Find:-

i)Whole root of(1-i)

ii)whole root of (1+i)

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Question

Find:-
i)Whole root of(1-i)
ii)whole root of (1+i)

Aakash Sharma · Accepted Answer

$Let \sqrt{1 - i} = x + i y \Rightarrow 1 - i = (x + i y)^{2} = x^{2} - y^{2} + 2 i x y \Rightarrow x^{2} - y^{2} = 1, 2 x y = - 1 ... (1)$

Now

${(x^{2} + y^{2})}^{2} = (x^{2} - y^{2})^{2} + 4 x^{2} y^{2} = 1 + 1 = 2 \Rightarrow x^{2} + y^{2} = \sqrt{2} ... (2)$

from (1) and (2) we get

$x^{2} = \frac{\sqrt{2} + 1}{2}, y^{2} = \frac{\sqrt{2} - 1}{2} \Rightarrow x = \pm \sqrt{\frac{\sqrt{2} + 1}{2},} y = \mp \sqrt{\frac{\sqrt{2} - 1}{2}} ∴ \sqrt{1 - i} = \pm (\sqrt{\frac{\sqrt{2} + 1}{2}} - \sqrt{\frac{\sqrt{2} - 1}{2}} i)$

Similarly

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

Find:- i)Whole root of(1-i) ii)whole root of (1+i)

Find:-

i)Whole root of(1-i)

ii)whole root of (1+i)