The smallest positive integer $ n $ for which $ \left(\frac{1+i}{1-i}\right)^{n}=1 $, is

Question

The smallest positive integer $ n $ for which $ \left(\frac{1+i}{1-i}\right)^{n}=1 $, is,

MARKS App · Accepted Answer

Given, 1 + i 1 - i n = 1 ⇒ 1 + i 1 - i × 1 + i 1 + i n = 1 ⇒ 1 + i 2 1 2 - i 2 n = 1 ⇒ 1 2 + i 2 + 2 i 1 - i 2 n = 1 ⇒ 2 i 2 n = 1 ⇒ i n = 1 Hence, n is an integer multiple of 4 . So, the smallest positive integer value of n is 4 .

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