Question

Given $ Z_1 = \frac{1}{2} + \frac{\sqrt{3}}{2}i, \quad Z_2 = -\frac{1}{2} - \frac{\sqrt{3}}{2}i, \quad \text{and} \quad w = Z_1 + Z_2, \quad \text{find} \, w. $

• A. \( w = 0 \)
• B. \( w = \sqrt{3}i \)
• C. \( w = 1 \)
• D. \( w = \sqrt{2} \)

Hint:

When adding complex numbers, simply add the corresponding real and imaginary parts separately.

Answer 1

The Correct Option is A

We are given two complex numbers: \[ Z_1 = \frac{1}{2} + \frac{\sqrt{3}}{2}i, \quad Z_2 = -\frac{1}{2} - \frac{\sqrt{3}}{2}i \] The problem asks us to find the sum \( w = Z_1 + Z_2 \). First, add the real parts of \( Z_1 \) and \( Z_2 \): \[ \frac{1}{2} + \left(-\frac{1}{2}\right) = 0 \] Next, add the imaginary parts of \( Z_1 \) and \( Z_2 \): \[ \frac{\sqrt{3}}{2}i + \left(-\frac{\sqrt{3}}{2}i\right) = 0i \] So, the sum \( w = Z_1 + Z_2 \) is: \[ w = 0 + 0i = 0 \]
Thus, the correct answer is \( 0 \).