To determine the sum of the complex numbers [tex]\( z_1 \)[/tex] and [tex]\( z_2 \)[/tex], follow these steps:
Given:
[tex]\[ z_1 = 2 + \sqrt{3} i \][/tex]
[tex]\[ z_2 = 1 - \sqrt{3} i \][/tex]
Step 1: Write down the formula for the sum of two complex numbers.
[tex]\[ z_{\text{sum}} = z_1 + z_2 \][/tex]
Step 2: Substitute the given values into the sum formula.
[tex]\[ z_{\text{sum}} = (2 + \sqrt{3} i) + (1 - \sqrt{3} i) \][/tex]
Step 3: Combine the real parts of [tex]\( z_1 \)[/tex] and [tex]\( z_2 \)[/tex]:
[tex]\[ \text{Real part of } z_{\text{sum}} = 2 + 1 = 3 \][/tex]
Step 4: Combine the imaginary parts of [tex]\( z_1 \)[/tex] and [tex]\( z_2 \)[/tex]:
[tex]\[ \text{Imaginary part of } z_{\text{sum}} = (\sqrt{3}) i - (\sqrt{3}) i = 0 i \][/tex]
Step 5: Combine the real and imaginary parts to write the final result:
[tex]\[ z_{\text{sum}} = 3 + 0 i = 3 \][/tex]
Thus, the sum of [tex]\( z_1 \)[/tex] and [tex]\( z_2 \)[/tex] is [tex]\( 3 \)[/tex].
So, the correct answer is:
[tex]\[ \boxed{3} \][/tex]
