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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Kolmogorov's Truncation Lemma (iii)

Probability with Martingales: In the definition of $f$, is that really $z$ and not $\lceil |z| \rceil$, $\lfloor |z| \rfloor + 1$ or something? How exactly do we have the part in the $\...
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2answers
294 views

Gnarly equality proof? Or not?

I have this gnarly equality which Mathematica's Reduce says it doesn't have the chops to handle: $\left.k\in \mathbb{Z}\land \left\lceil \frac{k-2}{2}\right\rceil \...
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2answers
30 views

Relation between a floor and a ceiling function for a problem

I was trying to formulate some problem. I want to find a relation between a floor and ceiling function. Suppose the Property 1 satisfies that it has $\lfloor \frac{n}{2} \rfloor$ number of $X$. Then ...
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0answers
22 views

Find when ceiling functions differ by one

For the function $$f(x) = (a/b)x + (c/b)$$ how do I find the smallest value x such that: $$\lceil f(x)\rceil < \lceil f(x) + kx\rceil $$ where x is a positive integer, a, b, and c are ...
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2answers
67 views

How would I compute this integral with a ceiling function?

It seems simple, but it is not, really. How would I calculate this ($\lceil a \rceil$ denotes the ceiling function)? $$\int{\lceil x+2 \rceil}\ln x\,\,dx$$ First, I noticed that $\lceil x+2 \rceil=\...
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88 views

Is there a non brute force way to solve this problem?

A friend of mine asked me to prove or disprove that: $$ \left\lceil\frac{2}{2^{1/n}-1}\right\rceil=\left\lfloor\frac{2n}{\ln 2}\right\rfloor \forall n \in \mathbb{Z^+} $$ First of all, I run a ...
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1answer
50 views

Help with equation that uses floor and ceiling functions

I have this equation and I've been stuck with it for a couple of hours. $\lfloor\log_2x\rfloor + 1 = \lceil\log_2(x+1)\rceil$ I've tried using this ceiling property: $\lceil x\rceil = n \Leftarrow\...
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1answer
64 views

Integrating $\int^b_a [x]\,dx+\int^b_a [-x]\,dx$

I came across a question today... Integrate $\int^b_a [x]\,dx+\int^b_a [-x]\,dx$ where [.] denotes greatest integer function is equal to Now this question is not helpful for me because in that ...
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1answer
38 views

How to solve mixed-integer problem?

I don't know how to solve this equation, $$\left\lceil\frac{x-A}{B}\right\rceil C + D x < E, \quad x\in \mathbb{Z}$$ In this equation, only $x$ is unknown and $x$ is integer, but $A,B,C,D,E$ are ...
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5answers
56 views

Floor-Ceil Properties

If $n$ and $k$ are integers, with $k$ different from zero: $$\left\lceil{\frac{n+1}{k}}\right\rceil = \left\lfloor{\frac{n}{k}}\right\rfloor + 1$$ How can I prove this property? I would appreciate ...
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36 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
46 views

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

$$\sum_{k=1}^{n}\left \lfloor \log _{m}k \right \rfloor$$ $$\sum_{k=1}^{n}\left \lceil \log_{m}k\right \rceil$$ I found myself stuck trying to solve these two summations but i can't make any progress....
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49 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
53 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
17 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 \dfrac{...
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1answer
26 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
42 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
74 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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46 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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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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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. $$\lceil{-3.14}\rceil=$$ $$\lfloor{-3.14}\...
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52 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) = \operatorname{ceil}(\operatorname{ceil}...
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1answer
58 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 x+...
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1answer
54 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
50 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
32 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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25 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
35 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
38 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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45 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
61 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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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
285 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 -1}...
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1answer
347 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) &= n-...
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2answers
55 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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91 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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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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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
225 views

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

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
92 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
39 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
57 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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380 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
56 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> \lceil\...
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1answer
276 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
243 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 ...
2
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0answers
45 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
29 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 p}\right\...