Properties of Argument
Introduction
In polar form, complex numbers are expressed as \( r(\cos(\theta) + i\sin(\theta)) \), directly incorporating both magnitude and argument. This representation reveals the angular aspect of complex arithmetic, particularly the arguments' behavior under operations like multiplication and division. In this section and upcoming sections we are going to see some very important and surprising applications of complex numbers.
Properties of Argument
Two most important properties of argument is about operations of multiplication and division.

The argument of the product of two complex numbers is the sum of their arguments.
\[ \text{arg}(z_1z_2) = \text{arg}(z_1) + \text{arg}(z_2) \]Proof:
Consider two complex numbers in polar form, \( z_1 = r_1(\cos\theta_1 + i\sin\theta_1) \) and \( z_2 = r_2(\cos\theta_2 + i\sin\theta_2) \). The product \( z_1z_2 \) is:
\( z_1z_2 = r_1r_2[(\cos\theta_1\cos\theta_2  \sin\theta_1\sin\theta_2) + i(\cos\theta_1\sin\theta_2 + \sin\theta_1\cos\theta_2)] \).
Using trigonometric identities, this can be simplified to:
\( z_1z_2 = r_1r_2[\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)] \).
Hence, \( \text{arg}(z_1z_2) = \theta_1 + \theta_2 \), which establishes the property:
\( \text{arg}(z_1z_2) = \text{arg}(z_1) + \text{arg}(z_2) \).
Geometrical Interpretation of Multiplication
We may interpret this multiplication in a pretty useful way. Multiplication of \( z_2 \) by \( z_1 \) has resulted in the rotation of \( z_2 \) by angle \( \theta_1 \) in the anticlockwise direction and its magnitude has changed from \( r_2 \) to \( r_1r_2 \).
Principal Argument with respect to multiplication
If \( \theta_1 \) and \( \theta_2 \) are principal arguments of two complex numbers \( z_1 \) and \( z_2 \), where both arguments are within the interval \((π, π]\), then adding \( \theta_1 \) and \( \theta_2 \) may result in an angle that does not lie within this principal range. Hence, while \( \arg(z_1z_2) = \arg(z_1) + \arg(z_2) \) holds for arguments in a general sense, the sum \( \arg(z_1) + \arg(z_2) \) may not be the principal argument of the product \( z_1z_2 \).
To find the principal argument of \( z_1z_2 \), one must ensure the result is within the principal range \((π, π]\). If \( \theta_1 + \theta_2 \) falls outside of this range, an adjustment of \( ±2kπ \), for some appropriate \(k\in\mathbb Z\), is necessary to correct it. Thus, the principal argument of the product, denoted simply as \( \arg(z_1z_2) \) when referring to its principal value, is the adjusted sum that falls within the specified interval.

The argument of the quotient of two complex numbers is the difference between the argument of the numerator and the argument of the denominator.
\[\text{arg}\left(\frac{z_1}{z_2}\right) = \text{arg}(z_1)  \text{arg}(z_2).\]Proof:
Let \( z_1 = r_1(\cos\theta_1 + i\sin\theta_1) \) and \( z_2 = r_2(\cos\theta_2 + i\sin\theta_2) \) be two complex numbers in polar form. The quotient \( \frac{z_1}{z_2} \) is:
\[ \frac{z_1}{z_2} = \frac{r_1}{r_2} \left( \frac{(\cos\theta_1 + i\sin\theta_1)(\cos\theta_2  i\sin\theta_2)}{\cos^2\theta_2 + \sin^2\theta_2} \right). \]Using trigonometric identities, we expand the numerator and use the fact that \( \cos^2\theta_2 + \sin^2\theta_2 = 1 \) to simplify to:
\[ \frac{z_1}{z_2} = \frac{r_1}{r_2} \left( \cos(\theta_1  \theta_2) + i\sin(\theta_1  \theta_2) \right). \]Therefore, the argument of the quotient is the difference of the arguments:
\[ \text{arg}\left(\frac{z_1}{z_2}\right) = \theta_1  \theta_2, \]which establishes the property:
\[ \text{arg}\left(\frac{z_1}{z_2}\right) = \text{arg}(z_1)  \text{arg}(z_2). \] 
If \( z \) is a real number, then the argument of \( z \), denoted as \( \arg(z) \), is either \( 0 \) or \( \pi \). The value depends on the sign of \( z \):
 If \( z \) is a positive real number, then \( \arg(z) = 0 \).
 If \( z \) is a negative real number, then \( \arg(z) = \pi \).

If \( z \) is purely imaginary, then \( \arg(z) \) is \( \frac{\pi}{2} \) or \( \frac{\pi}{2} \), depending on the sign of the imaginary part:
 If the imaginary part is positive, \( \arg(z) = \frac{\pi}{2} \).
 If the imaginary part is negative, \( \arg(z) = \frac{\pi}{2} \).

The argument of the product of multiple complex numbers \( z_1, z_2, \ldots, z_n \) is the sum of their individual arguments.
\[ \arg(z_1 z_2 \ldots z_n) = \theta_1 + \theta_2 + \ldots + \theta_n = \arg z_1 + \arg z_2 + \ldots + \arg z_n + 2k\pi, \] 
The argument of a power of a complex number \( z \) raised to an integer exponent \( n \) is \( n \) times the argument of \( z \), modulo \( 2\pi \). This can be stated as:
\[ \arg(z^n) = n \cdot \arg(z) + 2k\pi, \]where \( k \) is an integer such that the argument lies within the principal branch \( (\pi, \pi] \) if we are considering the principal value.
Proof:
To prove that \( \arg(z^n) = n \cdot \arg(z) \), consider \( z \) raised to the \( n \)th power, which is equivalent to \( z \) multiplied by itself \( n \) times:
\[ z^n = z \cdot z \cdot \ldots \cdot z. \]The argument of a product of complex numbers is the sum of the arguments of the individual numbers. Hence,
\[ \arg(z^n) = \arg(z \cdot z \cdot \ldots \cdot z) = \arg(z) + \arg(z) + \ldots + \arg(z), \]where \( \arg(z) \) is added \( n \) times. Therefore,
\[ \arg(z^n) = n \cdot \arg(z), \]taking into account that the resulting argument may need to be adjusted by \( 2k\pi \) to ensure it lies within the principal range if the principal argument is being considered.\

\(arg(\overline{z})=arg(z)\)
Proof:
To prove that \( \arg(\overline{z}) = \arg(z) \), consider the complex conjugate \( \overline{z} \), which can be expressed as:
\[ \overline{z} = \frac{z^2}{z}, \]since \( z^2 = z \overline{z} \).
The argument of a complex number is the angle it makes with the positive real axis. The argument of the product (or quotient) of two complex numbers is the sum (or difference) of their arguments.
So, the argument of \( \overline{z} \) is:
\[ \arg(\overline{z}) = \arg\left(\frac{z^2}{z}\right). \]Since \( z^2 \) is a nonnegative real number, its argument is \( 0 \) (or \( 2k\pi \) for any integer \( k \)), and the argument of \( z \) is \( \arg(z) \). Thus, we have:
\[ \arg(\overline{z}) = \arg(z^2)  \arg(z) = 0  \arg(z) = \arg(z), \]which establishes the property.