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WASpecialist MathematicsSyllabus dot point

Why does multiplying complex numbers rotate and scale, and how does polar form make this obvious?

Multiply and divide complex numbers in polar form, interpreting the result as rotation and scaling

WACE Specialist Unit 3 polar form arithmetic: multiplying moduli and adding arguments, dividing moduli and subtracting arguments, the geometric interpretation as rotation and dilation, and the conjugate in polar form, with a worked example.

Generated by Claude Opus 4.76 min answer

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  1. What this dot point is asking
  2. The multiplication rule
  3. The division rule
  4. Geometric interpretation
  5. Why polar form helps

What this dot point is asking

SCSA wants you to combine complex numbers in modulus-argument form and read off the geometric meaning: multiplication as a rotation plus dilation about the origin.

The multiplication rule

Write z1=r1 cis θ1z_1 = r_1\,\text{cis}\,\theta_1 and z2=r2 cis θ2z_2 = r_2\,\text{cis}\,\theta_2. Expanding the product and using the angle-sum identities cos⁡(θ1+θ2)\cos(\theta_1 + \theta_2) and sin⁡(θ1+θ2)\sin(\theta_1 + \theta_2) gives

So ∣z1z2∣=∣z1∣ ∣z2∣|z_1 z_2| = |z_1|\,|z_2| and arg⁡(z1z2)=arg⁡z1+arg⁡z2\arg(z_1 z_2) = \arg z_1 + \arg z_2 (up to a multiple of 2π2\pi, since the result may need adjusting back into the principal range).

The division rule

Dividing reverses the process:

z1z2=r1r2 cis(θ1−θ2).\frac{z_1}{z_2} = \frac{r_1}{r_2}\,\text{cis}(\theta_1 - \theta_2).

Moduli divide and arguments subtract. As a special case 1z=1r cis(−θ)\tfrac{1}{z} = \tfrac{1}{r}\,\text{cis}(-\theta), the reciprocal having reciprocal modulus and negated argument.

Geometric interpretation

Two clean cases worth memorising: multiplying by i=cis π2i = \text{cis}\,\tfrac{\pi}{2} rotates by a quarter turn anticlockwise with no scaling, and multiplying by −1=cis π-1 = \text{cis}\,\pi rotates by a half turn. The conjugate in polar form is r cis θ‾=r cis(−θ)\overline{r\,\text{cis}\,\theta} = r\,\text{cis}(-\theta), a reflection in the real axis.

Why polar form helps

Cartesian multiplication is algebraically heavy, but polar form reduces it to one product and one sum. This is also the engine behind de Moivre's theorem, which is just repeated multiplication of equal factors.