Multiplication and Division in Polar Form

This lesson explores how multiplication and division of complex numbers work geometrically in polar form. The key insight is that multiplying complex numbers multiplies their moduli and adds their arguments, and dividing subtracts arguments. This geometric perspective is beautiful and powerful.

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Multiplication in Polar Form

Let $z_1 = r_1(\cos\theta_1 + i\sin\theta_1)$z1​=r1​(cosθ1​+isinθ1​) and $z_2 = r_2(\cos\theta_2 + i\sin\theta_2)$z2​=r2​(cosθ2​+isinθ2​).

Their product is:

$z_1 z_2 = r_1 r_2 (\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2))$z1​z2​=r1​r2​(cos(θ1​+θ2​)+isin(θ1​+θ2​))

Proof: Expand using the angle addition formulae:

$z_1 z_2 = r_1 r_2 [(\cos\theta_1 \cos\theta_2 - \sin\theta_1 \sin\theta_2) + i(\sin\theta_1 \cos\theta_2 + \cos\theta_1 \sin\theta_2)]$z1​z2​=r1​r2​[(cosθ1​cosθ2​−sinθ1​sinθ2​)+i(sinθ1​cosθ2​+cosθ1​sinθ2​)]

$= r_1 r_2 [\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)]$=r1​r2​[cos(θ1​+θ2​)+isin(θ1​+θ2​)]

Geometric Interpretation

Multiplying by a complex number $w = r(\cos\theta + i\sin\theta)$w=r(cosθ+isinθ):

• Scales the distance from the origin by $r$r
• Rotates anticlockwise by angle $\theta$θ

This means complex multiplication is a spiral similarity (a combination of scaling and rotation).

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Division in Polar Form

$\frac{z_1}{z_2} = \frac{r_1}{r_2}(\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2))$z2​z1​​=r2​r1​​(cos(θ1​−θ2​)+isin(θ1​−θ2​))

Division divides moduli and subtracts arguments.

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Worked Examples