# Write the Values of the Square Root of −I. - Mathematics

Write the values of the square root of −i.

#### Solution

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

$\text { Squaring both the sides }$

$- i = x^2 + y^2 i^2 + 2ixy$

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

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

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

$\text { From } \left( ii \right),$

$x = \pm y$

$\text { From } \left( i \right),$

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

$\Rightarrow x^2 = \frac{1}{2}$

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

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

Concept: Concept of Complex Numbers - Square Root of a Complex Number
Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 13 Complex Numbers
Q 2 | Page 62