Question

Let \(z_1\) and \(z_2\) be two complex numbers, then :

• A. \(\left|\frac{z_1}{z_2}\right| < \frac{|z_1|}{|z_2|}\) for \(|z_2| \ne 0\)
• B. \(\left|\frac{z_1}{z_2}\right| \ne \frac{|z_1|}{|z_2|}\)
• C. \(|z_1 z_2| = |z_1||z_2|\)
• D. \(|z_1 z_2| < |z_1||z_2|\)

Hint:

Remember that the modulus operation is multiplicative and distributive over multiplication and division:
- \(|z_1 z_2| = |z_1| |z_2|\)
- \(\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}\)
However, it is NOT distributive over addition/subtraction, where the triangle inequalities apply:
- \(|z_1 + z_2| \le |z_1| + |z_2|\)
- \(|z_1 - z_2| \ge ||z_1| - |z_2||\)

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question tests the fundamental properties of the modulus of complex numbers.
We are asked to identify the correct identity concerning the modulus of two complex numbers \(z_1\) and \(z_2\).
Key Formula or Approach:
For any complex number \(z = x + iy\), the modulus is defined as \(|z| = \sqrt{x^2 + y^2}\).
Important algebraic properties of the modulus include:
- Product Property: \(|z_1 z_2| = |z_1| |z_2|\)
- Quotient Property: \(\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}\) (for \(z_2 \neq 0\))

Step 2: Detailed Explanation:

• Let us represent the complex numbers \(z_1\) and \(z_2\) in polar form:
\[ z_1 = r_1 e^{i\theta_1} \quad \text{and} \quad z_2 = r_2 e^{i\theta_2} \]
where \(r_1 = |z_1|\) and \(r_2 = |z_2|\) are their respective moduli.

• Now, we calculate the product of \(z_1\) and \(z_2\):
\[ z_1 z_2 = (r_1 e^{i\theta_1}) (r_2 e^{i\theta_2}) = (r_1 r_2) e^{i(\theta_1 + \theta_2)} \]

• Taking the modulus of both sides of the product:
\[ |z_1 z_2| = |(r_1 r_2) e^{i(\theta_1 + \theta_2)}| \]
Since the modulus of \(e^{i\phi}\) is always 1 for any real angle \(\phi\), we get:
\[ |z_1 z_2| = r_1 r_2 \]

• Substituting the values of \(r_1\) and \(r_2\) back into the equation:
\[ |z_1 z_2| = |z_1| |z_2| \]

• Let us evaluate the options:
- Option (A) claims inequality for the quotient, which is incorrect as they are equal.
- Option (B) says they are not equal, which is incorrect.
- Option (D) suggests the product modulus is strictly less, which is also incorrect.
- Option (C) states the exact modulus product property: \(|z_1 z_2| = |z_1||z_2|\).

Step 3: Final Answer:
The correct property is \(|z_1 z_2| = |z_1||z_2|\).