The function that maps a real number $x$ to the largest integer less than or equal to $x$ (often denoted by $\lfloor x\rfloor$). See also (ceiling-function).

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2answers
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 ...
-3
votes
2answers
109 views
+50

How does a C-constant in a substitution affect the result of integration?

I'm doing some pretty strange integrals (floor functions ones) and I think I should probably start asking some more complex questions regarding it. Since I now know how to integrate them, I have to ...
1
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0answers
49 views

Need to verify Real Analysis Proof

I wish to verify a proof; solutions to this exercise are not available. Lemma: If $S \subset \mathbb{Z}$ is bounded from above, it has a maximum element. Proof: If $S$ is bounded from above, it ...
3
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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. ...
3
votes
3answers
72 views

Limit of the floor function of $\frac{x}{\sin(x)}$

Alright this looks like a very simple problem at the first go. I need to find $$\lim_{x\rightarrow0^+} { \left\lfloor{\frac{x}{\sin (x)}}\right\rfloor}$$ So since I know the inner function's graph ...
1
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2answers
31 views

Proving associative property, floor function

I need to prove the following operation is associative: $x*y = xy \pmod 5$ I came up with the equation that $x*y=xy-5[\![xy/5]\!]$ I'm having difficulty proving that $x*(yz)=(xy)*z$. After ...
1
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2answers
50 views

$\lim_{x\to \infty}\frac{3}{x}\lfloor\frac{x}{4}\rfloor=\frac{3}{4}$

Prove that $\lim_{x\to \infty}\frac{3}{x}\lfloor\frac{x}{4}\rfloor=\frac{3}{4}$ If i put $x\to\infty$,the $\frac{3}{x}$ tends to zero and the $\lfloor\frac{x}{4}\rfloor$ tends to $\infty$.I do not ...
1
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0answers
31 views

Can this relation between the Merten's function be simplified?

Question Can this relationship be simplified further? $ \def\lf{\left\lfloor} \def\rf{\right\rfloor} $ $$ \sum_{i=1}^n \mu(i)(\lf \frac{n}{i}\rf \ln(\lf \frac{n}{i}\rf + 1) + \frac{1}{2}\ln(\lf ...
1
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0answers
26 views

Prove that $\lim_{n\to\infty}\sum_{k=1}^{n}\frac{\left\lfloor a_k(x)\right\rfloor}{10^k}=\frac{x}{19}$

Let $a(x)$ be the sequence defined as $$a_1(x)=\frac x2\\a_{n+1}(x)=\frac{20a_n(x)-19\left\lfloor a_n(x)\right\rfloor}{2}$$ for all $x,n\in\mathbb{N}$. Prove that ...
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1answer
42 views

The number of ordered pairs $(x,y)$ satisfying the equation

The number of ordered pairs $(x,y)$ satisfying the equation ...
-1
votes
2answers
86 views

Does $[0.9999…]=1$? [duplicate]

We all know that $0.99999...=1$ So does that imply $[0.99999...]=1?$ Or do we consider it as $0?$ My doubt is: any gif of the form $[0.xyz...]=0$. If $[0.99999...]=1$ won't that be contradicting? ...
2
votes
1answer
69 views

Expanding the floor function. $ \lfloor n/p \rfloor=?$

$n$ is a positive integer. Than we can write this. $$\lfloor n/2 \rfloor= \left(n+\frac{(-1)^{n}-1}{2}\right)\frac{1}{2}$$ I wonder can we do that simplifying with other dividers? Like can we do ...
0
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0answers
29 views

Floor Function vs Gaussian function

$\lfloor x \rfloor$ is often used to denote the Floor Function. However, in my country, we refer to them in a name that translates into English as Gaussian function. This is because Gauss introduced ...
1
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0answers
34 views

Is $\lim\limits_{m\to\infty}\frac{1}{m}\sum_{n=1}^m \cos\left(\frac{2\pi nj}{s}\right)$ “useful”?

In this answer it is explained that a big reason that the nth prime formula discussed with (13) and (14) isn't "useful" is because $\lfloor\frac{j}{s}\rfloor-\lfloor\frac{j-1}{s}\rfloor$ doesn't have ...
6
votes
1answer
164 views

Compute $\displaystyle\lim_{x\to 0}\dfrac{x}{[x]}$.

When I take left hand limit of the function $\displaystyle\lim_{x\to 0}\frac{x}{[x]}$,then $\displaystyle\lim_{h\to 0^{-}}\frac{-h}{[-h]}=\lim_{h\to 0^{-}}\frac{-h}{-1}=0$ where $0<h<1$ and [ ] ...
4
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0answers
61 views

Theorems Involving Integration (confirmation)

I came up with these few theorems and I am curious whether or not my hypothesis are true. I don't really have a method for proving these, mind you. I'm more considering these as things I've noticed to ...
1
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2answers
53 views

How to prove ceiling and floor inequality more 'formally'?

The inequality in question is below: $x - 1 < \lfloor x\rfloor \le x \le \lceil x \rceil < x + 1 $ Essentially, I must prove the above for every real number $x$. To begin this proof, I broke ...
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1answer
92 views

Equation involving floor function: $ x^n-\lfloor x \rfloor=n $ [closed]

Given n a natural number, find $x$ (positive real number) such that: $$ x^n-\lfloor x \rfloor=n, $$ where $ \lfloor x \rfloor $ represents the value of the floor function in x.
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2answers
87 views

The number of real roots of $x\lfloor x\rfloor+187=\lfloor x^{2}\rfloor+\lfloor x\rfloor$

Find the number of real roots of the equation $x\lfloor x\rfloor+187=\lfloor x^{2}\rfloor+\lfloor x\rfloor$. I search other methods except checking case by case.
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0answers
15 views

Reasoning about Floor Functions of Floor Functions [duplicate]

Does it follow that: $$\left\lfloor\frac{\lfloor\frac{a}{b}\rfloor}{c}\right\rfloor = \left\lfloor\frac{\lfloor\frac{a}{c}\rfloor}{b}\right\rfloor = \left\lfloor\frac{a}{bc}\right\rfloor$$ I believe ...
1
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2answers
94 views

Prove $\int_0^n \lfloor x\rfloor \,dx = \frac{n (n-1)}{2}$

Prove $\int_0^n \lfloor x\rfloor \,dx = \frac{n (n-1)}{2}, \text{ where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$ and $n \in \mathbb{Z}^+$.}$ I am working through ...
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3answers
40 views

How to solve $n^2-n<\lfloor\frac{n^2}{4}\rfloor$ ,$n \in \mathbb{N}$?

How to solve this inequality? $$n^2-n<\lfloor\frac{n^2}{4}\rfloor \\ n \in \mathbb{N}$$ I can solve normal inequalities but this includes floor function with which I am not familiar. and because ...
1
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5answers
57 views

floor functions [closed]

Hi all, I have a questions about solving floor functions equation, how to solve this problem? $$\lfloor x\times \lfloor x \rfloor \rfloor = 1$$ * and how we can find range and domain for $$f(x) = ...
3
votes
1answer
39 views

Find $\lfloor z \rfloor $ given that $z=(\{\sqrt{3}\}^2-2\{\sqrt{2}\}^2)/(\{\sqrt{3}\}-2\{\sqrt{2}\})$

Let $\lfloor x \rfloor$ denote the greatest integer function, and $\{x\}=x-[x]$ the fractional part of $x$. If $$z=\cfrac{\{\sqrt{3}\}^2-2\{\sqrt{2}\}^2}{\{\sqrt{3}\}-2\{\sqrt{2}\}}$$ find $\lfloor z ...
2
votes
3answers
59 views

Manual Graph Sketching For $\lfloor(x)\rfloor+\sqrt{x-\lfloor(x)\rfloor}$

How to prove $$\lfloor x\rfloor+\sqrt{x-\lfloor x \rfloor}$$ is continous for all $x$?And how to plot the nature of the graph manually? I can prove that at integer points it is continous.But what ...
4
votes
1answer
75 views

Calculating the value of $\lfloor(1+\sqrt{2})^{2n}\rfloor$

Problem: Calculate the value of $\lfloor(1+\sqrt{2})^{2n}\rfloor$ where $n$ is an arbitrary non-negative integer and $\lfloor x\rfloor$ indicates the largest integer not greater than $x$. What I ...
5
votes
5answers
549 views

A valid floor function trick?

Given $x\in\mathbb R_+$ and $m,n\in\mathbb Z_+$, is it true that $$\bigg\lfloor\frac{\lfloor \frac{x}{m}\rfloor}{n}\bigg\rfloor=\bigg\lfloor \frac{x}{mn}\bigg\rfloor?$$ Thanks for at least three ...
6
votes
2answers
87 views

I would like to calculate $\lim_ {n \to \infty} {\frac{n+\lfloor \sqrt{n} \rfloor^2}{n-\lfloor \sqrt{n} \rfloor}}$

I would like to calculate the following limit: $$\lim_ {n \to \infty} {\frac{n+\lfloor \sqrt{n} \rfloor^2}{n-\lfloor \sqrt{n} \rfloor}}$$ where $\lfloor x \rfloor$ is floor of $x$ and $x ∈ R$. Now I ...
3
votes
4answers
48 views

Assistance in finding $\lim_{n\to\infty} \frac{n-\lfloor \sqrt n \rfloor^2}{n}$

I am trying to prove $$\lim_{n\to\infty} \frac{n-\lfloor \sqrt n \rfloor^2}{n} = 0$$ given $n>0$. But I'm having difficulties dealing with the floor function. I tried splitting apart the limit ...
0
votes
2answers
35 views

If we have a sequence of 1's, why is the partial sum of it equal to a floor function?

If $a_n = 1$ then why is $$A_n = \sum_1^n a_n = 1+ 1+ ... + 1$$ equal to the floor function $x + O(1)$? I got this from Wikipedia's page regarding the Abel summation formula. Thanks,
2
votes
1answer
44 views

Why does the floor function $x \mapsto \lfloor x \rfloor$ have expansion $x + O(1)$?

Shouldn't it just be the largest previous integer? Why is there a remainder term $O(1)$? Thanks, Edit: I am working on a problem that uses the Abel summation formula, and the integration part of ...
1
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1answer
49 views

Summation of Greatest Integer $ \sum_{k=1}^{n} \left \lfloor{\sqrt{k}}\right \rfloor $

I want to know the concept behind summing up a Greatest Integer function like $\sum_{k=1}^{n} \left \lfloor{x}\right \rfloor $ can be evaluated by simply writing the series but something like $$ ...
0
votes
1answer
19 views

Difference between the integral of floor and the triangular series

I figured out that $$\int_0^x \lfloor t \rfloor \cdot dt= \frac{x\cdot(x-1)}{2} \qquad when \; x=\lfloor x \rfloor$$ But what would the equation look like when $x \neq \lfloor x \rfloor$? I ...
1
vote
1answer
78 views

What is $\lfloor(\sqrt 3 +1)^5\rfloor$? (without a calculator)

The question states: Let $[x]$ be the greatest integer less than or equal to $x$. If $x=(\sqrt 3 +1)^5$, then $[x]$ is equal to $75$ $50$ $152$ $151$ When I punch it out in the ...
1
vote
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 ...
0
votes
3answers
86 views

$\lfloor\frac{1}{x}\rfloor\cdot x^2$ continuous at $0$

I am trying to prove, that this function k is: 1)continuous at $0$ and 2) noncontinuous in all $x_0=\frac{1}{n}, n\in\{\Bbb Z\}\setminus\{0\} $ the function k is given through: $$ k:== ...
1
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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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3answers
83 views

Calculate limit with floor function [closed]

How can I proceed to find the following limit: $$\lim_{n \rightarrow \infty }(n\sqrt 2-\lfloor n\sqrt2 \rfloor) $$ where $n$ is a natural number. Please if there is no limit, would you provide a ...
0
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1answer
40 views

What is the jump series representation for $f(x)\lfloor g(x) \rfloor$?

Basically, what I wish to find is the general notation for a series that when subtracted to a greatest integer function, fixes the graph into a stable line. I was thinking that the jump series for ...
0
votes
1answer
57 views

Integral/Derivation of Modulo/Greatest Integer

I've been trying to find the integral and derivatives of the modulo function. For those of you who are not aware, it is a primarily Computer Science based operator that is defined as the remainder ...
5
votes
2answers
57 views

Solve $x-\lfloor x\rfloor= \frac{2}{\frac{1}{x} + \frac{1}{\lfloor x\rfloor}}$

Could anyone advise me how to solve the following problem: Find all $x \in \mathbb{R}$ such that $x-\lfloor x\rfloor= \dfrac{2}{\dfrac{1}{x} + \dfrac{1}{\lfloor x\rfloor}},$ where $\lfloor *\rfloor$ ...
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0answers
9 views

Computing a sum of functions of $\xi$-adhering numbers: $\big\lfloor{\xi \over\lfloor{\xi/\beta}\rfloor}\big\rfloor=\beta$

Assume the existence of a $\xi \in Z^+$. A $\beta \in Z^+ ; 1\le\beta\le \xi$ is said to be $\xi$-adhering if $$\big\lfloor{\xi \over\lfloor{\xi/\beta}\rfloor}\big\rfloor=\beta$$ Let $f(\xi)$ be ...
2
votes
2answers
50 views

Proof involving Floor function

Prove: $\forall l \in \mathbb{Z}: \forall r \in \mathbb{R}: 0 \le r \lt 1 \Rightarrow\lfloor l+r\rfloor = l$ My attempt: Let: $l \in \mathbb{Z}, r \in \mathbb{R}.$ Assume: $0 \le r \lt 1.$ ...
2
votes
1answer
34 views

Help with proof regarding generalization of a function: $f(x)=2x+1-2^{\lfloor \log_2x\rfloor+1}$

Consider the below function: $$f(x)=2x+1-2^{\lfloor \log_2x\rfloor+1}$$ Let $f^x(x)$ refer to the composition of $f(x)$, $x$ number of times. Now after having observed some patterns, I've ...
0
votes
3answers
52 views

The sum of integer parts of $2k/n$ over $k=1,\dots,n-1$

Let $$ S_n= \left[\frac{2}{n}\right]+\left[\frac{4}{n}\right]+\cdots+\left[\frac{2(n-1)}{n}\right] $$ where $n$ is an odd integer such that $n\geqslant 3$, and $[x]$ is the integer part of $x$. ...
2
votes
0answers
40 views

$x-\left\lfloor x \right\rfloor +\frac { 1 }{ x } -\left\lfloor \frac { 1 }{ x } \right\rfloor =1\implies x$ is irrational. [duplicate]

If $x-\left\lfloor x \right\rfloor +\frac { 1 }{ x } -\left\lfloor \frac { 1 }{ x } \right\rfloor =1$, then $x$ is irrational. I am thinking of using the contrapositive: If $x$ is rational, then ...
2
votes
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. $$\lceil{-3.14}\rceil=$$ ...
6
votes
3answers
94 views

Solve for $x$: $x\lfloor x + 2 \rfloor +\lfloor 2x - 2 \rfloor +3x =12$

Solve for $x$: $$x\lfloor x + 2 \rfloor +\lfloor 2x - 2 \rfloor +3x =12$$ My attempt: I have changed this equation into the fractional part function. so that we know $0 \leq \{x\}<1$. I have final ...
0
votes
1answer
93 views

If $x - \lvert x \rvert + \frac{1}{x} - \lvert \frac{1}{x} \rvert = 1$ then $x$ is irrational [duplicate]

For every real number $x$, if $x - \lvert x \rvert + \frac{1}{x} - \lvert \frac{1}{x} \rvert = 1$ then $x$ is irrational If $x$ equals $\sqrt{2}$ I get an inequality... So is this claim false?