Frequently-Used Properties of the Floor Function

: The paper collects 42 frequently-used properties of the floor function, including 35 ones from other literatures and 7 newly added-and-proved ones. The collected properties cover basic inequalities, basic identities, conditional inequalities, conditional equalities and practical formulas. The paper is helpful for scholars of mathematics and computer science and technology in reading and writing scientific works, reasoning and designing algorithms.


Introduction
The floor function, which is also called the greatest integer function (see in [1]), is a function that takes an integer value. For arbitrary real number x, the floor function of x, denoted by ⌊ ⌋, is defined by an inequality of − 1 < ⌊ ⌋ ≤ or equivalently ⌊ ⌋ ≤ < ⌊ ⌋ + 1. The floor function frequently occurs in many aspects of mathematics and computer science. However, as I stated in article [2], except the Graham's book [3], it is hard to find another book or a literature that introduces in general the know-of of the function although people can find something via the Internet, e.g., the wikipedia [4]. Since Graham's book was first published 30 year's ago and its following-up editions made few modifications on the part of the floor function, it is necessary to sort out the properties of the function as a reference for researchers.
In 2017 and 2019, I proved respectively several formulas for the function and made brief summaries on the frequently-used properties by my work together with certain formulas collected from previous literatures, as seen in [5] and [6]. In the past two years, I proved several new results and thus I put them together with the 2019 summary to form this literature.

Definition and Notation
The floor function of real number x is denoted by symbol ⌊ ⌋ that satisfies ⌊ ⌋ ≤ < ⌊ ⌋ + 1; the fraction part of x is denoted by symbol {} x that satisfies = ⌊ ⌋ + { }; the ceiling function of x is denoted by symbol basic equalities.

Basic Inequalities
In the following inequalities, x and y are real numbers by default.
(P1) [1] ny n x y x y with n being a positive integer , where n is a positive integer.

Conditional Inequalities
In the following inequalities, x and y are real numbers, and n is an integer.
x y x y

Basic Equalities
In the following equalities, x and y are real numbers, m and n are integers.
(P14) [3], [7] [3] It needs +   2 log 1 N binary bits to express decimal integer N in its binary expression. A positive integer n with base b has (P28) [11] Let N be an integer; then  −  o o (P32) [12] Let  and x be positive real numbers; then it holds Particularly, if  is a positive integer, say  = n , then it yields [12] For arbitrary positive real numbers  , x and y with  xy , it holds [12]. For arbitrary odd integer  7 n , it holds − +   2 1 1 log 2 n n (P35) [13] For positive integer k and real number  0

Some New Results
Here lists some newly found and proved equalities and inequalities.
x for positive numbers  and x . Particularly, Proof. See the following three steps: (P37) For an arbitrary positive integer k and an arbitrary odd integer  1 Remark on (P37). The condition that N is odd is mandatory because this property does not hold for an . Actually, when Remark on (P38). The condition that N is even is mandatory because this property does not hold for an odd integer N. A simple counterexample is . Readers can confirm the general cases by referring to the Remark on (P37).
(P39) For arbitrary positive real numbers x and y satisfying ≥ , it holds Proof. First is to prove the necessity as following reasoning: Proof. By property of T 3 tree,  − 21 is on level  −2 of T 3 (see in [14]). That is

Motivation of This Paper
In writing a paper related with mathematics, computer science, physics and so on, mathematical reasoning plays a major role in the whole procedure. During a reasoning procedure, some minor evidences such as one or more formulas are often required to keep the reasoning correct. Since our primary middle school we have remembered tens of identities, inequalities, theorems and axioms in our minds. The things we have remembered do help us to write an excellent paper of science and technology. However, it is not so fortunate for researchers who research the number theory, the graph theory and the related subjects because they often have to face the floor function, which frequently occurs in the reasoning but does not have many citable formulas. As I mentioned in [15], the mathematical reasoning or modeling involved with the floor function always requires quite a lot of special skills related with inequalities together with discrete mathematics and it is of quite individuality because the function is defined with an inequality,  [6]. In this paper, I collected 42 properties among which the newly added 7 ones are proved in previous subsection. I am sure these collected properties are helpful for certain people to cite. At least I myself frequently look over them when I was writing a paper related with the issue.
The motivation that I say so many words here is to show something on how to agitate interest in scientific research. An old thing like the floor function might contain a lot of new work and any new work cannot be lack of some old things. I hope the background and history of this paper are educationally meaningful.