Truncated division
$$ {\displaystyle q=\left[{\frac {a}{n}}\right]} $$
$$ {\displaystyle r=a-n\left[{\frac {a}{n}}\right]} $$
- [] 表示 rounding toward zero，向零取整。
red: quotient (q) and green: remainder (r) as functions of dividend (a), using truncated division
Floored division
$$ {\displaystyle q=\left\lfloor {\frac {a}{n}}\right\rfloor } $$
$$ {\displaystyle r=a-n\left\lfloor {\frac {a}{n}}\right\rfloor} $$
- ⌊⌋ 表示 rounding down，向负无穷取整。
red: quotient (q) and green: remainder (r) as functions of dividend (a), using floored division
$$ {\displaystyle q=\operatorname {sgn}(n)\left\lfloor {\frac {a}{\left|n\right|}}\right\rfloor ={\begin{cases}\left\lfloor {\frac {a}{n}}\right\rfloor &{\text{if }}n>0\\\left\lceil {\frac {a}{n}}\right\rceil &{\text{if }}n<0\\\end{cases}}} $$
$$ {\displaystyle r=a-|n|\left\lfloor {\frac {a}{\left|n\right|}}\right\rfloor } $$
- sgn 表示符号函数。
- ⌊⌋ 表示 rounding down，向负无穷方向取整。
- ⌈⌉ 表示 rounding up，向正无穷方向取整。
red: quotient (q) and green: remainder (r) as functions of dividend (a), using euclidean division
Rounded division
$$ {\displaystyle q=\operatorname {round} \left({\frac {a}{n}}\right)} $$
$$ {\displaystyle r=a-n\operatorname {round} \left({\frac {a}{n}}\right)} $$
- round 表示 rounding half to even，四舍五入取整。
red: quotient (q) and green: remainder (r) as functions of dividend (a), using rounded division
Ceiled division
$$ {\displaystyle q=\left\lceil {\frac {a}{n}}\right\rceil } $$
$$ {\displaystyle r=a-n\left\lceil {\frac {a}{n}}\right\rceil } $$
- ⌈⌉ 表示 rounding up，向正无穷方向取整。
red: quotient (q) and green: remainder (r) as functions of dividend (a), using ceiled division