Properties of Argument

Arguments add under multiplication and subtract under division — modulo 2π. Also: arg(z̄) = −arg(z), and arg(zⁿ) = n·arg(z). These hold as general arguments; for principal arguments, add 2kπ corrections as needed to stay in (−π, π].

Arguments behave like logarithms under multiplication: they add. This is not a coincidence — in Euler's form, the argument literally sits in the exponent.

Setup

Write $z_1$ and $z_2$ in Euler's form:

$$z_1 = r_1 e^{i\theta_1}, \quad r_1 = |z_1|,\; \theta_1 = \arg(z_1)$$ $$z_2 = r_2 e^{i\theta_2}, \quad r_2 = |z_2|,\; \theta_2 = \arg(z_2)$$

Property 1 — Argument of a Product

$$z_1 z_2 = r_1 e^{i\theta_1} \cdot r_2 e^{i\theta_2} = r_1 r_2\, e^{i(\theta_1 + \theta_2)}$$

The argument of $z_1 z_2$ is $\theta_1 + \theta_2$:

$$\arg(z_1 z_2) = \arg(z_1) + \arg(z_2) \pmod{2\pi}$$

The modulo $2\pi$ is necessary because argument is defined up to multiples of $2\pi$. For principal arguments, add $2k\pi$ to land in $(-\pi, \pi]$. $\blacksquare$

Property 2 — Argument of a Quotient

$$\frac{z_1}{z_2} = \frac{r_1 e^{i\theta_1}}{r_2 e^{i\theta_2}} = \frac{r_1}{r_2}\, e^{i(\theta_1 - \theta_2)}$$ $$\arg\!\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2) \pmod{2\pi} \qquad \blacksquare$$

Property 3 — Argument of Conjugate

$$\bar{z} = r e^{-i\theta}$$ $$\arg(\bar{z}) = -\arg(z)$$

Conjugation negates the argument — reflection across the real axis reverses the angle. $\blacksquare$

Property 4 — Argument of a Power

Applying Property 1 repeatedly, or directly from Euler's form:

$$z^n = r^n e^{in\theta} \implies \arg(z^n) = n\,\arg(z) \pmod{2\pi}$$

This is De Moivre's theorem restated in terms of argument. $\blacksquare$

The Modulo Caveat

These properties hold as general arguments (elements of the set $\{\theta + 2k\pi : k \in \mathbb{Z}\}$). For principal arguments $\operatorname{Arg}(z) \in (-\pi, \pi]$, corrections are needed.

Example: $z_1 = e^{i \cdot 3\pi/4}$, $z_2 = e^{i \cdot 3\pi/4}$. Then $\arg(z_1) + \arg(z_2) = 3\pi/2$, which lies outside $(-\pi, \pi]$. The principal argument of the product is $3\pi/2 - 2\pi = -\pi/2$.

In JEE problems, always check whether the sum/difference of arguments needs adjustment to re-enter $(-\pi, \pi]$.