Properties of Principal Argument of Complex Numbers

Argument Properties in Complex Number Operations

How does the argument behave when complex numbers are multiplied or divided?

These properties are very useful, especially when working with polar or exponential forms.

Suppose we have two complex numbers:

Where $\theta_1$ is one argument of $z_1$ and $\theta_2$ is one argument of $z_2$.

Argument of Product

The argument of the product of two complex numbers ($z_1 \times z_2$) is the sum of the arguments of the individual complex numbers.

Mathematically, the relationship between the sets of arguments is:

This means if $\theta_1$ is an argument of $z_1$ and $\theta_2$ is an argument of $z_2$,

then $\theta_1 + \theta_2$ is one of the arguments of $z_1 z_2$.

To find the Principal Argument $\text{Arg}(z_1 z_2)$:

1. Calculate $\text{Arg}(z_1) + \text{Arg}(z_2)$.
2. If the result is already within the range $[0^\circ, 360^\circ)$ (or $[0, 2\pi)$), that is the Principal Argument.
3. If the result is outside the range, add or subtract multiples of $360^\circ$ (or $2\pi$) to bring it into the range.

Argument of Quotient

The argument of the quotient of two complex numbers ($\frac{z_1}{z_2}$, with $z_2 \neq 0$) is the difference between the argument of the numerator complex number ($z_1$) and the argument of the denominator complex number ($z_2$).

The relationship between the sets of arguments:

This means if $\theta_1$ is an argument of $z_1$ and $\theta_2$ is an argument of $z_2$,

then $\theta_1 - \theta_2$ is one of the arguments of $\frac{z_1}{z_2}$.

To find the Principal Argument $\text{Arg}\left(\frac{z_1}{z_2}\right)$:

1. Calculate $\text{Arg}(z_1) - \text{Arg}(z_2)$.
2. If the result is already within the range $[0^\circ, 360^\circ)$ (or $[0, 2\pi)$), that is the Principal Argument.
3. If the result is outside the range, add or subtract multiples of $360^\circ$ (or $2\pi$) to bring it into the range.

Using Argument Properties

Given two complex numbers:

Find the Principal Argument of $z_1 \times z_2$ and $\frac{z_1}{z_2}$.

Solution:

We know the Principal Arguments are:

1. Argument of Product ($z_1 \times z_2$):

   Sum of Principal Arguments:

   $$\text{Arg}(z_1) + \text{Arg}(z_2) = 45^\circ + 95^\circ = 140^\circ$$

   Since $140^\circ$ is already within the range $[0^\circ, 360^\circ)$, then:

   $$\text{Arg}(z_1 \times z_2) = 140^\circ$$

   The set of all arguments is $\{140^\circ + k \cdot 360^\circ : k \in \mathbb{Z}\}$

2. Argument of Quotient ($\frac{z_1}{z_2}$):

   Difference of Principal Arguments:

   $$\text{Arg}(z_1) - \text{Arg}(z_2) = 45^\circ - 95^\circ = -50^\circ$$

   Since $-50^\circ$ is outside the range $[0^\circ, 360^\circ)$, we need to add $360^\circ$:

   $$-50^\circ + 360^\circ = 310^\circ$$

   Therefore:

   $$\text{Arg}\left(\frac{z_1}{z_2}\right) = 310^\circ$$

   The set of all arguments is $\{-50^\circ + k \cdot 360^\circ : k \in \mathbb{Z}\}$, which is the same as $\{310^\circ + k \cdot 360^\circ : k \in \mathbb{Z}\}$.

Exercise

Given $z_a = 4(\cos 120^\circ + i\sin 120^\circ)$ and $z_b = 2(\cos 50^\circ + i\sin 50^\circ)$. Find:

1. $\text{Arg}(z_a \times z_b)$
2. $\text{Arg}\left(\frac{z_a}{z_b}\right)$

Answer Key

Given $\text{Arg}(z_a) = 120^\circ$ and $\text{Arg}(z_b) = 50^\circ$.

1. Argument of Product:

   $$\text{Arg}(z_a) + \text{Arg}(z_b) = 120^\circ + 50^\circ = 170^\circ$$

   Since $170^\circ \in [0^\circ, 360^\circ)$, then $\text{Arg}(z_a z_b) = 170^\circ$.

2. Argument of Quotient:

   $$\text{Arg}(z_a) - \text{Arg}(z_b) = 120^\circ - 50^\circ = 70^\circ$$

   Since $70^\circ \in [0^\circ, 360^\circ)$, then $\text{Arg}\left(\frac{z_a}{z_b}\right) = 70^\circ$.