The function that maps a real number $x$ to the largest integer not greater than $x$. See also (ceiling-function).

learn more… | top users | synonyms

0
votes
2answers
14 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 ...
0
votes
5answers
55 views

Prove that for all positive integers $x$, $\left\lfloor \frac{x^2 +2x + 2}{4}\right\rfloor =\left\lfloor \frac{x^2 + 2x + 1}{4}\right\rfloor$.

Title says it all, basically. I believe it to be true that $$\left\lfloor \dfrac{x^2 + 2x + 2}{4} \right\rfloor=\left\lfloor \dfrac{x^2 + 2x + 1}{4} \right\rfloor$$ for all positive integers $x$. I ...
0
votes
1answer
45 views

Is $\frac{1}{\lfloor \frac{a}{b} \rfloor}=\lceil \frac{b}{a} \rceil$??

Is $\frac{1}{\lfloor \frac{a}{b} \rfloor}=\lceil \frac{b}{a} \rceil$? assuming that $\frac{a}{b}>1$.
0
votes
3answers
46 views

Show that $\, 0 \leq \left \lfloor{\frac{2a}{b}}\right \rfloor - 2 \left \lfloor{\frac{a}{b}}\right \rfloor \leq 1 $

How can I prove that, for $a,b \in \mathbb{Z}$ we have $$ 0 \leq \left \lfloor{\frac{2a}{b}}\right \rfloor - 2 \left \lfloor{\frac{a}{b}}\right \rfloor \leq 1 \, ? $$ Here, $\left \lfloor\,\right ...
4
votes
0answers
114 views

Closed form solution for $\left \lfloor{\frac{a}{x}}\right \rfloor$ = $\left \lfloor{\frac{b}{x}}\right \rfloor$?

I need to find the smallest value of $x$ such that: $\left \lfloor{\frac{a}{x}}\right \rfloor$ = $\left \lfloor{\frac{b}{x}}\right \rfloor$ EDIT: where $0 < x < a < b$, and $x \in ...
0
votes
1answer
22 views

Property of greatest integer function

I came across the following mathematical statement in a proof. Can somebody tell me which property of greatest integer function makes it possible? $x + y - \lfloor x + y \rfloor + z - \lfloor x + y - ...
-3
votes
0answers
43 views

Prove that the function is surjective [closed]

$f(x)=⌊x⌋$ Prove that the function is surjective
1
vote
1answer
67 views
+200

Is there a simple closed form of $|\alpha(\sqrt{n}-\left\lfloor \sqrt{n} \right \rfloor) + \beta(\sqrt{n}-\left\lfloor \sqrt{n} \right \rfloor)|$?

Let $d_n(x)$ denote the $n$'th digit after the decimal point in $x$. Examples: $d_8(e) = 2,\;d_5(\pi) = 9$ $\alpha(x)$ and $\beta(x)$ are defined this way: $$d_n(\alpha(x)) = \left\{ ...
0
votes
3answers
35 views

$\lfloor x\rfloor + \lfloor y\rfloor \leq \lfloor x+y\rfloor$ for every pair of numbers of $x$ and $y$

Give a convincing argument that $\lfloor x\rfloor + \lfloor y\rfloor \leq \lfloor x+y\rfloor$ for every pair of numbers $x$ and $y$. Could someone please explain how to prove this? I attempted to say ...
0
votes
0answers
12 views

Number of multiples of 6 between -600 and 3400

Could someone please verify if I did this correctly? The number of multiples of 6 between 3400 and -600 should be the floor of (3400 / 6) - the floor of (-600 - 1 / 6) + 1 (to account for zero. Is ...
0
votes
1answer
20 views

How to plot bivariate function involving modulus and floor functions?

I need to plot: $\displaystyle\large|||x|-2|-1|+|||y|-2|-1|=1$ $\displaystyle\large\left\lfloor\frac{|3x+4y|}{5}\right\rfloor+\left\lfloor\frac{|4x-3y|}{5}\right\rfloor=3$ either for finding area ...
0
votes
3answers
45 views

The solutions of the following equation

Please consider the following equation: $$\left\lfloor x+\frac{1}{x}\right\rfloor=\frac{2x}{3}$$ where $\lfloor x\rfloor$ is the largest integer not greater than $x$. It is clear that it has not a ...
0
votes
2answers
47 views

How to calculate with $\lceil \; \;\; \rceil$

I have a problem calculating with ceils. So If I have $\frac{\lceil \frac{n}{2} \rceil}{np}$, this is not the same as $\frac{\lceil \frac{1}{2} \rceil}{p}$. So do you have some rules how to ...
0
votes
1answer
40 views

Calculate floor of a non-negative value, without using ceiling, round, modulo, abs or if

I want to calculate floor of a non-negative value, without using ceiling, round, modulo, abs or if. I can calculate ceiling of a non-negative value, without using floor, round, modulo, abs or if: ...
2
votes
1answer
50 views

How prove this$\lfloor \sqrt{2x-\lfloor\sqrt{2x}\rfloor}\rfloor=\lfloor\frac{\sqrt{8x+1}-1}{2}\rfloor$

Question: let $x\ge 0$, show that $$\lfloor \sqrt{2x-\lfloor\sqrt{2x}\rfloor}\rfloor=\lfloor\dfrac{\sqrt{8x+1}-1}{2}\rfloor$$ My idea: let $\lfloor \sqrt{2x}\rfloor =m$ then ...
7
votes
1answer
140 views

A formula for $\lfloor n\rfloor+\left\lfloor \frac n2\right\rfloor+ \left\lfloor \frac n3\right\rfloor+\ldots+\left\lfloor \frac nk\right\rfloor$?

Is there any formula to calculate: $$\lfloor n\rfloor+\left\lfloor \frac n2\right\rfloor+ \left\lfloor \frac n3\right\rfloor+\ldots+\left\lfloor \frac nk\right\rfloor$$ with $n$ and $k$ positive ...
0
votes
3answers
50 views

Finding the fourier series of floor function

Find the fourier series for $f(x)=\cases{x-[x]\quad x\in\mathbb{R\setminus Z} \\ \frac 1 2\quad x\in\mathbb{Z}}$ on $[-\pi,\pi]$ and its values for $x=1.5,3,5$. In order to find the series I need ...
8
votes
3answers
118 views

Is it true that $\left\lfloor\sum_{s=1}^n\operatorname{Li}_s\left(\frac 1k \right)\right\rfloor\stackrel{?}{=}\left\lfloor\frac nk \right\rfloor$

While studying polylogarithms I observed the following. Let $n>0$ and $k>1$ be integers. Is the following statement true? $$\left\lfloor \sum_{s=1}^n \operatorname{Li}_s\left( \frac{1}{k} ...
0
votes
1answer
53 views

Measurability of the floor function

Let $u(x)=⌊x⌋$, i.e the largest integer not greater than $x$ . Determine $\{u≥a\}$ for all $a\in \mathbb{R}$. Show that $u$ is Borel-measurable. Can anyone help me with this problem?
0
votes
2answers
38 views

Proving that $x-\left\lfloor \frac{x}{2} \right\rfloor\; =\; \left\lfloor \frac{x+1}{2} \right\rfloor$

How would you prove that if $x$ is an integer, then $$x-\left\lfloor \frac{x}{2} \right\rfloor\; =\; \left\lfloor \frac{x+1}{2} \right\rfloor$$ I tried to start by saying that if $x$ is an even ...
1
vote
1answer
140 views

For every $x\in\mathbb R$ and $\varepsilon$ > 0 , there exist $\,q,q'\in\mathbb Q$, such that $q<x<q'$ and $\left |q-q' \right |< \varepsilon$

I'm asked to prove that for every $\varepsilon$ > 0 , there exists two rational numbers $q$ and $q'$ such that $q<x<q'$ and $\left |q-q' \right |<\varepsilon$ where $x$ is a real number. ...
1
vote
2answers
56 views

Proving the general formula [nx] where [.] is the floor function.

I've been trying to solve a exercise that asks me to prove the following generalization for the floor function: $$\lfloor nx\rfloor = \sum_{k=0}^{n-1} {\lfloor x + \frac kn \rfloor}$$ I've already ...
0
votes
3answers
32 views

Limit, Greatest Integer function?

Q. Find $\lim _{x\to 0}\left(1-x+\left[x-1\right]+\left[1-x\right]\right)$ where $\left[y\right]$ denotes the greatest integer function not exceeding 'y'.
0
votes
1answer
11 views

$U\sim \mathcal U(0,a)\overset{?}{\implies}U-\lfloor U \rfloor \sim \mathcal U (0,a-\lfloor a \rfloor$)

Suppose $U\sim \mathcal U(0,a)$ for some $a>0$. Is it true that $U-\lfloor U \rfloor \sim \mathcal U (0,a-\lfloor a \rfloor$)? How can I prove this? If $a\in \mathbb N$ then the following ...
0
votes
2answers
75 views

Evaluating $\displaystyle\sum_{x=a}^b \left\lfloor {\frac{k}{x}} \right\rfloor$

I'm trying to find a nicer form to evaluate this sum, but the floor function is throwing me off. $$\sum_{x=a}^b \left\lfloor {\frac{k}{x}} \right\rfloor$$ This is the most I've been able to do so ...
2
votes
0answers
43 views

Find the result of $ \sum_{x=1}^\infty \frac{1}{x} \log \frac{kx}{\lfloor kx \rfloor}_o$

I would like to calculate the sum $$\displaystyle \sum_{x=1}^\infty \frac{1}{x} \log \frac{kx}{\lfloor kx \rfloor}_o$$ where $k=\sqrt{2}+1$, $x$ is an odd integer and $\lfloor z \rfloor_o$ indicates ...
3
votes
1answer
75 views

Is this proof that $\lfloor x \rfloor \geq n \left\lfloor \frac{x}{n} \right\rfloor$ correct?

In this text the fractional part of a real $x$ shall be denoted $\{x\}$, such that $x = \lfloor x \rfloor + \{x\}$. Theorem: $$ \forall x \in \mathbb{R}_{\geq 0} \forall n \in \mathbb{N}_{\geq 1} : ...
2
votes
0answers
35 views

Is there a cycle for modulo operation on a floor?

I have a floor $${\left\lfloor\frac{n}{i}\right\rfloor},$$ where i varies from 1 to n (n can be upto $10^{10}$), and there's another number given 'm' (m upto $10^{5}$). Does ...
11
votes
3answers
479 views

Very challenging: max{floor,ceil}=?

I spotted a pattern while trying to generalize a problem. (EDIT: said problem has been removed from this post to avoid confusion. EDIT(2): Here is the problem again: ...
1
vote
1answer
46 views

How prove or disprove $\lfloor a^m\rfloor \neq \lfloor b^n\rfloor$ for all $m,n\in\mathbb{N}$

Question: prove or disprove: there exsit irrational $a>1,b>1$ such that for all positive integers $m,n$, $$\lfloor a^m\rfloor \neq \lfloor b^n\rfloor$$ Now I can't prove this ...
0
votes
2answers
60 views

Continuous differentiable spline or function resembling floor

I'd need any (real-valued) function (whatever meets the following description at least approximately) continuous and thrice differentiable everywhere (or twice if 3 not possible), with the following ...
3
votes
2answers
45 views

A sum of difference of floors

I have the sum ( $M$ is any integer $> 1$ ): $$ \sum_{h = 1}^{M}\left(\,\left\lfloor\, 2M + 1 \over h\,\right\rfloor -\left\lfloor\, 2M \over h\,\right\rfloor\,\right) $$ and looking for a way to ...
2
votes
0answers
59 views

$\sum_{k=1}^n \lfloor kx \rfloor =$ ?

Let $x$ be a positive real number and $n$ a positive integer , then how may we evaluate $\sum_{k=1}^n \lfloor kx \rfloor $ ? If a closed form doesn't exist then can we at least find an asymptotic ...
2
votes
1answer
60 views

Prove $f$ isn't continuous at $\frac{1}{\pi}$

Let $f(x)=\left\lfloor {\sin {1 \over x}} \right\rfloor$ (meaning floor of $\sin x$). I need to prove that $f(x)$ isn't continuous at $x=\frac{1}{\pi}$. Proof: For a nehiborhood of $\frac{1}{\pi}$: ...
1
vote
0answers
123 views

Is this function $f^n_b : \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}^n$ a surjection?

In this text the fractional part of a real $x$ shall be denoted $\{x\}$, such that $x = \lfloor x \rfloor + \{x\}$. Let the set of functions $f^n_b : \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}^n$ be ...
4
votes
4answers
85 views

Integration of some floor functions

Can anyone please answer the following questions ? 1) $\int$ $ \left \lfloor{x}\right \rfloor $ $dx$ 2) $\int$ $ \left \lfloor{\sin(x)}\right \rfloor $ $dx$ 3) $\int_0^2$ $\left ...
0
votes
5answers
63 views

Injective function proof involving floor function

Let $f : \Bbb{Z} \to \Bbb{Z}$ and $g : \Bbb{Z} \to \Bbb{Z}$ be functions defined by $f(x)=3x+1$ and $g(x)=\lfloor\frac{x}{2}\rfloor$. Is $g(f(x))$ one-to-one? So, $g(f(x)) = ...
0
votes
1answer
42 views

Writing the floor function as a contour integral

The function $f(z)=\frac{\pi}{\sin \pi z}$ has simple poles of residue 1 at the integers. Hence, by the residue theorem, I consider the interesting idea of drawing a (perhaps rectangular, for example) ...
1
vote
3answers
96 views

Solving equations with dozens of ceil and floor functions efficiently?

I am tackling a problem which uses lots of equations in the form of: $$ q_i(x) = c_i(x) + \sum_{j=1}^N \left\lfloor \frac{q_i(x)}{P_j} \right\rfloor C_j $$ where $q_i(x)$ is the only unknown, ...
1
vote
2answers
70 views

Is there also an other way to show the equality?

I want to show that: $$ \left\lfloor \frac{n}{2}\right\rfloor + \left\lceil \frac{n}{2} \right\rceil=n$$ That's what I have tried: $ \left\lfloor \frac{n}{2}\right\rfloor=\max \{ m \in \mathbb{Z}: ...
0
votes
1answer
55 views

Find Limit floor (sin x) / floor(x) as x approaches 0.

I am unable to evaluate this limit. The floor function is giving me trouble. Any help will be appreciated. And please edit it so that it looks readable.
0
votes
2answers
33 views

Quick floor function

This isn't true, right? $$k\left\lfloor\frac n {2k}\right\rfloor\leq \left\lfloor\frac n k\right\rfloor$$ $2<k\leq \left\lfloor\dfrac {n-1} 2\right\rfloor$, $n>4$, $k,n$ are coprime.
0
votes
2answers
29 views

Floor Function Bound?

I am trying to prove or disprove the following bound: $2+\left(n-\left\lfloor\dfrac n k\right\rfloor k\right)\ge \left\lfloor\dfrac n k\right\rfloor$, where $2<k\le \left\lfloor\dfrac {n-1} ...
1
vote
1answer
60 views

Algebraic manipulation of floors and ceilings

I am trying to solve the summation $$ \sum_{n=3}^{\infty} \frac{1}{2^n} \sum_{i=3}^{n} \left\lceil\frac{i-2}{2}\right\rceil $$ I will list some of the simplifications that I've found so far, and ...
3
votes
1answer
68 views

Number of solutions to equation involving floor-function

For school I have to solve some problems involving floor-functions, and I have no clue how to solve this one: for a given $n$, calculate the number of $k$'s, $k\lt n$ such that the number of multiples ...
1
vote
1answer
57 views

A problem of sum floors

let $n$ be a positive integer, prove that $$\sum_{i=0}^{\left\lfloor\frac{n}{3}\right\rfloor}\left\lfloor\frac{n-3i}{2}\right\rfloor=\left\lfloor\frac{n^2+2n+4}{12}\right\rfloor.$$ It looks like we ...
0
votes
3answers
70 views

How to derive this inequality?

How to derive the following inequality for all positive integers $n \geq 2$? $$ \frac{n!}{n^n} \leq \left(\frac{1}{2}\right)^k,$$ where $k$ denotes the greatest integer less than or equal to $\dfrac ...
0
votes
1answer
97 views

limits involving greatest integer function

What can you say about the following limit: $$\lim_{x\rightarrow 0}\left(\left[\dfrac{100x}{\sin x}\right]+\left[\dfrac{99\sin x}{x}\right]\right)$$ where [.] represents the greatest integer function ...
-2
votes
1answer
50 views

Comparing floor and ceiling fractions

Is the following true for all integers x>1: $\lfloor{\frac{2x}{3}}\rfloor \geq \lceil \frac{x}{2}\rceil$
4
votes
1answer
49 views

Prove that $\lceil \frac{\sqrt{n^2+1+\sqrt{n^2}}}{\sqrt{n^2+3+\sqrt{n^2+2}}-\sqrt{n^2+2+\sqrt{n^2+1}}}\rceil = 2n^2+n+3$

First, the question: Prove that $$\Bigg\lceil \frac{\sqrt{n^2+1+\sqrt{n^2}}}{\sqrt{n^2+3+\sqrt{n^2+2}}-\sqrt{n^2+2+\sqrt{n^2+1}}}\Bigg\rceil = 2n^2+n+3$$ The motive to this question is the ...