If $a,b\in\mathbb R$ with $b\ne0$, how precisely is division with remainder of $a$ by $b$ defined?
It should be something like $a=bc+r$ for some $c\in\mathbb Z$ and $r\in[0,|b|)$.
However, I'm interested in the case $b=1$. If, for example, $a=-0.1$, I would expect that $a\operatorname{mod}1=0.9$. So, $c=-1$ and $r=0.9$. However, $c=0$ and $r=-0.1$ would also been possible if we would allow $r$ to be negative. Does it matter? And is $c$ unique?
EDIT: I guess we need to choose $c=\lfloor\frac ab\rfloor$.