First Principles AI

Solution

\begin{aligned} & \boxed{\text{Given: } \left(\frac{1+i}{1-i}\right)^{2n}=1.} \\ \\ & \boxed{\text{First simplify } \frac{1+i}{1-i} \text{ using conjugate } (1+i):} \\ \\ & \frac{1+i}{1-i}\cdot\frac{1+i}{1+i} = \frac{(1+i)^2}{(1-i)(1+i)} \\ \\ & \boxed{\text{Expand numerator:}} \\ \\ & (1+i)^2 = 1+2i+i^2 = 1+2i-1=2i \\ \\ & \boxed{\text{Simplify denominator:}} \\ \\ & (1-i)(1+i)=1^2+1^2=2 \\ \\ & \Rightarrow \frac{1+i}{1-i} = \frac{2i}{2}=i \\ \\ & \boxed{\text{So the condition becomes:}} \\ \\ & i^{2n}=1 \\ \\ & i^{2n}=(i^2)^n = (-1)^n \\ \\ & \boxed{(-1)^n=1 \text{ when } n \text{ is even.}} \\ \\ & \boxed{\text{Least positive even } n = 2.} \end{aligned}

\boxed{n=2}