How Do You Multiply and Divide Complex Numbers in Polar Form?

To multiply complex numbers in polar form, multiply their moduli and add their arguments. To divide them, divide their moduli and subtract their arguments.

This is IB Mathematics: Analysis and Approaches HL, sub-topic AHL 1.13: Polar and Euler form. It can be assessed on Paper 1, Paper 2, or HL Paper 3.

Write two complex numbers as

$$z_1=r_1\operatorname{cis}\theta_1, \qquad z_2=r_2\operatorname{cis}\theta_2,$$

where $r_1,r_2$ are the moduli, $\theta_1,\theta_2$ are the , and .

These rules follow from the geometric interpretation of polar form. A modulus represents a scale factor, while an argument represents a rotation about the origin. Multiplication combines the scale factors and rotations; division reverses the second scale factor and rotation.

Algebraically, multiplication uses the angle-addition identities. Expanding $\operatorname{cis}\theta_1\operatorname{cis}\theta_2$ gives $\cos ({\theta }_1+{\theta }_2)+i\sin ({\theta }_1+{\theta }_2)$, which is . Division is multiplication by the reciprocal: . This explains both the modulus division and argument subtraction.

For example, let

$$z_1=2\operatorname{cis}\left(\frac{\pi}{3}\right), \qquad z_2=3\operatorname{cis}\left(-\frac{\pi}{6}\right).$$

For multiplication,

$$z_1z_2=2(3)\operatorname{cis}\left(\frac{\pi}{3}-\frac{\pi}{6}\right)=6\operatorname{cis}\left(\frac{\pi}{6}\right).$$

For division,

$$\frac{z_1}{z_2}=\frac{2}{3}\operatorname{cis}\left(\frac{\pi}{3}+\frac{\pi}{6}\right)=\frac{2}{3}\operatorname{cis}\left(\frac{\pi}{2}\right).$$

In this example, multiplication scales by $3$ and rotates by $-\pi/6$ relative to $z_1$; division scales by $1/3$ and rotates by $\pi/6$.

Notice that subtracting the negative argument in the division produces addition. Arguments differing by an integer multiple of $2\pi$ represent the same complex number. Therefore, adjust the final argument when a specified principal range is required, commonly $-\pi<\theta\leq\pi$.

A common misconception is to add or subtract the moduli. This is incorrect: moduli are multiplied or divided, whereas arguments are added or subtracted. The same rules apply in Euler form because $r\operatorname{cis}\theta=re^{i\theta}$.

Exam technique: Show the modulus calculation and argument calculation separately before stating the final polar or Euler form. Use exact angles on Paper 1, and ensure that your GDC is in radian mode on Papers 2 and 3.