Question

Which of the following is not a possible value of \[ \sqrt{i}+\sqrt{-i}? \]

• A. $2i$
• B. $\sqrt2\,i$
• C. $\sqrt2$
• D. $-\sqrt2$

Hint:

For roots of complex numbers, convert to polar form first. It makes finding square roots extremely easy.

Answer 1

The Correct Option is A

Concept: Complex numbers have two square roots. Therefore all possible combinations of square roots of $i$ and $-i$ must be considered.

Step 1: Find square roots of $i$. \[ i=e^{i\pi/2}. \] Hence \[ \sqrt{i} = \pm e^{i\pi/4} = \pm\frac{1+i}{\sqrt2}. \]

Step 2: Find square roots of $-i$. \[ -i=e^{-i\pi/2}. \] Thus \[ \sqrt{-i} = \pm e^{-i\pi/4} = \pm\frac{1-i}{\sqrt2}. \]

Step 3: Compute all possible sums. Possible values are \[ \frac{1+i}{\sqrt2}+\frac{1-i}{\sqrt2} = \sqrt2, \] \[ -\sqrt2, \] \[ \frac{1+i}{\sqrt2}-\frac{1-i}{\sqrt2} = \sqrt2\,i, \] \[ -\sqrt2\,i. \] Thus the possible values are \[ \pm\sqrt2,\quad \pm\sqrt2\,i. \] The value $2i$ never occurs. Therefore, \[ \boxed{2i}. \]