integer function: floors and ceilings

these two are fundamental notations, which are defined as follow(x arbitrary real number):
floor[x]=the greatest integer less than or equal to x;
ceiling[x]=the least integer greater than or equal to x;
there are several properties which accompany these two definition.
if x integer then floor[x]= ceiling[x];
if x isn't integer then ceiling[x]-floor[x]=1;
floor[-x]=-ceiling[x];
ceiling[-x]=-floor[x];
x-1<*floor[x]<=x<=ceiling[x]from the previous rule, we can have a set of extended rules;
floor[x]=n, iff n<=xceiling[x]=n, iff n-1floor[x+n]=floor[x]+n, if n integer;
etc...