What is the intuition behind medians (and quartiles) being defined as the value of the $(n+1)/p$ th observation?

Question

When calculating the median of an un-grouped distribution, we take the $(n+1)/2$ the value.

For quartiles it's $\text{I} \cdot (n+1)/4$.

For deciles it's $\text{I} \cdot (n+1)/10$.

What is the intuition behind the $1$ added to $n$? More generally, why do we find the midpoint of numbers $1$ to $n$ at $(n+1)/2$?