Exercise 825Algebra › Complex numbersL1Added 3 Nov 2026

Argument of a product

#argument#product#additive property

Statement

Show that $\arg(zw)=\arg(z)+\arg(w)$ modulo $2\pi$ .

Hint

Write $z=r_1 e^{i\theta_1}$ , $w=r_2 e^{i\theta_2}$ .

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Trigonometric form. $z=r_1 e^{i\theta_1}$ , $w=r_2 e^{i\theta_2}$ , with $r_i=|z_i|$ , $\theta_i=\arg(z_i)$ .

Product. $zw=r_1r_2 e^{i(\theta_1+\theta_2)}$ , so $|zw|=r_1r_2$ and $\arg(zw)\equiv\theta_1+\theta_2\pmod{2\pi}$ .

Conclusion. $\arg(zw)\equiv\arg(z)+\arg(w)\pmod{2\pi}$ .