# Why is $e^{i2\pi Nf_snT}=1$?

Why is $e^{i2\pi Nf_snT}=1$ in snippet from book?

Because $Tf{_s}$=1 and $N$ and $n$ are integers. So the exponent becomes 2$\pi$ times an integer, say $K$. But $e{^{i\theta}} = \cos(\theta)+i\sin(\theta)$. Therefore $e{^{i2K\pi}} = 1$.
A try without maths: $x(t)$ gives you your location in the complex plane at time $t$.
The answer is: because when you walk in circles ($2\pi$) an integer number of times, whatever your rotational speed, or the length of your steps, you finally up end at the same place as in the beginning, like in the latin palindromic sentence:
• It's the first time I see the ecce in that palindrome. Apr 9, 2016 at 14:27