Questions tagged [ceiling-and-floor-functions]
This tag is for questions involving the greatest integer function (or the floor function) and the least integer function (or the ceiling function).
2,156
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Integration including the floor function
I'm aware of the way to quantify the following integral in the following way, however I'm trying to find another way to express the given integral. Especially when the function $f(x)$ is not ...
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$ \left\lfloor\frac{x - \pi(z)^*}{n} \right\rfloor\geq\left\lfloor\frac{x-z^*}{m}\right\rfloor $ whenever $n\mid m$ and $\pi:\Bbb{Z}/m\to\Bbb{Z}/n$✨
If $z \in \Bbb{Z}/n$, w let $z^* =$ the standard residue or in other words the least non-negative integer equal to $z$ modulo $n$. Suppose that $n \mid m$ for some two positive integers $n,m$.
If $\pi ...
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This alternating sum of fractional floor functions over the divisors of primorial is always a non-decreasing function (the general case).
Define the family of functions for $n \geq 1$.
$$
f_n(x) = \sum_{d \mid p_n\#}(-1)^{\omega(d)}\sum_{0 \leq r \lt d \\ r^2 = 1 \pmod d}\left\lfloor \frac{x - r}{d}\right\rfloor
$$
Conjecture. In ...
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How can we prove that this alternating summation involving fractional floor functions is non-decreasing?
Question. How can we prove that this function is non-decreasing?
That is:
$$
f: \Bbb{R} \to \Bbb{Z} \\
f(x) = [\frac{x}{1}] - [\frac{x-1}{2}] - [\frac{x - 1}{3}] - [\frac{x - 2}{3}] - [\frac{x - 1}{...
- 20.8k
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floor function, algebra
a sequence $(a_n)_{n\ge0}$ is defined by $a_0=1$ and $$a_{n+1}=a_{{\lfloor}\frac{7n}
{9}{\rfloor}}+a_{{\lfloor}\frac{n}{9}{\rfloor}}$$ for $n\ge0$.prove that there is an $n$ such that $a_n<\frac{n}...
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what is the inverse function of 2[x]-x ,where[x] is the floor function? [closed]
$f(x) = 2[x]-x$,
$[x]$ is the floor function,
the domain of the function is $[-1,2]$,
its codomain is $\Bbb R$
$$
f(x) = \begin{cases}
-2-x & \text{if } -1 \leq x < 0 \\
-x & \text{if } 0 \...
- 13
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26
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Integrate $\int_{0}^{n}{[2^x]}$, where $n$ is a natural number [duplicate]
Source: MAT $2014$. Integrate the following -
$$\int_{0}^{n}{[2^x]}$$
I thought of reversing $\frac{d}{dx}(a^x) = a^xln(a)$. So I divide and multiply by $ln(2)$. I end up with the final answer being
$$...
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floor function algebra [duplicate]
prove for all real numbers $x$ we have the following inequality $$\sum_{k=1}^n \left(\frac{{\lfloor}kx{\rfloor}}{k} \right)\le {{\lfloor}nx{\rfloor}} $$
i can easily prove for $n=1$,how do i approach ...
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Evaluating $\int_0^{100}{e^{\{x\}}}dx$ on Desmos
I was watching this video, which shows the following result:
$$\int_0^{100}{e^{\{x\}}}dx=100(e-1)$$
I thought to check the result on Desmos, but it's giving me a different answer! *Note: I am using $\{...
- 306
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What notation should I use to denote the sum of multiples of k smaller than n
I want the clearest, simplest way to denote, for example, the sum of every multiple of 3 smaller than 62 which i could denote like: $$ \sum_{i=0}^{20}3i =0+3+6+…+60 $$ which was the way I found to ...
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Can we write down a formula for the simultaneous images of the $C_q$?
Fix $n \in \Bbb{N}$.
Define $C_2(x) = 2x, \ C_3(x) = 3x$, $C_5(x) = 2 [\frac{x + 2}{3}] + [\frac{ x + 1}{3}] + 2 [ \frac{x}{3}]$, more generally for prime $p_n \geq q \geq 5$ define $C_q(x) = 2[\frac{ ...
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How to solve the floor function equation $\lfloor2x\rfloor+\lfloor3x\rfloor+\lfloor7x\rfloor=2008$
So I was yet again looking on Michael Penn's Youtube channel to see if there were any math equations on his channel that I thought that I might be able to solve when I came across this member only ...
- 1,651
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Fractional part function and greatest integer function
Let $f (x)= \{x\} [x]$; $g(x)=ax^2$ the sum of all real solutions of equation satisfying $f(x) = g(x)$ is 420 (where a is (+ve) rational number), then a is equal to (where [.] and {.} represents ...
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Prove that if $m$ is a positive integer and $x$ is a real number...
Q: Prove that if $m$ is a positive integer and $x$ is a real number, then
$$
\lfloor mx \rfloor=\lfloor x \rfloor+\lfloor x+\frac{1}{m} \rfloor+\lfloor x+\frac{2}{m} \rfloor+...+\lfloor x+\frac{m-1}{m}...
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Prove that if $m$ is a positive integer and $x$ is a real number, then
Q: Prove that if $m$ is a positive integer and $x$ is a real number, then
$$
\lfloor mx \rfloor=\lfloor x \rfloor+\lfloor x+\frac{1}{m} \rfloor+\lfloor x+\frac{2}{m} \rfloor+...+\lfloor x+\frac{m-1}{m}...
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A question on the Greatest Integer function
Let $[x]$ denote the greatest integer less than or equal to $x$ for $x \in \mathbb{R}$.
I started by exploring the following question:
Is
$$
\left[\frac{x}{n}\right]-\left[\frac{y}{n}\right] = \left[\...
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Solving $\int_{0}^{1}\sin\lfloor\frac{1}{x}\rfloor dx$
This is just for fun
I know that without the floor function, the solution to this integral would be $\sin{1}-\operatorname{Ci}{1}$
My first idea to solve this is by creating an infinite summation.
$$\...
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Why does $\{x\in \mathbb{R}: \lfloor y-x\rfloor<y-M, \mbox{ for all $y>M$ } \}=(M,\infty)$?
Let $M$ be a fixed real number. During our research, we have obtained the following identities.
$$
\{x\in \mathbb{R}: \lfloor y-x\rfloor<y-M, \mbox{ for all $y>M$ } \}=(M,\infty),
$$
$$
\{x\in \...
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Can $f([x])$ be continuous?
There is a problem I encountered which has two functions $f(x)$ and $g(x)$ such that $g(f(x))=x$ and $f(\lfloor g(x) \rfloor)=x, \quad \forall x\geq 0$. I am not giving any other details here, because ...
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Proving an equality involving the floor function
Show that
$\left \lfloor \dfrac{ \lfloor x \rfloor+m }{n} \right \rfloor =
\left \lfloor \dfrac{x+m}{n} \right \rfloor$, where m is an
arbitrary integer and $ n$ is an integer $> 0$.
Let us start ...
- 2,572
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votes
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88
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Solving a tough equation with the floor function
Solve the equation $$\left\lfloor\frac{x-2}{x+1}\right\rfloor=\bigg|\frac{2-x}{3}\bigg|.$$
My attempt
Let us consider the case where $x$ is an integer, such that $x=n$, then we have:
$$\frac{n-2}{n+1}=...
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Q: Find a formula for $\sum_{k=0}^m⌊\sqrt{k}⌋$, when m is a positive integer. [duplicate]
Q: Find a formula for $\sum_{k=0}^m⌊\sqrt{k}⌋$, when m is a positive integer.
I checked the answer, and it is
$\frac{n(n+1)(2n+1)}{3}+\frac{n(n+1)}{2}+(n+1)(m-(n+1)^2+1)$, where $n=⌊\sqrt{m}⌋-1$.
I am ...
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Prove or disprove that $\forall x \in\mathbb{R}, \lfloor x^2 \rfloor = \lfloor x \rfloor^2$
Statement: Prove or disprove that $\forall x \in\mathbb{R}, \lfloor x^2 \rfloor = \lfloor x \rfloor^2$.
What I've tried: Let $x$ be a particular but arbitrary chosen real number. By definition of ...
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"Concrete Mathematics" error? Floor/Ceiling Sums, p. 88
No matter how I look at it, the following derivation from "Concrete Mathematics" seems wrong:
Book text (The black box is irrelevant to my question)
After cancelling identical terms, the ...
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Formula for numbers divisible in an interval
Assume we have three natural numbers $x$, $y$, and $z$, with $z>y>x$.
I want to find a formula, which gives us the number of natural numbers divisible by $x$, in the interval $[y,z]$.
From some ...
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Proving that $\lfloor n\rfloor -\left\lfloor\frac{n}{2^{k+1}}\right\rfloor=\lfloor n\rfloor$
How can I prove this?
$$\lfloor n\rfloor -\left\lfloor\frac{n}{2^{k+1}}\right\rfloor=\lfloor n\rfloor$$
I considered the base case, $k=0$
$$\lfloor n\rfloor -\left\lfloor\frac{n}{2}\right\rfloor=\...
- 2,572
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Prove that a floor function series is equal to $n\lfloor x\rfloor$
For each $k=1, 2, \dotsc$ , n let $S_k = \left\lfloor \frac{x+0}{k}\right\rfloor + \left\lfloor \frac{x+1}{k} \right\rfloor + \left\lfloor\frac{x+2 }{k} \right\rfloor + \dotsb + \left\lfloor \frac{x+(...
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$\lfloor 1/2+\sqrt{n+1/2}\rfloor =\lfloor 1/2+\sqrt{n+1/2020}\rfloor$
For every $n$ positive odd number show that $\lfloor 1/2+\sqrt{n+1/2}\rfloor=\lfloor 1/2+\sqrt{n+1/2020}\rfloor$
Here is my second solution: Let's first consider the expression $\sqrt{n+1/2}$ and $\...
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Showing that a sequence containing the floor function follows an equality [duplicate]
Show that $\Big \lfloor \frac{n+1}{ 2 } \Big \rfloor + \Big \lfloor
\frac{n+2}{ 2^2 } \Big \rfloor + \Big \lfloor \frac{n+2^2}{ 2^3 } \Big
\rfloor + \dotsb + \Big \lfloor \frac{n+2^k}{ 2^{k+1} } \Big ...
- 2,572
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About the Sequence $\psi_{n+1}=\left \lfloor{\psi_n \zeta}\right \rfloor$
Let $\zeta$ be the positive root of $x^2-2023x-1=0$
Define a sequence $\psi_i$ such that $\psi_0=1$ and
$$\psi_{n+1}=\left \lfloor{\psi_n \zeta}\right \rfloor
, n\geq0$$
$(
\left \lfloor{x}\right \...
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Looking for the asymptotic expansion of $\sum_{k = \lceil{\sqrt{n}}\rceil}^{\lfloor{m/2}\rfloor} \left[{\sqrt{{k}^{2} - n} \in \mathbb{Z}}\right]$
Looking for the asymptotic expansion to as high order as possible of the sum $$S \left({m,n}\right) = \sum_{k = \lceil{\sqrt{n}}\rceil}^{\lfloor{m/2}\rfloor} \left[{\sqrt{{k}^{2} - n} \in \mathbb{Z}}\...
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When do millimeters and inches line up nicely?
Let the set of all inch measures $\bigl( \mathbb{IMS} \bigr)$ be the set of unitized real numbers constructed from the set of strings $\begin{Bmatrix} f(x) + “ \,\,\,\,\mathtt{inch(es)}” : x \in \...
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Solving the integral $\int\limits^{\infty}_{1}\frac{x-\sqrt{\lfloor x^2\rfloor}}{x}dx$
This problem was just for fun
Here was what I managed to come up with:
The integral is approximately equal to $0.242070053984$.
The integral equals
$$\sum_{n=1}^{\infty}\int_{\sqrt{n}}^{\sqrt{n+1}}...
- 391
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Convergence of floor-function
I am given an expression using the Gauss-brackets [] (assuming this is meant as floor-function).
Let $0 \leq s < t\leq 1$ and $n\in\mathbb{N}$. I want to know against what the following expression ...
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Closed form of $a_1=x$, $a_n=\left\lfloor\dfrac{(n+2)a_{n-1}-2}{n}\right\rfloor,n\ge 2$
Find the closed form of the following sequence of integers: $a_1=x>0$,
$$a_n=\left\lfloor\dfrac{(n+2)a_{n-1}-2}{n}\right\rfloor, n \ge 2$$
I calculated a few terms, and $a_1=x$, $a_2=2x-1$, $a_3=...
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Integrating a formula involving quantisation (floor function)
I have a gamma correction function of the following form:
$$f(x) = x^\gamma \text{ for } 0 \le x \le 1, \gamma \in \mathbb{R}^+$$
For example, a simplified approximation of the sRGB electro-optical ...
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167
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Greatest Integer, Least Integer functions
Two positive real numbers, $a$ and $b$, are expressed as the sum of $m$ positive real numbers and $n$ positive real numbers respectively as follows:
$a = s_1 + s_2 + s_3 + s_4 + \cdots + s_m$
$b = t_1 ...
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Flooring Number Theory
How should one approach the floor function of a binomial expansion mod an integer? For instance, what is
$$\left\lfloor\left(\sqrt{2}+\sqrt{3}\right)^{2020}\right\rfloor$$
mod $101$?
I tried to expand ...
0
votes
0
answers
45
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How Many Multiples of $m$ are between $1$ and $n$ inclusive
"How many numbers between $1$ and $n$ — inclusive — are multiples of $m$?". Perhaps it's very easy to find the answer to such a basic question; hence I do not need an answer to the question, ...
- 1,377
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0
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Floor function and hexagonal numbers
While playing around with squares, I wondered about the sum of square roots of all natural numbers between two perfect squares(both inclusive). After taking the floor value of the expression for first ...
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0
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10
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Keeping the amplitude of a formula constant through a change of frequency
There is a function $f(x)=\frac{1}{\lfloor x \rfloor}\times x$ where the amplitude changes and goes to zero for large positive values, while the frequency remains the same for the appearance of the ...
- 101
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Solve for variable inside and outside of ceiling operation
I have this equation.
2⌈C / E⌉ + C ≤ S + 2
I need to find the largest integer C satisfying the equation given S and E. I proceeded to "solve" the equation for C (math below is wrong):
...
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Is this result about the floor function true?
I want to know if the following result is true or not.
Prove that for all positive integer $n$, $$\left\lfloor\sqrt{n}+\frac{1}{\sqrt{n}+\sqrt{n+2}}\right\rfloor \leq \sqrt{n}$$
I drew a graph of $$...
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Methods to prove statements with floor (or ceiling) expressions?
It is simple to prove by induction that
$$
\label{eq1}\tag{1}
n + \left\lfloor \tfrac 3 5 n \right\rfloor + \left\lfloor \tfrac {n+2} 5 \right\rfloor
\;=\;
\left\lfloor \tfrac 9 5 n \right\rfloor
$$
...
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2
answers
63
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How to solve function with floor: $a\left(\big \lfloor\frac {a} {2\pi}\big \rfloor + 1\right) = 100$
I've never worked with equations that contains floor. I wonder how would you solve $$a\left(\big \lfloor\frac {a} {2\pi}\big \rfloor + 1\right) = 100$$
I looked it up in the internet but I don't get ...
0
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2
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65
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Can floor(floor(a/b)/c) ever be different to floor(a/(bc))? [duplicate]
Is there any mathematical proof that the following equality holds true always (across the natural number domain)?
$\lfloor{\frac{\lfloor^a/_b\rfloor}{c}}\rfloor = \lfloor\frac{a}{bc}\rfloor \quad \...
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Why does limit: $\lim_{x\to\infty}\prod\limits^{\lfloor x\rfloor}_{n=1}\frac{\lceil x\rceil}{\lfloor x\rfloor}=e$?
This is for all $x\in\mathbb{R}$ and $x\notin\mathbb{Z}$ because it equals $1$ for all positive integers.
I was just messing around with floor and ceiling functions in Desmos when I came upon this. I ...
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$p = s(\left \lfloor t\right \rfloor+\frac{\sin(\frac{\pi\left \lfloor t\right \rfloor}{t})}{\sin(\frac{\pi}{t})})$ , make $t$ the subject?
I've wanted to know the solution to this for years since I couldn't figure out how to get the answer, not even from wolfram alpha and this has also made me question whether if it was even possible to ...
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2
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Where is $g(x)=x\lfloor x+1 \rfloor$ continuous?
Determine the points of continuity of $$g(x)=x\lfloor x+1 \rfloor$$
I know the point of continuity of $f(x)=\lfloor x+1 \rfloor$ are $x \in (a,a+1)$ where $a \in \mathbb{Z}$. But how do I determine ...
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I came up with a proof for a Putnam Question, and I would appreciate feedback or direction to a resource that would give feedback.
The question above is Question A4 on the 2015 Putnam. I have written the solution I had come up with down below. I ended up coming up with the right answer, but my proof is long winded and hard to ...
