Find the smallest positive integer value of n for which (1+i)^n/(1-i)^n-2 is a real number.

Question

Find the smallest positive integer value of n for which \(\frac{(1+i)^n}{(1-i)^{n-2}}\) is a real number.

Answer 1

Let us write the given complex number as Z = \(\frac{(1+i)^n}{(1-i)^{n-2}}\)

Multiplying and dividing with (1-i)² 

We know that i^2k is real for k≥0. 

∴ The least positive integral of n is 1.