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Complex Numbers — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsComplex Numbers1 Mark
Question
Write the values of the square root of -i.
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Answer

Let $\Rightarrow\sqrt-{\text{i}}=\text{x}+\text{iy}$
(Squaring both sides)
$\Rightarrow-\text{i}=\text{x}^2+\text{y}^2\text{i}^2+2\text{ixy}$
$\Rightarrow2\text{xy}=-1 \ ...(\text{i})$
and $\text{x}^2-\text{y}^2=0 \ ...(\text{ii})$
By equation (ii), we find that x and y are of opposite sign.
From equation (ii),
$\text{x}=\pm\text{y}$
From equation (i),
$2(\text{x})(\text{-x})=\frac{-1}{2}$
$\Rightarrow\text{x}=\pm\frac{1}{\sqrt2}$ $\Big[$ Since x and y have opposite signs, $\text{y}=-\frac{1}{\sqrt{2}}$ when $\text{x}=\frac{1}{\sqrt{2}}$ and vice versa $\Big]$
$\therefore \sqrt-{\text{i}}=\pm\frac{1}{\sqrt{2}}(1-\text{i})$

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