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Some properties of floor function
05-01-2010, 03:01 PM
Post: #1
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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