Questions tagged [floor-function]

The floor function, also known as the greatest integer function, maps a real number $x$ to the greatest integer less than or equal to $x$ (often denoted $\lfloor x \rfloor$). See also (ceiling-function).

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Calculation of $x$ in $x \lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor = 88$

How can I calculate real values of $x$ in $x \lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor = 88$, where $\lfloor x\rfloor$ is the floor function? My attempt: Let $\lfloor x\lfloor x\lfloor x\...
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Sum of floor of harmonic progression: $\sum_{i=1}^n\lfloor\frac ni\rfloor=2\sum_{i=1}^k\lfloor\frac ni\rfloor-k^2$ for $k=\lfloor\sqrt n\rfloor$

This question is actually from a programming question that a formula is required to compute maths faster. (Please note that computer frequently rounds down to the nearest integer, thus the floor ...
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Generalization of $\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor$

I have been asking the following question at MSE with an answer: $\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor$ is true? I found this relational ...
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$\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor$ is true?

I found the following relational expression by using computer: For any natural number $n$, $$\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor.$$ Note ...
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Proof of the greatest integer theorem: for every real number $x$ there exists a unique greatest integer less than or equal to $x$ [duplicate]

To define the function $f(x)=|[x]|$ where $|[x]|$ is the greatest integer that is less or equal to $x$, we need to prove that indeed such an integer exists. In other words, $$\forall x\in \mathbb{R}...
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Floor Inequalities

Proving the integrality of an fractions of factorials can be done through De Polignac formula for the exponent of factorials, reducing the question to an floored inequality. Some of those inequalities ...
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Evaluating $\sum_{i=0}^{m-1} [ \frac{b + ia}m ]$

Let $a,b\in\mathbb{Z}$ and $m\in\mathbb{Z}_{>1}$ Evaluate $[\frac {b}{m}] + [\frac {(b+a)}{m}]+ [\frac {(b+2a)}{m}]+ [\frac {(b+3a)}{m}]+ [\frac {(b+4a)}{m}]+ [\frac {(b+5a)}{m}]+.....+ [\frac {(b+...
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1answer
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Let $f(x)= \frac {1}{\sqrt{|[|x|-1]|-5}}$ where $[ .]$ is greatest integer function, Find domain of $f(x)$ ??

Problem: Let $$f(x)= \dfrac {1}{\sqrt{|\bigl[|x|-1\bigr]|-5}}$$ where $\bigl[.\bigr]$ is greatest integer function. Find domain of $f(x)$. Solution: The function $f$ is defined for $|\bigl[|x|-...
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Closed form of $\sum\limits_{i=1}^n\left\lfloor\frac{n}{i}\right\rfloor^2$?

Does $\displaystyle\sum_{i=1}^n\left\lfloor\dfrac{n}{i}\right\rfloor^2$ admit a closed form expression?
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Show that if $x\geq 0$ and $n$ is a positive integer, then $\sum_{k=0}^{n-1}\left\lfloor {x+\frac{k}{n}}\right\rfloor=\lfloor {nx}\rfloor$ [duplicate]

I need help with this question, where $\lfloor x\rfloor$ means the floor function of $x$. Show that if $x\geq 0$ and $n$ is a postive integer, then $$\sum_{k=0}^{n-1}\left\lfloor x+\frac{k}{n} \right\...
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Efficient computation of $\sum_{k=1}^n \lfloor \frac{n}{k}\rfloor$

I realize that there is probably not a closed form, but is there an efficient way to calculate the following expression? $$\sum_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor$$ I've noticed $$\sum_{k=...
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1answer
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Integral of a floor function

Let $f(x) = \lfloor 1-x^2 \rfloor$ with $x \in [-2,2]$. Calculate: $$F(x) = \int_{-2}^{x}f(t)dt$$ I know that: $$f(x) = \begin{cases} -3 & : x \in [-2,-\sqrt{3})\\ -2 & : x \...
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How to solve an definite integral of floor valute function?

How do you prove this identity: $$\int_0^{n^2}\lfloor\sqrt{t}\rfloor dt = \frac{1}{6}n(n-1)(4n+1)$$ I'd very much appreciate your help on this one!
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Prove the following ceiling and floor identities?

Could someone help me prove these identities? I'm completely lost: $$\begin{align*} &(1)\quad \left\lceil \frac{\left\lceil \frac{x}{a} \right\rceil} {b}\right\rceil = \left\lceil {\frac{x}{ab}}\...
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How do the floor and ceiling functions work on negative numbers?

It's clear to me how these functions work on positive real numbers: you round up or down accordingly. But if you have to round a negative real number: to take $\,-0.8\,$ to $\,-1,\,$ then do you take ...
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For what $c > 0$ is $S = \lfloor \left( n+1 \right)^c \rfloor - \lfloor n^c \rfloor$ non-decreasing?

I recently solved a practical sequence problem, but got curious and tried to generalize it. Let $$ S_{n, c} = \lfloor \left( n+1 \right)^c \rfloor - \lfloor n^c \rfloor $$ be a set of sequences ...
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Integration of an integral part of x?

If $ f(x)={\left\lfloor x^2\right\rfloor -\left\lfloor x\right\rfloor ^2}$,where ${\left\lfloor x\right\rfloor }$ denotes the greatest integer $\le x$ then $\int_1^2 f(x)dx?$ please give some hint. ...
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Prove $2\lfloor x\rfloor \le \lfloor2x\rfloor$

I am trying to prove $$2\lfloor x\rfloor \le \lfloor2x\rfloor$$ which in turn will yield a prove for a homework question. I thought it is a simple prove but I can't figure it out. Maybe it is just a ...
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Theta bound about $\sum \lfloor {\sqrt{n}}\rfloor$

$$S_k=\sum_{n=1}^{k^2-1}\lfloor\sqrt{n}\rfloor $$ Can somebody give me an idea about the steps I should follow? Initially I thought $$n^{1/2}\log(n) \leq n^{1/2}\leq n^{3/2}$$ so $\Theta(f(n))=S_k ...
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When does $\lfloor (n-1)x \rfloor + \lfloor x \rfloor = \lfloor nx \rfloor$?

I am trying to find the conditions under which $\lfloor(n-1)x\rfloor + \lfloor x \rfloor = \lfloor nx \rfloor$. The trivial case is whenever $x \in \mathbb{Z}$. If $n = 2$, then $x - \lfloor x \rfloor ...
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Floor function properties: $[2x] = [x] + [ x + \frac12 ]$ and $[nx] = \sum_{k = 0}^{n - 1} [ x + \frac{k}{n} ] $

I'm doing some exercises on Apostol's calculus, on the floor function. Now, he doesn't give an explicit definition of $[x]$, so I'm going with this one: DEFINITION Given $x\in \Bbb R$, the integer ...
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How to solve an equation involving the floor of a number?

I'm looking for a solving procedure for this type of exercises. If $[x]$ represents the floor of $x$, solve the equation: $$\left[\frac{6x+5}8\right]=\frac{15x-7}5$$ Choose the correct ...
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2answers
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Proof of greatest integer theorem: floor function is well-defined

I have to prove that $$\forall x \in \mathbb{R},\exists\text{ exactly ONE }n \in \mathbb{Z} \text{ s.t. }n \leq x < n+1\;.$$ I'm done with proving that there are at least one integers for the ...
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How do we prove that $\lfloor0.999\cdots\rfloor = \lfloor 1 \rfloor$?

Are the floor functions of $0.999\cdots$ and 1 equal? It is true that $0.999\cdots=1$ but how does one justifies the integer part of $0.999\cdots$ being 1 , where it is not, or alternatively without ...
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Integral and derivative of $\lfloor x\rfloor$ and $x - \lfloor x\rfloor$

I've always assumed by graphical inspection that $\int (x - \lfloor x\rfloor)\mathrm dx = \dfrac{(x - \lfloor x\rfloor)^2 + \lfloor x\rfloor}{2}$ (W|A) and $\int \lfloor x\rfloor\mathrm dx = x\...
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$2 \lfloor x \rfloor \leq \lfloor 2x \rfloor \leq 2 \lfloor x \rfloor +1$

May I know the standard proof technique to prove such kind of inequalities. $2 \lfloor x \rfloor \leq \lfloor 2x \rfloor \leq 2 \lfloor x \rfloor +1$ Thanks!
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1answer
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The floor of $\sqrt{2x}+1/2$ is the ceiling of $(\sqrt{1+8x}-1)/2$

I've been working on this for a while now and I can't figure out how to prove it: $$\left\lfloor \sqrt{2x} + \frac{1}{2}\right\rfloor = \left\lceil \frac{\sqrt{1+8x}-1}{2}\right\rceil.$$ Here $x$ is ...
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How to show $\lim_{n \to \infty} a_n = \frac{ [x] + [2x] + [3x] + \dotsb + [nx] }{n^2} = x/2$?

This question came from the prelim exam I took last month. I have a proof that seems a bit unwieldy to me (posted as an answer), so I'm opening it up to ask if there are other ways of showing this. ...
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How to prove $\left \lceil \frac{n}{m} \right \rceil = \left \lfloor \frac{n+m-1}{m} \right \rfloor$?

everybody, how can I prove that, for natural $m$ and $n$, $$ \left \lceil \frac{n}{m} \right \rceil = \left \lfloor \frac{n+m-1}{m} \right \rfloor \; ? $$ Thanks a lot.
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Is this a justified expression for the integral of the floor function?

Mathematica seems to agree with me in general with saying that $\displaystyle\int \lfloor x \rfloor dx = \frac{\lfloor x\rfloor (\lfloor x\rfloor-1)}{2}+\lfloor x\rfloor \{ x \}+C = \frac{\lfloor x\...
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Integral of floor function: $\int \,\left\lfloor\frac{1}{x}\right\rfloor\, dx$

How would you go about solving integral of a floor? The particular problem I have is: $$\int \,\left\lfloor\frac{1}{x}\right\rfloor\, dx$$
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Floor of Square Root Summation problem

I have problem calculating the following summation: $$ S = \sum_{j=1}^{k^2-1} \lfloor \sqrt{j}\rfloor. $$ As far as I understand the mean of that summation it will be something like $$1+1+1+2+2+2+2+2+...
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Identity involving Euler's totient function: $\sum \limits_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}$

Let $\varphi(n)$ be Euler's totient function, the number of positive integers less than or equal to $n$ and relatively prime to $n$. Challenge: Prove $$\sum_{k=1}^n \left\lfloor \frac{n}{k} \right\...
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For $n \in \mathbb{N}$ $\lfloor{\sqrt{n} + \sqrt{n+1}\rfloor} = \lfloor{\sqrt{4n+2}\rfloor}$

This is Exercise 3.20 from Apostol's book. Many of them seem tough and here is one of them which I am struggling with. For $n \in \mathbb{N}$, prove that this identity is true: $$\Bigl\lfloor{\sqrt{...
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Proving $\sum\limits_{k=0}^{n-1} \Bigl[x + \frac{k}{n}\Bigr] = [nx]$ [duplicate]

Right, this is an exercise in Apostol, which I am not being able to solve. I was able to prove this result for a small case, that is the case when $n=2$, $[x] + \Bigl[x + \frac{1}{2}\Bigr]=[2x]$, but ...