Question

Write the values of the square root of −i.

Answer 1

\[\text { Let } \sqrt{- i} = x + iy\]

\[\text { Squaring both the sides }\]

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

\[ \Rightarrow 2xy = - 1 . . . \left( i \right)\]

\[\text { and }x^2 - y^2 = 0 . . . \left( ii \right)\]

\[\text { Equation } \left( ii \right)\text {  shows that x and y are of opposite sign }. \]

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

\[x = \pm y\]

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

\[2\left( x \right)\left( - x \right) = \frac{- 1}{2}\]

\[ \Rightarrow x^2 = \frac{1}{2}\]

\[ \Rightarrow x = \pm \frac{1}{\sqrt{2}} \left[ \text { Since x and y have opposite signs, y } = - \frac{1}{\sqrt{2}} \text { when} x = \frac{1}{\sqrt{2}}\text { and vice versa } \right]\]

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