Step 1 : Understanding the Question:
This problem explores the properties of complex numbers that lie on the unit circle in the complex plane. Given that all the complex numbers have a modulus of 1, we want to find an equivalent expression for the modulus of their sum.
Step 2 : Key Formulas and Approach:
We can solve this by utilizing the properties of complex conjugates and moduli. For any complex number \( z \):
\[ |z|^2 = z \bar{z} \]
If a complex number lies on the unit circle, then \( |z| = 1 \), which implies:
\[ z \bar{z} = 1 \implies \bar{z} = \frac{1}{z} \]
We also use the property that the modulus of a complex number is equal to the modulus of its conjugate:
\[ |z| = |\bar{z}| \]
Step 3 : Detailed Explanation:
Given that \( |z_k| = 1 \) for all \( k = 1, 2, \ldots, n \), we can write the conjugate of each number as its reciprocal:
\[ \bar{z}_k = \frac{1}{z_k} \]
We want to analyze the expression \( |z_1 + z_2 + \ldots + z_n| \). Since the modulus of a complex sum is equal to the modulus of its overall conjugate:
\[ |z_1 + z_2 + \ldots + z_n| = |\overline{z_1 + z_2 + \ldots + z_n}| \]
Use the property that the conjugate of a sum is equal to the sum of the individual conjugates:
\[ |\overline{z_1 + z_2 + \ldots + z_n}| = |\bar{z}_1 + \bar{z}_2 + \ldots + \bar{z}_n| \]
Substitute \( \bar{z}_k = \frac{1}{z_k} \) into this expression:
\[ |\bar{z}_1 + \bar{z}_2 + \ldots + \bar{z}_n| = \left| \frac{1}{z_1} + \frac{1}{z_2} + \ldots + \frac{1}{z_n} \right| \]
This shows that \( |z_1 + z_2 + \ldots + z_n| \) is mathematically equivalent to \( \left| \frac{1}{z_1} + \frac{1}{z_2} + \ldots + \frac{1}{z_n} \right| \).
Step 4 : Final Answer:
The given expression is equal to \( \left| \frac{1}{z_1} + \frac{1}{z_2} + \ldots + \frac{1}{z_n} \right| \), which corresponds to option (C).