Floor
For any $$x \in \mathbb{ R }$$, we define the floor of $$x$$ denoted by $$\lfloor x \rfloor$$ as $$\max \left( \left\{ m \in \mathbb{ Z } : m \le x \right\} \right)$$. Note that $$\lfloor \cdot \rfloor : \mathbb{ R } \to \mathbb{ Z }$$

The max exists as it is a subset of $$\mathbb{ Z }$$ which is bounded above, its explicit value can be by using the archimedian property to find the smallest $$n$$ such that $$n \gt x$$ then taking $$m = n - 1$$.

From here we can see that $$\lfloor \pi \rfloor = 3$$

Fractional Part of Floor
Let $$x \in \mathbb{ R }$$ then we define $\left\rfloor x \right\lfloor := x - \left\lfloor x \right\rfloor$

In other words $$x = \left\lfloor x \right\rfloor + \left\rfloor x \right\lfloor$$

Fractional Part of Floor Bound
For any $$x \in \mathbb{ R }$$ $\left\rfloor x \right\lfloor \in [0, 1)$
Ceiling
For any $$x \in \mathbb{ R }$$, we define the ceiling of $$x$$ denoted by $$\lceil x \rceil$$ as $$\min \left( \left\{ n \in \mathbb{ Z } : n \ge x \right\} \right)$$, here $$\lceil \cdot \rceil : \mathbb{ R } \to \mathbb{ Z }$$

And that $$\lceil \pi \rceil = 4$$

Floor of an Integer is Itself
Suppose $$x \in \mathbb{ Z }$$, then $$\lfloor x \rfloor = x$$
The greatest integer less than or equal to $$x$$ is $$x$$.
Ceiling of an Integer is Itself
Suppose $$x \in \mathbb{ Z }$$, then $$\lceil x \rceil = x$$
The smallest integer greater than or equal to $$x$$ is $$x$$.
Floor of a Non Integer is Smaller
Suppose that $$a \in \mathbb{ R } \setminus \mathbb{ Z }$$, then $\lfloor a \rfloor \lt a$
Ceiling of a Non Integer is Greater
Suppose that $$a \in \mathbb{ R } \setminus \mathbb{ Z }$$ then $\lceil a \rceil \gt a$
The floor of a Sum of a Real and an Integer
Suppose that $$x \in \mathbb{ R }$$ and $$n \in \mathbb{ Z }$$ then we have $\left\lfloor x + n \right\rfloor = \left\lfloor x \right\rfloor + n$
A Number is a Perfect Square if its Square Root is an Integer
Let $$n \in \mathbb{ N } _ 1$$ then $$n$$ is a perfect square if and only if $$\sqrt{ n } \in \mathbb{ N } _ 1$$
Counting Squares with Floor
The number of squares in the set $$\left[ 1, \ldots , n \right]$$ is given by $\left\lvert a \in \left[ 1, \ldots , n \right] : \left\lfloor \sqrt{ a } \right\rfloor = \sqrt{ a } \right\rvert$
When Flooring Tells us Two Numbers Divide Eachother
Suppose $$a, b \in \mathbb{ Z }$$ then $a \mid b \iff \left\lfloor \frac{a}{b} \right\rfloor = \frac{a}{b}$
Counting Multiples with Floor
Let $$n, d \in \mathbb{ N } _ 1$$ then there are $$\left\lfloor \frac{n}{d} \right\rfloor$$ multiples of $$d$$ within the set $$\left[ 1, \ldots , n \right]$$
Largest Prime Power Dividing the Factorial
Suppose that $$n \in \mathbb{ N } _ 2$$ and that $$p \in \mathbb{ P }$$ such that $$p \mid n!$$ then $p ^ { \sum _ { i \in \mathbb{ N } _ 1 } \left\lfloor \frac{n}{p ^ i} \right\rfloor } \mid n!$ and $$\sum _ { i \in \mathbb{ N } _ 1 }\left\lfloor \frac{n}{p ^ i} \right\rfloor$$ is the largest integer with this property
Prime Factorization of the Factorial
Suppose that $$n \in \mathbb{ N } _ 1$$ then $n! = \prod _ { p \in \mathbb{ P } } p ^ { \sum { i \in \mathbb{ N } _ 1 } \left\lfloor \frac{n}{p ^ i} \right\rfloor }$
Double Sum of Divisors Equation
For any arithmetic $$f: \mathbb{ N } _ 1 \to C$$ we have $\sum _ { k = 1 } ^ { n } \sum _ { d \mid k } f \left( d \right) = \sum _ { e = 1 } ^ { n } \left\lfloor \frac{n}{e} \right\rfloor f \left( e \right)$