Free Pearson Edexcel International A-level Mathematics · FP2

De Moivre, roots and complex loci

What you’ll learn

Use exponential and trigonometric forms to calculate powers and roots.
Interpret simple loci and transformations on Argand diagrams.

Understand the idea

Writing $z=r(\cos\theta+i\sin\theta)=re^{i\theta}$ separates magnitude and direction. Multiplication multiplies moduli and adds arguments; division divides moduli and subtracts arguments. De Moivre's theorem extends this pattern to integer powers.

An nth root has modulus $r^{1/n}$ and one of $n$ equally spaced arguments. On an Argand diagram, $|z-a|=k$ is a circle and $|z-a|=|z-b|$ is a perpendicular bisector. A fixed argument describes a ray, with any undefined starting point excluded; substitution also lets you track simple maps such as $w=z^2$ .

Key formulae

$(r(\cos\theta+i\sin\theta))^n=r^n(\cos n\theta+i\sin n\theta)$

$z_k=r^{1/n}e^{i(\theta+2k\pi)/n},\quad k=0,\ldots,n-1$

$e^{i\theta}=\cos\theta+i\sin\theta$

$|z-a|=R$

Complex roots are evenly spaced on a circle

Conditions. Fourth roots of 1 are 1,i,−1,−i, all on the unit circle. The connected diamond joins these four discrete roots as a positional guide; its interior points are not additional roots. Equal coordinate-unit scales.

Real part / dimensionless; Imaginary part / dimensionless.

Each root has modulus 1, and adjacent arguments differ by 2π/4 =π/2. Raising any of the four points to the fourth power gives 1. The circle alone contains infinitely many points, but the root set has exactly four.

Open the interactive lesson to view plotted curves and coordinate tables.

Worked example

Find all solutions of $z^3=8$ .

Write $8=8e^{2k\pi i}$ .
The roots have modulus $2$ and arguments $0,2\pi/3,4\pi/3$ .
Convert each root to Cartesian form using sine and cosine.

Answer: $2$ , $-1+i\sqrt3$ , $-1-i\sqrt3$ .

Open the interactive lesson to try original questions and recall. Build a revision plan.