Question

Let $z_1$ and $z_2$ be two arbitrary complex numbers with non-zero modulus. Which of the following conditions is FALSE?

• A. $|z_1 + z_2|>|z_1| + |z_2|$
• B. $0 \leq |z_1 + z_2|<\infty$
• C. $|z_1 + z_2| \leq |z_1| + |z_2|$
• D. $|z_1 z_2| = |z_1||z_2|$

Hint:

Remember: For complex numbers, the triangle inequality is fundamental: $|z_1 + z_2| \leq |z_1| + |z_2|$. Equality holds only when $z_1$ and $z_2$ are in the same direction.

Answer 1

The Correct Option is A

- We recall the triangle inequality for complex numbers:
\[ |z_1 + z_2| \leq |z_1| + |z_2| \]
This inequality is always true for any two complex numbers. Hence, option (C) is always valid.
- Option (B) states that the modulus of a complex number is always non-negative and finite.
Indeed, for any $z \in \mathbb{C}$, $|z| \geq 0$ and $|z|<\infty$, since modulus measures distance from the origin.
Therefore, (B) is also correct.
- Option (D) uses the multiplicative property of modulus:
\[ |z_1 z_2| = |z_1| \cdot |z_2| \]
This is a standard property of modulus and is always true. Hence, (D) is correct.
- Option (A) suggests that:
\[ |z_1 + z_2|>|z_1| + |z_2| \]
But this contradicts the triangle inequality, which tells us that $|z_1 + z_2|$ can never exceed $|z_1| + |z_2|$.
At best, equality holds when $z_1$ and $z_2$ point in the same direction in the complex plane.
Hence, (A) is FALSE.
Therefore, the false statement is (A).