Find the square root of the following complex numbers: -i - Vidyadip

Question

Find the square root of the following complex numbers: $-\text{i}$

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Answer

Let $\text{z}=-\text{i}$ Then $|\text{z}|=|-\text{i}|$ $=|-1|\times|\text{i}| \ (\because|\text{z}_1\text{z}_2|=|\text{z}_1|\times|\text{z}_2|)$ $=1\times\text{i} \ (\because|\text{i}|=1)$ $=1$ $\therefore\sqrt{-\text{i}}=\pm\Bigg\{\sqrt{\frac{1+0}{2}}-\text{i}\sqrt{\frac{1-0}{2}}\Bigg\} \ (\because\text{y}<0)$ $=\pm\Big\{\frac{1}{\sqrt{2}}-\frac{\text{i}}{\sqrt{2}}\Big\}$ $=\pm\frac{1}{\sqrt{2}}(1-\text{i})$

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