Mathematics · Ch 2 — Complex Numbers

Euler's Form of the Complex Number

2.7.2

Euler's Form of the Complex Number

Euler's formula identifies the trigonometric bracket in polar form with a complex exponential:

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

Substituting this into the polar form $z=r(\cos\theta+i\sin\theta)$ gives the compact Euler (exponential) form

z=re^{i\theta}.

Note

When performing multiplication, or finding powers or roots of complex numbers, the Euler form can be used exactly as the polar form is — multiplying two such exponentials just adds the exponents, $r_1e^{i\theta_1}\cdot r_2e^{i\theta_2}=r_1r_2\,e^{i(\theta_1+\theta_2)}$ , matching the polar-form multiplication rule from §2.7.1 term for term. This exponential viewpoint is what makes de Moivre's theo …

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