Complex number operations and De Moivre’s theorem interactive visualisation
Raising a complex number to a power just spins
De Moivre’s theorem reveals that complex number exponentiation is pure geometry — the argument is multiplied by n and the modulus is raised to the power n. Change the power and watch zⁿ wind outward along its spiral.
De Moivre visualisation
n = 3
z = 1.2cis(40°). Watch how zⁿ rotates and scales as you change n
z³ = 1.73cis(120°)
Power (n)
12345678
Angle multiplied by
3
Modulus raised to
r³
In polar form, raising a complex number to a power is remarkably elegant — if z = r·cis(θ), then zⁿ = rⁿ·cis(nθ). The modulus gets raised to the power normally, while the argument simply multiplies by n. This is De Moivre’s theorem, which transforms exponentiation into two simple operations: one multiplication for the angle, one exponentiation for the distance. The geometric interpretation is immediate — the angle is multiplied by n, so the point turns through nθ instead of θ, and its distance from the origin becomes rⁿ. This explains why complex roots are evenly spaced around a circle — if zⁿ = w, then the n solutions differ by rotating 360°/n each time.
Know This
De Moivre’s theorem states that (r·cis θ)ⁿ = rⁿ·cis(nθ), meaning complex exponentiation multiplies the argument by n and raises the modulus to the nth power.