Gujarat 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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