Question

The least positive integer $n$ for which ${\left(\frac{1+i}{1-i}\right)}^n$ represents a real number is

• A. $4$
• B. $8$
• C. $12$
• D. $2$

Answer 1

The correct option is D $2$

${\left(\frac{(1+i)(1+i)}{(1-i)(1+i)}\right)}^n$ (Dividing and multiplying by $(1+i)$)

=$\frac{1}{2^n}$ . $(1+i{)}^{2n}$ -(1)

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

From (1) and (2),

$\frac{1}{2^n}$ . $(1+i{)}^{2n}$ = ${(\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}})}^n$

=$cos$$n\pi /2$$+isin$$n\pi /2$

To be a real number, an imaginary part should be $0$

$\therefore$ $n=2$ so that $sin$ component becomes $0.$

Hence, option D is correct.