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Question
Find the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
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Solution
`((1+i)/(1-i))^n`
Simplify the base
`(1+i)/(1-i)`
`= ((1+i)(1+i))/((1-i)(1+i)) = (1+i)^2/(1^2-i^2)`
(1 + i)2 = 1 + 2i + i2 = 1 + 2i − 1 = 2i
12 − i2 = 1 − (−1) = 2
`(1+i)/(1-i) = (2i)/2 = i`
`((1+i)/(1-i))^n = i^n`
Find the least positive n
n = 2
n = 4 ...
n = 2
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