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 — multiply the angle by n, multiply the distance by rⁿ. Drag the slider to change the power and watch z³ trace its path.
De Moivre Visualisation
n = 3
Watch how z^n rotates and scales as you change n
z³ = 8cis(270°)
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 point rotates n times around the origin while its distance from the origin changes by the factor 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.