If-1-i-then-what-is-the-value-of-i-

Question Number 14002 by Joel577 last updated on 26/May/17

If \sqrt{- 1} = i, then

what is the value of \sqrt{i} ?

Answered by ajfour last updated on 26/May/17

i = e^{i (\frac{π}{2} + 2 n π)}

\sqrt{i} = e^{i (\frac{π}{4} + n π)}

= \cos (\frac{π}{4} + n π) + i \sin (\frac{π}{4} + n π)

i f n i s e v e n w e h a v e

\sqrt{i} = \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}}

i f n i s o d d,

\sqrt{i} = - (\frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}})

s o \sqrt{i} = \pm (\frac{1 + i}{\sqrt{2}}) .

Commented by tawa tawa last updated on 26/May/17

God bless you sir \dots . .

Answered by RasheedSindhi last updated on 26/May/17

Let \sqrt{i} = x + iy

{(x + iy)}^{2} = i

x^{2} + 2 xyi - y^{2} = 0 + i

x^{2} - y^{2} = 0 \land 2 xy = 1

(x + y) (x - y) = 0 \land xy = \frac{1}{2}

(x = - y ∣ x = y) \land xy = 1 / 2

∙ If x = y \Rightarrow x^{2} = 1 / 2 \Rightarrow x = \pm \frac{1}{\sqrt{2}}

y = \pm \frac{1}{\sqrt{2}}

∙ If x = - y \Rightarrow - x^{2} = 1 / 2 \Rightarrow x = \pm \frac{i}{\sqrt{2}}

y = \mp \frac{i}{\sqrt{2}}

∙ \sqrt{i} = x + iy = \pm \frac{1}{\sqrt{2}} + i (\pm \frac{1}{\sqrt{2}})

= \pm \frac{1 + i}{\sqrt{2}}

∙ \sqrt{i} = x + iy = \pm \frac{i}{\sqrt{2}} + i (\mp \frac{i}{\sqrt{2}})

= \pm \frac{i}{\sqrt{2}} + (\mp \frac{i^{2}}{\sqrt{2}}) = \pm \frac{1 + i}{\sqrt{2}}

Hence in both cases

\sqrt{i} = \pm \frac{1 + i}{\sqrt{2}}

Commented by chux last updated on 26/May/17

wow \dots \dots thanks

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