To find the sum of the complex numbers [tex]\(\sqrt{3} - i\)[/tex] and [tex]\(2 - \sqrt{3} i\)[/tex], we need to add their real and imaginary parts separately.
1. Identify Real and Imaginary Parts:
- For the first complex number [tex]\(\sqrt{3} - i\)[/tex]:
- Real part = [tex]\(\sqrt{3}\)[/tex]
- Imaginary part = [tex]\(-1\)[/tex]
- For the second complex number [tex]\(2 - \sqrt{3} i\)[/tex]:
- Real part = [tex]\(2\)[/tex]
- Imaginary part = [tex]\(-\sqrt{3}\)[/tex]
2. Sum of the Real Parts:
- Real part of the sum = [tex]\(\sqrt{3} + 2\)[/tex]
3. Sum of the Imaginary Parts:
- Imaginary part of the sum = [tex]\(-1 + (-\sqrt{3})\)[/tex]
- Imaginary part of the sum = [tex]\(-1 - \sqrt{3}\)[/tex]
4. Combine the results:
- The sum of the complex numbers is [tex]\( (\sqrt{3} + 2) + (-1 - \sqrt{3})i \)[/tex].
Simplifying:
- The real part simplifies to: [tex]\( 2 + \sqrt{3} \)[/tex]
- The imaginary part simplifies to: [tex]\(-1 - \sqrt{3}\)[/tex]
Thus, the sum is [tex]\( (2 + \sqrt{3}) - (1 + \sqrt{3})i \)[/tex].
So, the correct answer is:
[tex]\[
\boxed{(2+\sqrt{3})-(1+\sqrt{3}) i}
\][/tex]
