Product and power in exponential form

In exponential form the product, quotient and power reduce to operations on the moduli and the arguments.

Property — Product and power in exponential form

Let $z_1=\rho_1 e^{i\theta_1}$ and $z_2=\rho_2 e^{i\theta_2}$. Then: $z_1 \cdot z_2 = \rho_1\rho_2\,e^{i(\theta_1+\theta_2)}, \qquad \frac{z_1}{z_2} = \frac{\rho_1}{\rho_2}\,e^{i(\theta_1 - \theta_2)}.$ And for raising to a power (De Moivre’s formula): $z^n = \rho^n\,e^{in\theta} = \rho^n\bigl(\cos(n\theta) + i\sin(n\theta)\bigr).$

Geometric interpretation. Multiplying by a complex number corresponds to rotating (by $\theta$) and rescaling (by $\rho$). If $|z_2|=1$, the multiplication is a pure rotation. It is precisely this interpretation that underlies representing the transformations of the plane with complex numbers.

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Topics: Complex numbers
Concepts: De Moivre · Exponential form
Methods: De Moivre · Numeri complessi polare
Skills: Calculate · Use formulae
People: Abraham de Moivre