I have come across the following formula of the inverse DFT:
$x[n] = \frac{1}{N} \sum\limits_{k=0}^{N-1}X[k] e^{j \frac{2 \pi k n}{N}}$
The formula of the normal DFT seems intuitive to me as far as my understanding goes but I can't wrap my head around the inverse.
I have read here that switching the sign for positive is not even necessary and that the $1/N$ can even be omitted, which makes even less sense to me.
So I actually can't see any point of switching the sign at all, since it is only the direction in which we are moving along the complex circle. And moving along this or that direction seems quite similiar to me.
Also what is the point of $1/N$ if it might not be needed? I mean when it not needed, doing one DFT and IDFT aftwards, would have different results. Also the fact that we can use $\frac{2 \pi k n}{N}$ in forwards and backwards puzzles me a little.
So what would be an intuitive understanding here?
