how to prove that floor(-x) = -ceiling(x) and that -floor(x) = ceiling(-x) show steps
Let x= k+p where k is an integer and 0<=p<1 (obviously k is first integer less than x) floor(-x)=floor(-k-p) and since k or -k is an integer we can take it out of floor(or ceiling) floor(-x)=-k +floor(-p) floor(-p)=-1 when -1<-p<0 and floor(-p)=0 when p=0 -ceiling(x)=-ceiling(k+p)=-k - ceiling(p) ceiling(p)=1 when 0<p<1 and ceiling(p)=0 when p=0 then floor(-p)=-ceiling(p) thus we proved floor(-x)=-ceiling(x) ceiling(-x)=celing(-k-p)= -k + celing(-p) ceiling(-p)=0 when -1<-p<=0 -floor(x)=-floor(k+p)=-k - floor(p) -floor(p)=0 when 0<=p<1 then ceiling(-p)=-floor(p) thus we proved ceiling(-x)=-floor(x)
