Floor and ceil are defined in the following way, respectively,
⌊x⌋=the greatest integer less than or equal to x{\displaystyle \lfloor x\rfloor ={\mbox{the greatest integer less than or equal to }}x}
⌈x⌉=the least integer greater than or equal to x{\displaystyle \lceil x\rceil ={\mbox{the least integer greater than or equal to }}x}