How is the solving method to find Square root of 1+i is different from Solving method of to find the Square root of 1-i ?

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Solution

Dear student
$Square root of 1 + i let z = 1 + i . Then Re (z) = 1 and |z| = \sqrt{1^{2} + {(1)}^{2}} = \sqrt{2} So, \sqrt{1 + i} = \pm \{\sqrt{\frac{|z| + Re (z)}{2}} + i \sqrt{\frac{|z| - Re (z)}{2}}\} [As Im (z) > 0] = \pm \{\sqrt{\frac{\sqrt{2} + 1}{2}} + i \sqrt{\frac{\sqrt{2} - 1}{2}}\} and Square root of 1 - i let z = 1 - i . Then Re (z) = 1 and |z| = \sqrt{1^{2} + {(- 1)}^{2}} = \sqrt{2} So, \sqrt{1 - i} = \pm \{\sqrt{\frac{|z| + Re (z)}{2}} - i \sqrt{\frac{|z| - Re (z)}{2}}\} [As Im (z) < 0] = \pm \{\sqrt{\frac{\sqrt{2} + 1}{2}} - i \sqrt{\frac{\sqrt{2} - 1}{2}}\}$
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