Question

Let $z_1$ and $z_2$ be two non-zero complex numbers. Then

• A. Principle value of arg (z1z2) may not be equal to principle value of argz1 + principle value of argz2
• B. Principal value of arg(z1z2) = principal value of argz1 + principal value of argz2
• C. principal value of arg(z1/z2) = principal value of argz1- principal value of arg z2
• D. principal value of arg (z1/z2) may not be argz1- argz2

Answer 1

The Correct Option is A, D

Let \( z_1 \) and \( z_2 \) be two non-zero complex numbers, and let \( \arg(z) \) represent the argument of the complex number \( z \).

Step 1: Product of complex numbers
For two complex numbers \( z_1 \) and \( z_2 \), we know that: \[ \arg(z_1 z_2) = \arg(z_1) + \arg(z_2) \] However, the principal value of the argument is restricted to the interval \( (-\pi, \pi] \), so when calculating the principal value of \( \arg(z_1 z_2) \), it may not be exactly equal to \( \arg(z_1) + \arg(z_2) \) due to the periodicity of the argument function. Hence, the principal value of \( \arg(z_1 z_2) \) may not equal the sum of the principal values of \( \arg(z_1) \) and \( \arg(z_2) \). 

Step 2: Division of complex numbers
Similarly, for the division of two complex numbers \( z_1 \) and \( z_2 \), we know that: \[ \arg\left( \frac{z_1}{z_2} \right) = \arg(z_1) - \arg(z_2) \] However, due to the periodicity of the argument function, the principal value of the argument of the quotient may not exactly equal the difference of the principal values of \( \arg(z_1) \) and \( \arg(z_2) \). 

Conclusion
Thus, both the principal value of \( \arg(z_1 z_2) \) and \( \arg\left( \frac{z_1}{z_2} \right) \) may not exactly equal the sum or difference of the principal values of \( \arg(z_1) \) and \( \arg(z_2) \), respectively, because the principal values are confined to the interval \( (-\pi, \pi] \), and wrapping may occur.

Answer:

\[ \boxed{\text{The principal value of } \arg(z_1 z_2) \text{ may not be equal to the principal value of } \arg(z_1) + \arg(z_2), \text{ and the principal value of } \arg\left( \frac{z_1}{z_2} \right) \text{ may not be equal to } \arg(z_1) - \arg(z_2).} \]