The argument of the product z1 z2 equals which expression?

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Multiple Choice

The argument of the product z1 z2 equals which expression?

Think of each complex number in polar form: z1 = r1 e^{iθ1} and z2 = r2 e^{iθ2}. Multiplying them gives z1 z2 = (r1 r2) e^{i(θ1 + θ2)}. The angle in the product’s polar form is the sum of the two angles, so arg(z1 z2) = θ1 + θ2. That means arg(z1 z2) = arg(z1) + arg(z2) (taken modulo 2π, depending on the convention for the principal value). Note this is undefined if either z1 or z2 is zero.

Think of each complex number in polar form: z1 = r1 e^{iθ1} and z2 = r2 e^{iθ2}. Multiplying them gives z1 z2 = (r1 r2) e^{i(θ1 + θ2)}. The angle in the product’s polar form is the sum of the two angles, so arg(z1 z2) = θ1 + θ2. That means arg(z1 z2) = arg(z1) + arg(z2) (taken modulo 2π, depending on the convention for the principal value). Note this is undefined if either z1 or z2 is zero.

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