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Find the Square Root of the Following Complex Number: −I - Mathematics

Find the square root of the following complex number:

i

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Solution

\[\sqrt{z} = \pm \left[ \sqrt{\frac{\left| z \right| + Re\left( z \right)}{2}} + i\sqrt{\frac{\left| z \right| - Re\left( z \right)}{2}} \right] , \text { if Im }(z) > 0\]

\[\sqrt{z} = \pm \left[ \sqrt{\frac{\left| z \right| + Re\left( z \right)}{2}} - i\sqrt{\frac{\left| z \right| - Re\left( z \right)}{2}} \right] , \text { if Im }(z) < 0\]

\[z = - i, Re\left( z \right) = 0, \left| z \right| = 1\]

\[\text { Here, Im }(z) < 0 \]

\[ \therefore \sqrt{z} = \pm \left[ \sqrt{\frac{\left| z \right| + Re\left( z \right)}{2}} - i\sqrt{\frac{\left| z \right| - Re\left( z \right)}{2}} \right]\]

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

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

Concept: Concept of Complex Numbers - Square Root of a Complex Number
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 13 Complex Numbers
Exercise 13.3 | Q 1.9 | Page 39
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