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MATH
[1251]

Methods from Polar form

All methods from Polar form.

Back to Polar form

Multiply and divide in polar form

If z1=r1eiθ1z_1=r_1e^{i\theta_1} and z2=r2eiθ2z_2=r_2e^{i\theta_2}, then

z1z2=r1r2ei(θ1+θ2)z_1z_2=r_1r_2e^{i(\theta_1+\theta_2)}

and, for z2≠0z_2\ne0,

z1z2=r1r2ei(θ1−θ2).\frac{z_1}{z_2} =\frac{r_1}{r_2}e^{i(\theta_1-\theta_2)}.

In particular,

(reiθ)−1=1re−iθ.(re^{i\theta})^{-1}=\frac1r e^{-i\theta}.

Geometrically, a product multiplies the moduli and adds the arguments and a quotient divides the moduli and reverses the second rotation.

To see why the angles add, multiply (cos⁡θ1+isin⁡θ1)(cos⁡θ2+isin⁡θ2)(\cos\theta_1+i\sin\theta_1)(\cos\theta_2+i\sin\theta_2) and use the angle-addition identities: its real and imaginary parts become cos⁡(θ1+θ2)\cos(\theta_1+\theta_2) and sin⁡(θ1+θ2)\sin(\theta_1+\theta_2). The quotient rule then follows by multiplying by the inverse.

  1. [1]:Polar form
  2. [2]:Perform product and quotient calculations
  3. [3]:Principal arguments of products and quotients
  4. [4]:De Moivre's formula

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Multiply and divide in polar form