Question

If \( Z_1 \) and \( Z_2 \) are two non-zero complex numbers, then which of the following is not true?

• A. \( |Z_1 Z_2| = |Z_1| |Z_2| \)
• B. \( Z_1 Z_2 = Z_1 \cdot Z_2 \)
• C. \( |Z_1 + Z_2| \geq |Z_1| + |Z_2| \)
• D. \( Z_1 + Z_2 = Z_1 + Z_2 \)

Hint:

In complex number properties, always remember that the modulus of a product is the product of the moduli, and the triangle inequality governs the relationship between the modulus of a sum and the sum of the moduli.

Answer 1

The Correct Option is D

Let's analyze the given options:

1. \( |Z_1 Z_2| = |Z_1| |Z_2| \): This is true. The modulus of a product of two complex numbers is the product of their moduli.
2. \( Z_1 Z_2 = Z_1 \cdot Z_2 \): This is also true. The multiplication of complex numbers is commutative, so the product can be written as either \( Z_1 Z_2 \) or \( Z_1 \cdot Z_2 \).
3. \( |Z_1 + Z_2| \geq |Z_1| + |Z_2| \): This is true based on the triangle inequality for complex numbers, which states that the modulus of the sum of two complex numbers is always less than or equal to the sum of their moduli.
4. \( Z_1 + Z_2 = Z_1 + Z_2 \): This is trivially true. It doesn't represent a non-trivial property of complex numbers; it just states that \( Z_1 + Z_2 \) is equal to itself.

Thus, the statement \( Z_1 + Z_2 = Z_1 + Z_2 \) is not an interesting or valid result in terms of complex number properties, and therefore, it is the answer.