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 Some properties of floor function
05-01-2010, 03:01 PM (This post was last modified: 05-02-2010 04:50 PM by elim.)
Post: #1
 elim Moderator Posts: 544 Joined: Feb 2010 Reputation: 0
Some properties of floor function
• $$[x]$$ is called the integer part of $$x$$ or floor function of $$x$$
• $$r(x)=x-[x]$$ is called the fractional part of $$x$$. Clearly $$0 \le r(x) < 1$$
• $$\left \lceil x\right \rceil = -[-x]$$ is called ceiling function of $$x$$
(0) $$x-1<[x]\le x, \ x \le \left\lceil x\right\rceil < x +1$$
(1) $$[x]+[y]\le [x+y] \le [x]+[y]+1$$
(2) $$\left [ \left [ x\right ]/m\right]=\left[x/m\right]$$ where $$x \in \mathbb{R},\quad m\in \mathbb{N}^+$$ this also implies that $$\left[\left[ n/a\right]/b\right] = \left[ n/(ab)\right]$$
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