What are ceiling and floor in mathematics?

[Neo]

floor
?x? means the floor of x, i.e. the largest integer less than or equal to x.
Ex: ?5? = 5, ?3.4? = 3, ?8.6? = 8, ??1.2? = ?2

ceiling
?x? means the ceiling of x, i.e. the smallest integer greater than or equal to x.
Ex: ?5? = 5, ?3.4? = 4, ?8.6? = 9, ??1.2? = ?1