information:extraction, structural:expansion, linguistic:paraphrasing
Relations among the functions

It is clear from the definitions that

In fact, for integers n, both floor and ceiling functions are the identity:

Negating the argument switches floor and ceiling and changes the sign:

and:

Negating the argument complements the fractional part:

The floor, ceiling, and fractional part functions are idempotent:

The result of nested floor or ceiling functions is the innermost function:

due to the identity property for integers.