I want to start on Dedekind sums tonight and so my first questions is why is the fractional part defined as $(0)$ rather than what we would commonly define it as, shown below as $(1)$.
The difference is obvious, so it should go without saying I am looking for deeper understanding in an algebraic sense.
$$((x))=\cases{x- \lfloor x \rfloor -1/2&$x \in \mathbb R \backslash \mathbb Z$\cr 0&$x \in \mathbb Z$\cr} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (0)$$
$$((x))=\cases{x- \lfloor x \rfloor &$x \in \mathbb R \backslash \mathbb Z$\cr 0&$x \in \mathbb Z$\cr} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\ (1)$$
the first definition was found here