Conference rusure::math

114.0. "Properties of floor" by HARE::STAN () Wed Aug 01 1984 23:50

	Properties of the Floor Function

Let [x] denote the floor of x, that is, the greatest integer
not exceeding x.  I give below some properties of the floor
function, culled from various sources.  These may be useful when
solving other problems, for example, the sqrt(2) recursion problem.

In the following, x and y denote real numbers and m and n denote integers.

1.	x - 1 < [x] <= x

2.	[x] <= x < [x] + 1

3.	[x+n] = [x] + n

4.	[ [x]/n ] = [x/n]		for n>0

5.	(m+1)/n <= [m/n] + 1		for n>0

6.	[2x] - 2[x] = 0 or 1   according as x-[x]<1/2 or >= 1/2

7.	If 0<a<1, then [x]-[x-a]= 0 or 1 according as x-[x]>=a or < a

8.	[x]+[y] equals either [x+y] or [x+y]-1

9.	[x]-[y] equals either [x-y] or [x-y]+1

10.	[2x] + [2y] >= [x] + [x+y] + [y]

11.	[ [nx]/n ] = [x]

12.	[x] + [x+1/n] + [x+2/n] + ... + [x+(n-1)/n] = [nx]

13.	[x] + [y] <= [x+y] <= [x] + [y] + 1

14.	[x][y] <= [xy] <= [x][y] + [x] + [y]	for x>0, y>0

15.	[n/2] - [-n/2] = n

16.	[x^(1/n)] = [ [x]^(1/n) ]		for x>0

17.	[x/n] + [(x+1)/n] + [(x+2)/n] + ... + [(x+n-1)/n] = [x]

18.	[ sqrt(2) [(1+1/sqrt(2))n + 1/2] ] = [(1+sqrt(2))n]