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

Question

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

✓

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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