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}