The function that maps a real number $x$ to the largest integer less than or equal to $x$ (often denoted by $\lfloor x\rfloor$). See also (ceiling-function).

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3
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36 views

Points Distributed evenly around a circle: how many points are in each region?

A circle of circumference $2$ is split into three arcs of length $\frac{2}{3}$ (so the regions are $[0,\frac{2}{3})$, $[ \frac{2}{3},\frac{4}{3})$, $[\frac{4}{3},2)$, $2$ identifies with $0$) and ...
0
votes
2answers
107 views

What is the derivative of floor function? [on hold]

What is the derivative of the next equation? $$f(x) = \left\lfloor\frac{c}{x}\right\rfloor\ \ \ \ \ \text{ where }\lfloor\cdot\rfloor\text{ is the floor function.}$$ $c,x$ are positive integers ...
1
vote
0answers
38 views

Explicit formula for floor(x)?

In number theory we have so-called explicit formula's in terms of the Riemann zeta zero's. For instance to count the sum of the logarithms of the primes below some given integer. ( second Chebyshev ...
3
votes
2answers
78 views

Determine if the function $1/\lfloor 1/x\rfloor$ is integrable on $[0,2]$

Is this function integrable on $[0,2]$? $$\cfrac{1}{\left\lfloor\cfrac{1}{x}\right\rfloor}$$ I have suspicion that it is, but I'm unsure of how I could determine if that's true.
-2
votes
2answers
59 views

Is $ \left\lfloor \frac{n+3}{2} \right\rfloor = \left\lfloor \frac{n}{2} \right\rfloor +1$? [closed]

$$ \left\lfloor \frac{n+3}{2} \right\rfloor = \left\lfloor \frac{n}{2} \right\rfloor +1$$ Is it correct? Thanks in advance.
1
vote
1answer
30 views

Would there be no input or input does not exist?

This problem is from Discrete Mathematics and Its Applications. And the definition of inverse from the book: For part 43 (c), would the inverse not exist? For the floor function, in terms of $f(a) ...
2
votes
1answer
31 views

Can we find a relation between the three integrers $m$, $j$ and $k$?

Let $r>4$ and $n>1$ positive integers and let $α$ be a positive real number. Let us define the following three positive integers: $$ \begin{align*} m &= \lfloor r^{(n+1)^2} \alpha \rfloor ...
0
votes
2answers
13 views

Find the number of digits of the number $k$ in function of $r$ and $n$

Let $α∈(0,1)$ be an irrational number with infinitely digits after the decimal point. Let $r>4$ and $n>1$ be positive integers. Let $$k=⌊r^{n²}α⌋$$ where $⌊.⌋$ is the floor function. My ...
1
vote
1answer
83 views

Inverse function for $y=\lfloor x\rfloor+x$

Find the inverse function of the following function: $y=\lfloor x\rfloor +x$ I have tried writing down $x$ as $\lfloor x\rfloor +\{x\}$ but didn't get anywhere with that. A proper hint ...
1
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1answer
41 views

Can we deduce that $⌊r^{n}α⌋≃r^{n}α$ when $r→∞$?

Let $α∈(0,1)$ be an irrational number and let $n≥1$ be a fixed positive integer. For any $r>4$ we define the positive integer $$k=⌊r^{n}α⌋$$ where $⌊.⌋$ denotes the floor function. My question is: ...
2
votes
2answers
38 views

Concrete Mathematics: Manipulation of Summation of Floor

On page 88 of Concrete Mathematics, there is a particular series of equations that I could not understand: $$ \begin{array} {lcl} \sum_{0 \le k < n} ([\{k\alpha\} < v] - v) & = & ...
1
vote
1answer
30 views

Borel-Stieltjes measure problem with floor functions.

We have 2 functions, $$\alpha(x)=\left\{ \begin{array} {cl} \lfloor x \rfloor + \log(1+x),& x\geq 0 \\ \lfloor x \rfloor, & x<0 \end{array} \right.$$ $$\beta(x)=\lfloor x \rfloor$$ ...
1
vote
1answer
109 views

How to solve $ \left \lfloor x^2 - x - 2 \right \rfloor = \left \lfloor x \right \rfloor $

I need some help to solve the next equation: $$ \left \lfloor x^2 - x - 2 \right \rfloor = \left \lfloor x \right \rfloor $$ Where $ \left \lfloor \cdot \right \rfloor $ is the floor function. What ...
-1
votes
4answers
50 views

How to solve equation with floor function [closed]

How to solve the next equation: $$ \left \lfloor x \right \rfloor = 2x $$ Where $ \left \lfloor \cdot \right \rfloor $ is the floor function.
9
votes
3answers
164 views

How prove this $\{a\}\cdot\{b\}\cdot\{c\}=0$ if $\lfloor na\rfloor+\lfloor nb\rfloor=\lfloor nc\rfloor$

Interesting problem Let $a,b,c$ be real numbers such that $$\lfloor na\rfloor+\lfloor nb\rfloor=\lfloor nc\rfloor$$ for all postive integers $n$. Show that: $$\{a\}\cdot\{b\}\cdot\{c\}=0$$ ...
0
votes
1answer
39 views

Why are these two floor sequences equal?

I want to convert from a base-10 integer number to its base-2 equivalent as has been shown in $\eqref{2}$. But after reviewing the MATLAB dec2bin function ...
1
vote
1answer
88 views

what value series $\sum_{r=1}^\infty\frac{\lfloor rp \rfloor}{2^r}$ converges?

I have been wondering about series of $$S=\sum_{r=1}^\infty\frac{\lfloor rp \rfloor}{2^r}$$ where p is a constant positive real number and $\lfloor\cdot\rfloor$ is floor function. I know it converges ...
10
votes
3answers
139 views

Is it possible to define $\lceil x\rceil$ in terms of $\lfloor\ldots\rfloor$?

I have tried to define the ceiling function of $x$ in terms of its floor function; I thought this might be easy, but it isn't. I can easily do this with a piecewise equation, but I need to do without ...
0
votes
1answer
38 views

For which constants $a$ function is continuous

Let $f(x)= \lfloor x \rfloor \cdot\cos{(a\cdot x)}$ where $x\in \mathbb{R}$. Find for which real constants $a$ function is continuous. We know function $ \lfloor x \rfloor$ is continuous apart from ...
2
votes
1answer
71 views

How to compute the integral $ \int_0^\pi \lfloor\cot (x)\rfloor dx $

I have been given to compute $$ \int_0^\pi \lfloor\cot (x)\rfloor dx$$ Where, $ \lfloor$ $\rfloor$ is floor function. Now since floor function is discontinuous we need to break out Integral in ...
1
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1answer
70 views

Proof, that the floor and ceiling functions exist

I want to proof the following, with elementary properties of the integers and reals: Let $x\in \mathbb{R}$. Then there are unique $p,q\in \mathbb{Z}$, such that: $$p\leq x < p+1\text{ and ...
0
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2answers
20 views

Rounding half-integers to approximate product

Given two half-integers $a,b\in\mathbb Z+\frac 1 2$, the integers nearest to each is given by $N_a=\{a-0.5,\ a+0.5\}$ and $N_b=\{b-0.5,\ b+0.5\}$. Is there a general method to find, for the two given ...
3
votes
2answers
72 views

How many numbers less than $x$ have a prime factor that is not $2$ or $3$

I am trying to figure out the number of integers greater than $1$ and less than or equal to $x$ that have a prime factor other than $2$ or $3$. For example, there are only two such integer less than ...
4
votes
0answers
78 views

Calculating integral of step function

My question is from Apostol's Vol. 1: One-variable calculus with introduction to linear algebra textbook. Page 70. Exercise 10. Given a positive integer $p$. A step function $s$ is defined on the ...
0
votes
2answers
13 views

Expressing n mod m in terms of floor values?

I'm trying to prove the expression: $$\left\lceil\frac{n}m\right\rceil = \left\lfloor n+m-\frac1m\right\rfloor\;,$$ where $n,m$ are integers` So I've come across this article (PDF) which gives a ...
1
vote
2answers
62 views

How do I solve floor function integral equation?

This equations comes from my other question, and I thought it was ok to create another question about the same exercise. So I have to solve the equation: $$\int_0^{\lfloor x\rfloor}\lfloor ...
0
votes
0answers
52 views

Proof of nearest integer equality

Let $N(n)$ be the nearest-integer function undefined on half-integers. There are many valid ways to define $N(n)$, I like to choose $N(n) =\arg \min_{z \in \mathbb{Z}} |n-z|$. Consider the function ...
1
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1answer
51 views

Exercise on graphing integral of floor function

My question is from Apostol's Vol. 1: One-variable calculus with introduction to linear algebra textbook. Page 70. Exercise 6. a) If $n$ is a positive integer, prove that $\int_0^n\lfloor ...
0
votes
1answer
26 views

Floor function and continuity

In the topic Proof concerning definite integral, I've received down-votes because I said that the function $f(x) = \lfloor x \rfloor$ is continuous in $[a, b]$, for $0 < a < b<1$. Why am I ...
2
votes
1answer
46 views

Recurrence equation - floor problem

I'm having trouble solving this recurrence equation: $$x(n) = x\left(\left\lfloor \frac n2\right\rfloor \right) + n,\quad x(1)=1$$ I`m trying to find non-recurrence equation: $$x(n) = 2n - 1$$ But ...
1
vote
2answers
46 views

Proof of a nearest-integer inequality

Let $N(y)$ be the nearest-integer function and undefined on half-integers. For all $r \in \mathbb R$ that are not half-integers, prove $$\forall{\ i \in \mathbb ...
0
votes
2answers
42 views

floor function problem

Prove that if $m$ and $n$ are positive integers, and $x$ is a real number, then: $$\left\lfloor\frac{\lfloor x\rfloor+n}m\right\rfloor = \left\lfloor\frac{x+n}m\right\rfloor$$
14
votes
3answers
460 views

Integer part of a sum (floor)

Let $\left(\, x_{n}\,\right)_{\,n\ \geq\ 1}$ be a sequence defined as follows: $$ x_{1}={1 \over 2014}\quad\mbox{and}\quad x_{n + 1}=x_{n} + x_{n}^{2}\,, \qquad\forall\ n\ \geq\ 1 $$ Compute the ...
2
votes
1answer
44 views

How to prove this apparent identity?

While solving the problem in my other question, I've come across an identity, which I've empirically found to be true, but can't seem to prove. Here I've simplified the problem a bit, so that the ...
2
votes
3answers
104 views

Proving Floor and Ceiling of a Rational Number

Suppose x,y $ \in \mathbb{Z}^+ $ Prove $\lceil x/y \rceil = \lfloor (x-1)/y \rfloor + 1$ I was considering using the definition of floor and ceiling to prove this. But this does not seem like a ...
7
votes
6answers
97 views

Solve $\lfloor \sqrt x \rfloor = \lfloor x/2 \rfloor$ for real $x$

I'm trying to solve $$\lfloor \sqrt x \rfloor = \left\lfloor \frac{x}{2} \right\rfloor$$ for real $x$. Obviously this can't be true for any negative reals, since the root isn't defined for such. My ...
2
votes
2answers
57 views

Math floor and limits? $\lim \limits_{n \to \infty}\frac{n}{2}\left\lfloor \frac{3}{n}\right\rfloor$

I have these equations: $$\lim \limits_{n \to \infty}\frac{n}{2}\left\lfloor \frac{3}{n}\right\rfloor$$ $$\lim \limits_{n \to \infty}\frac{2}{n}\left\lfloor \frac{n}{3}\right\rfloor$$ What is ...
1
vote
1answer
47 views

Strict upper and lower bounds of a sum (Big-Theta)

I am trying to find a function f(k) such that $S_k=\sum_{n=1}^{k^2-1}(\lfloor\sqrt{n}\rfloor)=\Theta(f(k))$. What I have done so far: Ignoring the floor asymptotically we get: ...
0
votes
5answers
81 views

What is an example of a bijective function f: Z to N that isn't piecewise?

Like without using if even or odd. Like how you can define a bijection $f\colon\mathbb{N}\to\mathbb{Z}$ by is $f(n)=\lfloor n/2\rfloor\cdot(-1)^n$.
1
vote
1answer
72 views

Proving Via delta epsilon $(n\sqrt2- \lfloor n\sqrt2\rfloor )$

$a_n = (n\sqrt2- \lfloor n\sqrt2\rfloor )$ Any hints on how to prove there is no limit?
3
votes
0answers
29 views

Distribution and convergence of the r.vs.: $X_n= \frac{ \lfloor nX \rfloor}{n}$

$X$ is an absolutely continous random variable, with a continous density function, and: $$X_n= \frac{ \lfloor nX \rfloor}{n}$$ What is the distribution of $X_n$, and what can we say about its ...
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votes
2answers
451 views

⎣⎡x⎤⎦=⎡x⎤ for all real numbers x

Let LHS = ⎣⎡x⎤⎦ and RHS = ⎡x⎤ Let us call n=⎡x⎤ Case 1: n is even, there exists k in Z such that n=2k I'm not sure how to go on or if I'm setting this up right.
1
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2answers
65 views

$x = c + (\lfloor \frac{y}{m} \rfloor \times m) - y$ resulting in negative number

(x + y) mod m = c or (x + y) - (floor((x + y)/m) * m) = c This should be reversible if x is smaller or equal to m so my try is: ...
0
votes
1answer
18 views

Finding the limit when there is the mantissa function involved

I can't seem to be able to use any known limits and/or variable change for this limit. I just need something to start. (log is natural logarithm and M(x) is the mantissa function.)
0
votes
1answer
49 views

Is this statement false? if so, how should I disprove it?

We define $\lfloor x\rfloor$ by $$\lfloor x\rfloor \in \mathbb{Z} \land \lfloor x\rfloor \leq x \land( \forall z \in \mathbb{Z}, z\leq x \Rightarrow z\leq\lfloor x\rfloor)$$ Prove or ...
1
vote
3answers
43 views

If $x>y$, then $\lfloor x\rfloor\ge \lfloor y\rfloor$, formal proof

For x ∈ ℝ, define by: ⌊x⌋ ∈ ℤ ∧ ⌊x⌋ ≤ x ∧ (∀z ∈ ℤ, z ≤ x ⇒ z ≤ ⌊x⌋). Claim 1.1: ∀x ∈ ℝ, ∀y ∈ ℝ, x > y ⇒ ⌊x⌋ ≥ ⌊y⌋. Assume, x, y ∈ ℝ # Domain assumption ...
2
votes
3answers
45 views

floor ceiling proof

Hi I would like to proof without using induction that: $$ \left\lceil\frac{n}{m}\right\rceil \leq \frac{n+m-1}{m} $$ I tried: $$ \left\lceil\frac{n}{m}\right\rceil \leq ...
1
vote
1answer
40 views

How should I approach solving this floor function?

Prove that for all $n \in \Bbb Z, \lfloor\sqrt {(n)}+ \sqrt {(n+1)} \rfloor = \lfloor \sqrt{4n+2}\rfloor$. There must be some algebraic substitution?
0
votes
0answers
20 views

Zeroes of the floor (greatest integer function)

I was just browsing Wolfram Alpha and looking at interesting graphs of functions when I noticed that it gave complex numbers for the numerical roots of $\lfloor x\rfloor$. Does anybody know why?
2
votes
3answers
46 views

Limit of floor function when $x$ goes infinity

Is it true that $\lim_{x \to \infty} (\left \lfloor{x}\right \rfloor -x) = 0$, or alternatively, $\lim_{x \to \infty} \left \lfloor{x}\right \rfloor=x$? If so, how can we prove it using ...