Question

Write the values of the square root of i.

Answer 1

\[\text { Let the square root of i be } x + iy . \]

\[ \Rightarrow \sqrt{i} = x + iy\]

\[ \Rightarrow i = x^2 + y^2 i^2 + 2ixy\]

\[ \Rightarrow i = x^2 - y^2 + 2ixy \left( \text { Squaring both the sides } \right)\]

\[\text { Comparing both the sides }: \]

\[ x^2 - y^2 = 0 . . . (i) \]

\[\text { and } 2xy = 1 . . . \left( ii \right)\]

\[\text { By equation (ii), we find that x and y are of the same sign } . \]

\[\text {From equation } \left( i \right), \]

\[ x = \pm y\]

\[ \therefore xy = \frac{1}{2}, x^2 = \frac{1}{2}\]

\[x = \pm \frac{1}{\sqrt{2}}, y = \pm \frac{1}{\sqrt{2}}\]

\[ \therefore \sqrt{i} = \pm \frac{1}{\sqrt{2}}\left( 1 + i \right)\]