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Complex Numbers — MATHS STD 11 Science — Question

Rajasthan 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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