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 the modulus properties:
- Multiplication and division are perfectly preserved: \(|z_1 z_2| = |z_1||z_2|\) and \(\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}\).
- Addition and subtraction follow the triangle inequality: \(|z_1 + z_2| \le |z_1| + |z_2|\) and \(|z_1 - z_2| \ge ||z_1| - |z_2||\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question tests basic properties of the modulus (or absolute value) of complex numbers.
The modulus of a complex number \(z = x + iy\) represents its distance from the origin in the complex plane, defined as \(|z| = \sqrt{x^2 + y^2}\).

Step 2: Detailed Explanation:
Let us analyze the standard properties of the modulus of complex numbers:
1. The modulus of the product of two complex numbers is always equal to the product of their individual moduli:
\[ |z_1 z_2| = |z_1||z_2| \]
We can prove this by letting \(z_1 = r_1 e^{i\theta_1}\) and \(z_2 = r_2 e^{i\theta_2}\) in polar form:
\[ z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)} \]
Taking the modulus of both sides gives:
\[ |z_1 z_2| = |r_1 r_2 e^{i(\theta_1 + \theta_2)}| = r_1 r_2 = |z_1||z_2| \]
Thus, Option (C) is a fundamental, always-true mathematical identity.
Let us examine why the other options are incorrect:
- For division, the modulus of the quotient is also equal to the quotient of the moduli:
\[ \left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|} \quad (\text{for } |z_2| \ne 0) \]
This directly invalidates both Option (A) and Option (B), which state that they are unequal or strictly less than.
- Option (D) states that the modulus of the product is strictly less than the product of the moduli, which is mathematically false because they are always equal.
Therefore, only Option (C) is correct.

Step 3: Final Answer:
The correct option is (C).