1. 底和顶的定义
分别定义底和顶为:
$$\left \lfloor x \right \rfloor = \mbox{小于等于}x\mbox{的最大整数}$$
$$\left \lceil x \right \rceil = \mbox{大于等于}x\mbox{的最小整数}$$
容易想到,有一些有意思的性质:
$$\left \lfloor x \right \rfloor = x \quad\Leftrightarrow \quad x\mbox{是整数} \quad\Leftrightarrow \quad \left \lceil x \right \rceil =x$$
$$\left \lfloor x \right \rfloor – \left \lceil x \right \rceil =[x\mbox{不是整数}]$$
$$x-1<\left \lfloor x \right \rfloor \leq x\leq \left \lceil x \right \rceil < x+1$$
$$\left \lfloor -x \right \rfloor = -\left \lceil x \right \rceil;\left \lceil -x \right \rceil=-\left \lfloor x \right \rfloor$$
$$\left \lfloor x+n \right \rfloor=\left \lfloor x \right \rfloor+n,\quad n\mbox{为整数}$$
同时也会有四条基本法则
$$\left \lfloor x \right \rfloor=n\Leftrightarrow n \leq x<n+1 \tag{a}$$
$$\left \lfloor x \right \rfloor=n\Leftrightarrow x-1 < n\leq x \tag{b}$$
$$\left \lceil x \right \rceil=n\Leftrightarrow n-1 < x\leq n \tag{c}$$
$$\left \lceil x \right \rceil=n\Leftrightarrow x\leq n<x+1 \tag{d}$$
称\(x\)和\(\left \lfloor x \right \rfloor\)之间的差为\(x\)的分数部分:
$$\left \{ x \right \}=x-\left \lfloor x \right \rfloor$$
这个值的范围是\([0,1)\)