Question

If \( z_1, z_2, \ldots, z_n \) are complex numbers such that \( |z_1| = |z_2| = \ldots = |z_n| = 1 \), then \( |z_1 + z_2 + \ldots + z_n| \) is equal to:

• A. \( |z_1 z_2 z_3 \ldots z_n| \)
• B. \( |z_1| + |z_2| + \ldots + |z_n| \)
• C. \( \left| \frac{1}{z_1} + \frac{1}{z_2} + \ldots + \frac{1}{z_n} \right| \)
• D. \( n \)

Hint:

Always look for symmetry in complex number problems involving the unit circle. The identity $z^{-1} = \bar{z}$ is a powerful tool for simplifying sums of complex numbers.

Answer 1

The Correct Option is C

Concept: For any complex number \( z \) such that \( |z| = 1 \), a fundamental property is that the conjugate \( \bar{z} \) is equal to the reciprocal \( \frac{1}{z} \). This follows from the definition \( |z|^2 = z \bar{z} = 1 \).

Step 1: Use the property of complex numbers on the unit circle. Since \( |z_k| = 1 \) for all \( k = 1, 2, \ldots, n \), we know that: \[ z_k \bar{z}_k = |z_k|^2 = 1 \implies \bar{z}_k = \frac{1}{z_k} \]

Step 2: Transform the given expression. We want to evaluate \( |z_1 + z_2 + \ldots + z_n| \). Using the property that the modulus of a complex number is equal to the modulus of its conjugate (i.e., \( |z| = |\bar{z}| \)): \[ |z_1 + z_2 + \ldots + z_n| = |\overline{z_1 + z_2 + \ldots + z_n}| \] \[ = |\bar{z}_1 + \bar{z}_2 + \ldots + \bar{z}_n| \]

Step 3: Substitute the reciprocal property. Substitute \( \bar{z}_k = \frac{1}{z_k} \) back into the expression: \[ |\bar{z}_1 + \bar{z}_2 + \ldots + \bar{z}_n| = \left| \frac{1}{z_1} + \frac{1}{z_2} + \ldots + \frac{1}{z_n} \right| \] Thus, \( |z_1 + z_2 + \ldots + z_n| = \left| \frac{1}{z_1} + \frac{1}{z_2} + \ldots + \frac{1}{z_n} \right| \).