Gujarat BoardEnglish MediumSTD 11 ScienceMATHSComplex Numbers1 Mark
Question
Write the values of the square root of i.
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Answer
Let the square root of i be $\text{x}+\text{iy}.$ $\Rightarrow\sqrt{\text{i}}=\text{x}+\text{iy}$ $\Rightarrow\text{i}=\text{x}^2+\text{y}^2\text{i}^2+2\text{ixy}$ $\Rightarrow\text{i}=\text{x}^2-\text{y}^2+2\text{ixy}$ (Squaring both sides) Comparing both the sides: $\text{x}^2-\text{y}^2=0 \ ...(\text{i})$ and $2\text{xy}=1 \ ...(\text{ii})$ By equation (ii), we find that x and y are of the same sign. From equation (i), $\text{x}=\pm\text{y}$ $\therefore\text{xy}=\frac{1}{2},\text{x}^2=\frac{1}{2}$ $\text{x}=\pm\frac{1}{2},\text{y}=\pm\frac{1}{\sqrt2}$ $\therefore \sqrt{\text{i}}=\pm\frac{1}{\sqrt{2}}(1+\text{i})$
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