Floor
For any x ∈ R , we define the floor of x denoted by ⌊ x ⌋ as max ( { m ∈ Z : m ≤ x } ) . Note that ⌊ ⋅ ⌋ : R → Z

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

From here we can see that ⌊ π ⌋ = 3

Fractional Part of Floor
Let x ∈ R then we define ⌋ x ⌊ := x − ⌊ x ⌋

In other words x = ⌊ x ⌋ + ⌋ x ⌊

Fractional Part of Floor Bound
For any x ∈ R ⌋ x ⌊ ∈ [ 0 , 1 )
Ceiling
For any x ∈ R , we define the ceiling of x denoted by ⌈ x ⌉ as min ( { n ∈ Z : n ≥ x } ) , here ⌈ ⋅ ⌉ : R → Z

And that ⌈ π ⌉ = 4

Floor of an Integer is Itself
Suppose x ∈ Z , then ⌊ x ⌋ = x
The greatest integer less than or equal to x is x .
Ceiling of an Integer is Itself
Suppose x ∈ Z , then ⌈ x ⌉ = x
The smallest integer greater than or equal to x is x .
Floor of a Non Integer is Smaller
Suppose that a ∈ R ∖ Z , then ⌊ a ⌋ < a
Ceiling of a Non Integer is Greater
Suppose that a ∈ R ∖ Z then ⌈ a ⌉ > a
The floor of a Sum of a Real and an Integer
Suppose that x ∈ R and n ∈ Z then we have ⌊ x + n ⌋ = ⌊ x ⌋ + n
A Number is a Perfect Square if its Square Root is an Integer
Let n ∈ N 1 then n is a perfect square if and only if n ∈ N 1
Counting Squares with Floor
The number of squares in the set [ 1 , … , n ] is given by | a ∈ [ 1 , … , n ] : ⌊ a ⌋ = a |
When Flooring Tells us Two Numbers Divide Eachother
Suppose a , b ∈ Z then a ∣ b ⟺ ⌊ a b ⌋ = a b
Counting Multiples with Floor
Let n , d ∈ N 1 then there are ⌊ n d ⌋ multiples of d within the set [ 1 , … , n ]
Largest Prime Power Dividing the Factorial
Suppose that n ∈ N 2 and that p ∈ P such that p ∣ n ! then p ∑ i ∈ N 1 ⌊ n p i ⌋ ∣ n ! and ∑ i ∈ N 1 ⌊ n p i ⌋ is the largest integer with this property
Prime Factorization of the Factorial
Suppose that n ∈ N 1 then n ! = ∏ p ∈ P p ∑ i ∈ N 1 ⌊ n p i ⌋
Double Sum of Divisors Equation
For any arithmetic f : N 1 → C we have ∑ k = 1 n ∑ d ∣ k f ( d ) = ∑ e = 1 n ⌊ n e ⌋ f ( e )