values in the range $(-\frac{\pi}{2}, \frac{\pi}{2})$. You must manually check the quadrant of $(a, b)$ to ensure the argument is correct.

3. Multiplication and Division in Polar Form

Arithmetic operations are often simpler in polar form.

Multiplication

To multiply two complex numbers in polar form, multiply their moduli and add their arguments:
$$z_1 z_2 = (r_1 \operatorname{cis} \theta_1)(r_2 \operatorname{cis} \theta_2) = r_1 r_2 \operatorname{cis}(\theta_1 + \theta_2)$$

Division

To divide two complex numbers in polar form, divide their moduli and subtract their arguments:
$$\frac{z_1}{z_2} = \frac{r_1 \operatorname{cis} \theta_1}{r_2 \operatorname{cis} \theta_2} = \frac{r_1}{r_2} \operatorname{cis}(\theta_1 - \theta_2)$$

Properties of Conjugates and Inverses

• Conjugate: If $z = r \operatorname{cis} \theta$, then $\bar{z} = r \operatorname{cis}(-\theta)$.
• Reciprocal: $\frac{1}{z} = \frac{1}{r} \operatorname{cis}(-\theta)$.

KEY TAKEAWAY: When multiplying or dividing, always check if the resulting argument is still in the principal range $(-\pi, \pi]$. If it exceeds this range, add or subtract $2\pi$ until it fits.

4. Summary of Special Cases

Some complex numbers on the axes can be written in polar form by inspection:

• Positive Real Axis: $z = a$ (where $a > 0$) $\implies z = a \operatorname{cis} 0$
• Negative Real Axis: $z = -a$ (where $a > 0$) $\implies z = a \operatorname{cis} \pi$
• Positive Imaginary Axis: $z = bi$ (where $b > 0$) $\implies z = b \operatorname{cis}\left(\frac{\pi}{2}\right)$
• Negative Imaginary Axis: $z = -bi$ (where $b > 0$) $\implies z = b \operatorname{cis}\left(-\frac{\pi}{2}\right)$

VCAA FOCUS: Questions often involve converting a result from a multiplication or division back into Cartesian form using exact values (e.g., $\sin \frac{\pi}{6}, \cos \frac{\pi}{4}$). Fluency with the unit circle is essential for these marks.