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Complex numbers
Complex numbers: polar form and De Moivre
CatalogueDPMathematicsMathematics: Analysis and Approaches HLNumber and algebra

Complex numbers: polar form and De Moivre

Use polar form to interpret multiplication, powers and roots as changes of scale and angle.

  • AA HL
  • Complex numbers
  • Papers 1 and 2
  • GDC optional
  • About 25 min

By the end of this lesson

  • convert between Cartesian and polar form
  • multiply and divide using modulus and argument
  • apply De Moivre's theorem to powers and roots

Why this matters. Cartesian form is convenient for addition. Polar form exposes rotation and scaling, which makes multiplication, division, powers and roots much clearer.

See modulus and argument

For $z=x+iy$, the modulus is the distance from the origin and the argument is the directed angle from the positive real axis.

$$z=r(\cos\theta+i\sin\theta)=r\operatorname{cis}\theta,\quad r=\sqrt{x^2+y^2}.$$
Argand-plane interpretationAn Argand plane showing the point z equals three plus three i, its modulus from the origin, and argument pi over four.z = 3 + 3ir = 3√2θ = π/4ReIm
The same complex number can be read as coordinates $(3,3)$ or as modulus $3\sqrt2$ and argument $\pi/4$.

Checkpoint

The point $-1+i\sqrt3$ lies in quadrant II. Which principal argument is correct?
Show explanation

The reference angle is $\pi/3$, and quadrant II gives $\theta=2\pi/3$.

Example 1

Convert to polar form

From coordinates to angle

Write $z=-2+2i$ in polar form using a principal argument.

Modulus
$$r=\sqrt{(-2)^2+2^2}=2\sqrt2$$
Quadrant and angle
$$\tan\theta=-1,\quad z\text{ is in quadrant II},\quad \theta=\frac{3\pi}{4}$$
Polar form
$$z=2\sqrt2\operatorname{cis}\frac{3\pi}{4}.$$

Multiply by rotating and scaling

If $z_1=r_1\operatorname{cis}\theta_1$ and $z_2=r_2\operatorname{cis}\theta_2$, then

$$z_1z_2=r_1r_2\operatorname{cis}(\theta_1+\theta_2),\qquad \frac{z_1}{z_2}=\frac{r_1}{r_2}\operatorname{cis}(\theta_1-\theta_2).$$

Checkpoint

Multiplication by $2\operatorname{cis}(\pi/3)$ does what to every non-zero point?
Show explanation

Moduli multiply by $2$ and arguments increase by $\pi/3$.

Example 2

Use De Moivre

A high power

Find $(1+i)^8$ exactly.

Since $1+i=\sqrt2\operatorname{cis}(\pi/4)$, De Moivre gives

$$(1+i)^8=(\sqrt2)^8\operatorname{cis}(2\pi)=16.$$
Why may the final answer be real?

An argument of $2\pi$ points along the positive real axis, so the imaginary component is zero.

Locate roots

The $n$ roots of $R\operatorname{cis}\phi$ have modulus $R^{1/n}$ and arguments

$$\theta_k=\frac{\phi+2k\pi}{n},\qquad k=0,1,\ldots,n-1.$$

They are equally spaced around a circle.

Checkpoint

The fourth roots of a non-zero complex number are separated by which angle?
Show explanation

Four roots divide a full turn into intervals of $2\pi/4=\pi/2$.

Example 3

Find every cube root

Roots of $-8i$

Write $-8i=8\operatorname{cis}(-\pi/2)$. Its cube roots have modulus $2$ and arguments

$$-\frac{\pi}{6},\quad -\frac{\pi}{6}+\frac{2\pi}{3}=\frac\pi2,\quad -\frac{\pi}{6}+\frac{4\pi}{3}=\frac{7\pi}{6}.$$

Therefore the roots are $2\operatorname{cis}(-\pi/6)$, $2i$, and $2\operatorname{cis}(7\pi/6)$.

Substituting any one of these into $z^3$ returns $-8i$.

In the exam

Quick check

Five questions to check the main decisions from this lesson.

Your score is not saved.

  1. What is $|3-4i|$?
    Show explanation

    $\sqrt{3^2+(-4)^2}=5$.

  2. When two complex numbers are multiplied in polar form, their arguments are…
    Show explanation

    Multiplication adds arguments and multiplies moduli.

  3. Which is the principal argument of $-i$?
    Show explanation

    $-i$ points down the imaginary axis.

  4. What is $(\operatorname{cis}\theta)^5$?
    Show explanation

    De Moivre's theorem multiplies the argument by the power.

  5. How many distinct fifth roots does a non-zero complex number have?
    Show explanation

    The roots are equally spaced with $k=0,1,2,3,4$.

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