The function that maps a real number $x$ to the smallest integer greater than or equal to $x$ (which is often denoted $\lceil x\rceil$. See also (floor-function).

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28 views

Floor and ceiling opposite property

For $x\in \mathbb{R}$ let's define $[x]$ as: $$ [x] = max \{ k\in \mathbb{Z}: k\leq x \} $$ and $[x]^{*}$ as: $$ [x]^{*} = min \{ k\in \mathbb{Z}: k\geq x \}. $$ Show that: $$ [x]^{*} = -[-x]. $$ So ...
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1answer
31 views

Summation of $\sum_{k=1}^{n}\left \lfloor \log _{m}k \right \rfloor$ and $\sum_{k=1}^{n}\left \lceil log_{m}k\right \rceil$

$$\sum_{k=1}^{n}\left \lfloor \log _{m}k \right \rfloor$$ $$\sum_{k=1}^{n}\left \lceil log_{m}k\right \rceil$$ I found myself stuck trying to solve these two summations but i can't make any progress. ...
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2answers
37 views

ceiling functions inequality

Please, help me in solving this ceiling function inequality. $ \lceil n/4 \rceil \ge 3$ I know the formal definiton of the ceiling functions: $\lceil x \rceil = n$ iff $n-1< x \le n $ ...
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1answer
50 views

Possible circular reasoning in textbook proof that $\lceil x+m\rceil=\lceil x\rceil +m$

The goal is to prove that $\lceil x+m\rceil=\lceil x\rceil +m$, where $x$ is a real number and $m$ is an integer. The book outlines the following proof: Write $x=n-\epsilon$, where $n$ is an ...
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1answer
13 views

Closed form expression for $n_j$ defined by $n_j=\lceil n_{j−1}/b\rceil$ clarification

I came across this answer to this question: Closed form expression for $n_j$ defined by $n_j=\lceil n_{j-1}/b \rceil$ I was hoping someone could clarify the following step: $q−1 \leqslant ...
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1answer
17 views

Derivatives and discontinuity with ceiling function

I'm reviewing a previous test, and this is bothering me: An example function: $f(x)=x-[x]$ at $a=3$, where $[]$ is the ceiling function. I'm supposed to find the left and right derivatives of ...
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1answer
27 views

Floor and Ceiling functions

I have been trying to proof ⌊log_2(⌈n/k⌉)⌋ = ⌊log_2(n/k)⌋, but I never learned any rules with floor and ceiling functions. I am not sure if this theorem is true either. So my question is: Is it safe ...
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1answer
61 views

Ceiling and Floor function

I believe that I have found a trigonometric expression for both the ceiling and floor function, and I seek confirmation that it is, indeed, correct. Update
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1answer
32 views

What technique should I apply to find the derivative of a ceiling or floor function e.g d/dx(x*⌈x⌉) and d/dx(x*⌊x⌋)?

May I know the technique to apply to find the derivative, whenever I see a ceiling function of floor function. Thank You! e.g $$ \frac{d}{dx}(x*\lceil x \rceil )$$ and $$ \frac{d}{dx}(x*\lfloor x ...
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1answer
31 views

Prove that $\left\lceil\frac xy\right\rceil = \left\lceil\frac {\lceil x\rceil}y\right\rceil$ with x element of R and y element of Z

How can one prove that $$\left\lceil\frac xy\right\rceil = \left\lceil\frac {\lceil x\rceil}y\right\rceil$$ with $x\in \Bbb R$ and $y\in \Bbb Z$.
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3answers
379 views

What does this 'L' and upside down 'L' symbol mean?

I need help finding out what the following symbols are called and what they do. I searched up math symbols but couldn't find them anywhere near there. $$\lceil{-3.14}\rceil=$$ ...
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1answer
46 views

Is it possible to prove that ceil(x/y) = ceil(ceil(x)/ceil(y))?

Let $\operatorname{ceil}(x)$ be the smallest integer greater than or equal to $x$. (Also denoted $\lceil x\rceil$.) Is it true that $\operatorname{ceil}(x/y) = ...
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1answer
53 views

Proof of $\lceil x \rceil + \lceil y \rceil \geq \lceil x + y \rceil$

I have been trying to prove $$ \lceil x \rceil + \lceil y \rceil \geq \lceil x + y \rceil $$ by using $$ x \leq \lceil x \rceil < x+1,\\ y \leq \lceil y \rceil < y+1,\\ x + y \leq \lceil ...
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1answer
34 views

Proofs regarding Ceilings and Floors

I understand the concept of what ceilings and floors when using numbers; however, when it comes to proofs I am so lost, I was wondering if anyone could shed some light on some useful links that could ...
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1answer
49 views

How would I prove $\lceil \frac{\binom{2^{n+1}}{2}}{2^{n+1}}\rceil = 2^n$ for all natural numbers n? [closed]

Any leads on how to prove $\lceil \frac{\binom{2^{n+1}}{2}}{2^{n+1}}\rceil = 2^n$?
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1answer
25 views

Inverting functions containing ceiling

I have a recurrence relation for a countably infinite sequence that contains the integers divisible by 5 but not by 7. The relation I came up with is: $5((n-1) + \lceil \frac{n}{6} \rceil)$ The ...
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0answers
22 views

Ceiling Expression

I might be over complicating things a bit, or I just want to verify that this is correct. two positive integers $m$ and $n$ such that $⌈(m+n)/12⌉ ≠ ⌈m/12⌉+⌈n/12⌉$, where $⌈⌉$ indicate the ceiling ...
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1answer
34 views

What is $\operatorname{frac(x)}$ or $\{x\}$?

I understand this is an opinion kind of a question...but still: Well I know that $\operatorname{frac(x)}$ or $\{x\}$ stands for the fractional part of $x$ but how is it exactly defined? Quoting ...
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1answer
32 views

ceiling and floor functions with powers

I have tried to prove this all day but couldn't come up with a convincing solution. Please i need help on how to prove this. Thanks $\lfloor n^{1/k}\rfloor$+1 =$\lceil(n+1)^{1/k}\rceil$ where k is ...
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1answer
35 views

Algebra to show Induction for $\lceil {x\over 2} \rceil \mapsto \lceil{(x+1)\over 2}\rceil $

$5^{\lceil \frac x 2 \rceil}$ (Which Operation?) x = $5^{\lceil \frac x2\ + 1 ) \rceil}$ $x$ can be any value that can help equate this for induction. I'm confused on how the algebra would work to ...
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0answers
30 views

Closed form for a recursive equation that include the ceiling function

Can someone help me with finding the closed form of g(n) in terms of n, A, and B? g(n<0)=0 ; g(0)=0 ; g(1)=0 ; g(n) = A + g(n-1) - ceiling[g(n-1)/B] , n>=2 , A and B are Natural numbers, ...
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1answer
52 views

Floor and Ceiling Series (I) [closed]

Prove or disprove: 1) $$\sum_{n=2}^{\infty}{\frac{1}{\lfloor n^2/2 \rfloor}} = \frac{1+\zeta(2)}{2}$$ 2) $$\sum_{n=1}^{\infty}{\frac{1}{\lceil\ n^2/2 \rceil}} = \frac{\zeta(2)}{2} + \frac{\pi}{2} ...
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0answers
25 views

How to pigeonhole the primes between $p_n$ and $p_{n+1}^2$ for twin prime conjecture?

For any full list of the primes up to the $n$th prime: $P = \{2, 3,5,\dots, p_n\}$, any natural number $q$ such that $ p_n \lt q \lt p_{n+1}^2$ that is not sieved by a prime in $P$ is also a prime. ...
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3answers
268 views

How to prove this identity of ceiling function?

My book writes down this identity of least integer function: $$\lceil x\rceil +\left\lceil x + \frac{1}{n}\right \rceil + \left\lceil x + \frac{2}{n}\right \rceil + \cdots +\left\lceil x + \frac{n ...
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1answer
210 views

Solving recurrence equation with floor and ceil functions

I have a recurrence equation that would be very easy to solve (without ceil and floor functions) but I can't solve them exactly including floor and ceil. \begin{align} k (1) &= 0\\ k(n) &= ...
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2answers
46 views

Prove the following floor function identity

The identity is this: $\lfloor 2x \rfloor + \lfloor 2y \rfloor \geq \lfloor x \rfloor + \lfloor y \rfloor + \lfloor x+y \rfloor$ I am truly stumped. Help is appreciated. Thank you!
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69 views

Concrete Mathematics Floor Sum

In Concrete Mathematics, after a short introduction of the floor and ceiling functions, the problem of the "Concrete Math Club Casino" is proposed. Essentially, how many numbers in $[1,1000]$ are ...
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29 views

Factorization by multiplying and representation as difference of two squares

Definition 1.$$R: \mathbb N \to \mathbb N: \ R(n) = \lceil\sqrt{n}\rceil^2-n.$$ This is the distance from $n$ to the smallest square greater or equal to $n$. Definition 2. Let $a$ be as positive ...
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4answers
109 views

Prove that $\left\lceil \frac{n}{m} \right\rceil =\left \lfloor \frac{n+m-1}{m} \right\rfloor$

On a discrete mathematics past paper, I must prove that $$\left\lceil \frac{n}{m} \right\rceil = \left\lfloor \frac{n+m-1}{m} \right\rfloor$$ for all integers $n$ and all positive integers $m$. ...
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1answer
154 views

Least integer function and Greatest Integer Function Without using ceil() and Floor()

I was wondering if there is any mathematical way to calculate Least Integer and Greatest integer without using predefined Ceil() and Floor() Function of Programming Language.
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2answers
67 views

Ceil () and Floor()

I already know the basic rules for the both functions: $$\text{Ceil}(2.5)=3\\ \text{Floor}(2.5)=2$$ But I could not understand the following these: $$\text{Ceil}(2.6, 0.25)=2.75\\ \text{Floor}(2.6, ...
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1answer
34 views

Find an expression for the ones digit of a positive integer using a floor or ceiling function.

I am trying to find out how to write an expression for the ones digit for any given positive integer. For example, if n = 326, the expression should evaluate to 6. The only things I've been able to ...
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1answer
42 views

Rounding off general equation

I would like to know if there is a way to express the "roundup" function of excel or "rounding off of a number to the nearest whole number" in an equation form. e.g. in excel: roundup $(2.13,0) = 3$, ...
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1answer
203 views

Proving statements about ceiling and floor functions.

Prove or disprove the statements below. (a) For all positive real numbers x and y, $\lfloor x \cdot y\rfloor ≤ \lfloor x\rfloor \cdot \lfloor y\rfloor $. (b) For all positive real numbers x and y, ...
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1answer
55 views

Can we deduce from the equality that $\max \{n-q,q-1\} \in \left[ \frac{n}{2},n \right)$?

Suppose that we have a uniform distributed random variable in $[0,n]$. We have the following: $$\max \{n-q,q-1\}=\left\{\begin{matrix} q-1, q > \lfloor \frac{n}{2}\rfloor & \to q-1> ...
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1answer
117 views

Proving two equations containing ceiling and floor function to be equal

The problem is to show that if $n$ is a positive integer, then $$1 = C(n, 0) < C(n, 1) < ... < C(n, \left \lfloor{ n/2} \right \rfloor) = C(n, \left \lceil {n/2} \right \rceil) > ... > ...
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3answers
149 views

How to prove that the $\lceil x y \rceil \le \lceil x\rceil\lceil y\rceil$ for real numbers $x, y$?

I know intuitively that this is true, but whenever I try and break apart that intuition to see where it's coming from, I essentially end up re-writing the assumption I'm trying to prove. I've tried ...
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6answers
1k views

What is ⌊0.9 recurring ⌋? [duplicate]

For a ceiling and floor function, the number is taken to 0 decimal places. Does this process mean that 0.9 recurring inside a floor function would go to 0? Or would the mathematician take 0.9 ...
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0answers
44 views

A relation with limits

Let $t \in \mathbb{R}_+$, $\varepsilon > 0$ and $p_\varepsilon \in [0, 1]$. There is in a book the following relation: $$ \underset{\varepsilon \rightarrow 0}{lim} (1 - p_\varepsilon)^{\lceil t / ...
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1answer
27 views

Product of a prime and other number expressed in two ways are equal

Let $p \in \mathbb P$ be an odd prime and let $1\leq a \leq p-1$ be such a number that $$a p = \left\lceil \sqrt{a p}\right\rceil ^2-\left\lceil \sqrt{\left\lceil \sqrt{a p}\right\rceil ^2-a ...
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1answer
37 views

Solve an Equation with Ceiling Function

Given the following Term $$n=\left\lceil{\frac{a}{b}}\right\rceil \quad \text{with} \quad a,b\in \mathbb{R}_{>0}$$ Is there a way to get $b$ if $a$ and $n$ are given? I. e. what is the solution ...
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1answer
28 views

$R = A \times B + C$, How retrieve A and C if C is negative?

For A = 18, B = 54 and $C = {-53, 53}$ $$R = A \times B + C$$ I can retrieve A and C from R with: $$ A = \lfloor R \div B \rfloor $$ $$ C={R} \pmod {B} $$ example, A = 18, C = 53: $$ R = 18 \times ...
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1answer
27 views

Log inequality- is $(\lceil\log x\rceil - \lfloor\log m\rfloor)\cdot m+2^{\lfloor\log m\rfloor+1}\leq m\cdot(\lceil\log\frac{x}{m}\rceil+2)$?

I'm having some hard times making a tight analysis of the memory requirements for my algorithm. I want to show the following inequality, which will show my data structure can use about 2 bits per ...
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2answers
56 views

Log inequality - is $\lceil\log x\rceil - \lfloor\log y\rfloor\leq \lceil\log\frac{x}{y}\rceil+1$

Is it true that $$\forall x>y\in\mathbb N:\lceil\log_2 x\rceil - \left\lfloor\log_2 y\right\rfloor\leq \left\lceil\log_2\frac{x}{y}\right\rceil+1$$? I reached this inequality when further ...
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1answer
34 views

Rounding errors for expressions with floor/ceil funcitons

In parallel computations I frequently use the following distribution of data to processors: having a sequence of $n$ elements $x_1,\ldots,x_n$ and $P$ processors $p_1,\ldots,p_P$, processor $p_k$ ...
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1answer
41 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) ...
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2answers
24 views

Matrix entries using a single function

Is it possible to write the following matrix as a single function, without any explicit if condition mathematically? For, $$1 \leq i,j\leq n,$$ $$A(i,j)=\begin{cases} 2 ~~\text{if}~~ i \neq j \\ 1 ...
12
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4answers
282 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
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2answers
202 views

Definite integral of ceiling function

I understand the basics of solving definite integrals but I have trouble with the following one because of the ceiling function. How can I solve this? $\int_0^3 [x^2] dx$
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1answer
254 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 ...