The function that maps a real number $x$ to the largest integer not greater than $x$. See also (ceiling-function).

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19 views

Rounding half-integers to approximate product

Given two half-integers $a,b\in\mathbb Z+\frac 1 2$, the integers nearest to each is given by $N_a=\{a-0.5,\ a+0.5\}$ and $N_b=\{b-0.5,\ b+0.5\}$. Is there a general method to find, for the two given ...
3
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2answers
64 views

How many numbers less than $x$ have a prime factor that is not $2$ or $3$

I am trying to figure out the number of integers greater than $1$ and less than or equal to $x$ that have a prime factor other than $2$ or $3$. For example, there are only two such integer less than ...
3
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0answers
42 views

Calculating integral of step function

My question is from Apostol's Vol. 1: One-variable calculus with introduction to linear algebra textbook. Page 70. Exercise 10. Given a positive integer $p$. A step function $s$ is defined on the ...
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2answers
11 views

Expressing n mod m in terms of floor values?

I'm trying to prove the expression: $$\left\lceil\frac{n}m\right\rceil = \left\lfloor n+m-\frac1m\right\rfloor\;,$$ where $n,m$ are integers` So I've come across this article (PDF) which gives a ...
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2answers
48 views

How do I solve floor function integral equation?

This equations comes from my other question, and I thought it was ok to create another question about the same exercise. So I have to solve the equation: $$\int_0^{\lfloor x\rfloor}\lfloor ...
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0answers
48 views

Proof of nearest integer equality

Let $N(n)$ be the nearest-integer function undefined on half-integers. There are many valid ways to define $N(n)$, I like to choose $N(n) =\arg \min_{z \in \mathbb{Z}} |n-z|$. Consider the function ...
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1answer
36 views

Exercise on graphing integral of floor function

My question is from Apostol's Vol. 1: One-variable calculus with introduction to linear algebra textbook. Page 70. Exercise 6. a) If $n$ is a positive integer, prove that $\int_0^n\lfloor ...
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1answer
23 views

Floor function and continuity

In the topic Proof concerning definite integral, I've received down-votes because I said that the function $f(x) = \lfloor x \rfloor$ is continuous in $[a, b]$, for $0 < a < b<1$. Why am I ...
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1answer
40 views

Recurrence equation - floor problem

I'm having trouble solving this recurrence equation: $$x(n) = x\left(\left\lfloor \frac n2\right\rfloor \right) + n,\quad x(1)=1$$ I`m trying to find non-recurrence equation: $$x(n) = 2n - 1$$ But ...
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2answers
40 views

Proof of a nearest-integer inequality

Let $N(y)$ be the nearest-integer function and undefined on half-integers. For all $r \in \mathbb R$ that are not half-integers, prove $$\forall{\ i \in \mathbb ...
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2answers
37 views

floor function problem

Prove that if $m$ and $n$ are positive integers, and $x$ is a real number, then: $$\left\lfloor\frac{\lfloor x\rfloor+n}m\right\rfloor = \left\lfloor\frac{x+n}m\right\rfloor$$
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3answers
362 views

Integer part of a sum (floor)

Let $\left(\, x_{n}\,\right)_{\,n\ \geq\ 1}$ be a sequence defined as follows: $$ x_{1}={1 \over 2014}\quad\mbox{and}\quad x_{n + 1}=x_{n} + x_{n}^{2}\,, \qquad\forall\ n\ \geq\ 1 $$ Compute the ...
2
votes
1answer
42 views

How to prove this apparent identity?

While solving the problem in my other question, I've come across an identity, which I've empirically found to be true, but can't seem to prove. Here I've simplified the problem a bit, so that the ...
2
votes
3answers
100 views

Proving Floor and Ceiling of a Rational Number

Suppose x,y $ \in \mathbb{Z}^+ $ Prove $\lceil x/y \rceil = \lfloor (x-1)/y \rfloor + 1$ I was considering using the definition of floor and ceiling to prove this. But this does not seem like a ...
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6answers
90 views

Solve $\lfloor \sqrt x \rfloor = \lfloor x/2 \rfloor$ for real $x$

I'm trying to solve $$\lfloor \sqrt x \rfloor = \left\lfloor \frac{x}{2} \right\rfloor$$ for real $x$. Obviously this can't be true for any negative reals, since the root isn't defined for such. My ...
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vote
1answer
33 views

Strict upper and lower bounds of a sum (Big-Theta)

I am trying to find a function f(k) such that $S_k=\sum_{n=1}^{k^2-1}(\lfloor\sqrt{n}\rfloor)=\Theta(f(k))$. What I have done so far: Ignoring the floor asymptotically we get: ...
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5answers
78 views

What is an example of a bijective function f: Z to N that isn't piecewise?

Like without using if even or odd. Like how you can define a bijection $f\colon\mathbb{N}\to\mathbb{Z}$ by is $f(n)=\lfloor n/2\rfloor\cdot(-1)^n$.
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1answer
71 views

Proving Via delta epsilon $(n\sqrt2- \lfloor n\sqrt2\rfloor )$

$a_n = (n\sqrt2- \lfloor n\sqrt2\rfloor )$ Any hints on how to prove there is no limit?
3
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0answers
28 views

Distribution and convergence of the r.vs.: $X_n= \frac{ \lfloor nX \rfloor}{n}$

$X$ is an absolutely continous random variable, with a continous density function, and: $$X_n= \frac{ \lfloor nX \rfloor}{n}$$ What is the distribution of $X_n$, and what can we say about its ...
0
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2answers
434 views

⎣⎡x⎤⎦=⎡x⎤ for all real numbers x

Let LHS = ⎣⎡x⎤⎦ and RHS = ⎡x⎤ Let us call n=⎡x⎤ Case 1: n is even, there exists k in Z such that n=2k I'm not sure how to go on or if I'm setting this up right.
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0answers
21 views

Floor and Fractional Part function properties?

Floor and Fractional Part function properties? e.g frac(4.3)=3 Prove the following: If $\lfloor a+x\rfloor=\lfloor b+x\rfloor, \space \space \forall x\in \mathbb{R}$, then $a=b$. If $\lfloor ...
1
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2answers
62 views

$x = c + (\lfloor \frac{y}{m} \rfloor \times m) - y$ resulting in negative number

(x + y) mod m = c or (x + y) - (floor((x + y)/m) * m) = c This should be reversible if x is smaller or equal to m so my try is: ...
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1answer
14 views

Finding the limit when there is the mantissa function involved

I can't seem to be able to use any known limits and/or variable change for this limit. I just need something to start. (log is natural logarithm and M(x) is the mantissa function.)
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1answer
48 views

Is this statement false? if so, how should I disprove it?

We define $\lfloor x\rfloor$ by $$\lfloor x\rfloor \in \mathbb{Z} \land \lfloor x\rfloor \leq x \land( \forall z \in \mathbb{Z}, z\leq x \Rightarrow z\leq\lfloor x\rfloor)$$ Prove or ...
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3answers
41 views

If $x>y$, then $\lfloor x\rfloor\ge \lfloor y\rfloor$, formal proof

For x ∈ ℝ, define by: ⌊x⌋ ∈ ℤ ∧ ⌊x⌋ ≤ x ∧ (∀z ∈ ℤ, z ≤ x ⇒ z ≤ ⌊x⌋). Claim 1.1: ∀x ∈ ℝ, ∀y ∈ ℝ, x > y ⇒ ⌊x⌋ ≥ ⌊y⌋. Assume, x, y ∈ ℝ # Domain assumption ...
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3answers
43 views

floor ceiling proof

Hi I would like to proof without using induction that: $$ \left\lceil\frac{n}{m}\right\rceil \leq \frac{n+m-1}{m} $$ I tried: $$ \left\lceil\frac{n}{m}\right\rceil \leq ...
1
vote
1answer
38 views

How should I approach solving this floor function?

Prove that for all $n \in \Bbb Z, \lfloor\sqrt {(n)}+ \sqrt {(n+1)} \rfloor = \lfloor \sqrt{4n+2}\rfloor$. There must be some algebraic substitution?
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0answers
13 views

Zeroes of the floor (greatest integer function)

I was just browsing Wolfram Alpha and looking at interesting graphs of functions when I noticed that it gave complex numbers for the numerical roots of $\lfloor x\rfloor$. Does anybody know why?
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3answers
41 views

Limit of floor function when $x$ goes infinity

Is it true that $\lim_{x \to \infty} (\left \lfloor{x}\right \rfloor -x) = 0$, or alternatively, $\lim_{x \to \infty} \left \lfloor{x}\right \rfloor=x$? If so, how can we prove it using ...
0
votes
5answers
62 views

Prove that for all positive integers $x$, $\left\lfloor \frac{x^2 +2x + 2}{4}\right\rfloor =\left\lfloor \frac{x^2 + 2x + 1}{4}\right\rfloor$.

Title says it all, basically. I believe it to be true that $$\left\lfloor \dfrac{x^2 + 2x + 2}{4} \right\rfloor=\left\lfloor \dfrac{x^2 + 2x + 1}{4} \right\rfloor$$ for all positive integers $x$. I ...
0
votes
1answer
49 views

Is $\frac{1}{\lfloor \frac{a}{b} \rfloor}=\lceil \frac{b}{a} \rceil$??

Is $\frac{1}{\lfloor \frac{a}{b} \rfloor}=\lceil \frac{b}{a} \rceil$? assuming that $\frac{a}{b}>1$.
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vote
3answers
53 views

Show that $\, 0 \leq \left \lfloor{\frac{2a}{b}}\right \rfloor - 2 \left \lfloor{\frac{a}{b}}\right \rfloor \leq 1 $

How can I prove that, for $a,b \in \mathbb{Z}$ we have $$ 0 \leq \left \lfloor{\frac{2a}{b}}\right \rfloor - 2 \left \lfloor{\frac{a}{b}}\right \rfloor \leq 1 \, ? $$ Here, $\left \lfloor\,\right ...
4
votes
0answers
124 views

Closed form solution for $\left \lfloor{\frac{a}{x}}\right \rfloor$ = $\left \lfloor{\frac{b}{x}}\right \rfloor$?

I need to find the smallest value of $x$ such that: $\left \lfloor{\frac{a}{x}}\right \rfloor$ = $\left \lfloor{\frac{b}{x}}\right \rfloor$ EDIT: where $0 < x < a < b$, and $x \in ...
0
votes
1answer
30 views

Property of greatest integer function

I came across the following mathematical statement in a proof. Can somebody tell me which property of greatest integer function makes it possible? $x + y - \lfloor x + y \rfloor + z - \lfloor x + y - ...
1
vote
1answer
87 views

Is there a simple closed form of $|\alpha(\sqrt{n}-\left\lfloor \sqrt{n} \right \rfloor) + \beta(\sqrt{n}-\left\lfloor \sqrt{n} \right \rfloor)|$?

Let $d_n(x)$ denote the $n$'th digit after the decimal point in $x$. Examples: $d_8(e) = 2,\;d_5(\pi) = 9$ $\alpha(x)$ and $\beta(x)$ are defined this way: $$d_n(\alpha(x)) = \left\{ ...
0
votes
3answers
40 views

$\lfloor x\rfloor + \lfloor y\rfloor \leq \lfloor x+y\rfloor$ for every pair of numbers of $x$ and $y$

Give a convincing argument that $\lfloor x\rfloor + \lfloor y\rfloor \leq \lfloor x+y\rfloor$ for every pair of numbers $x$ and $y$. Could someone please explain how to prove this? I attempted to say ...
0
votes
0answers
26 views

Number of multiples of 6 between -600 and 3400

Could someone please verify if I did this correctly? The number of multiples of 6 between 3400 and -600 should be the floor of (3400 / 6) - the floor of (-600 - 1 / 6) + 1 (to account for zero. Is ...
0
votes
1answer
20 views

How to plot bivariate function involving modulus and floor functions?

I need to plot: $\displaystyle\large|||x|-2|-1|+|||y|-2|-1|=1$ $\displaystyle\large\left\lfloor\frac{|3x+4y|}{5}\right\rfloor+\left\lfloor\frac{|4x-3y|}{5}\right\rfloor=3$ either for finding area ...
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3answers
47 views

The solutions of the following equation

Please consider the following equation: $$\left\lfloor x+\frac{1}{x}\right\rfloor=\frac{2x}{3}$$ where $\lfloor x\rfloor$ is the largest integer not greater than $x$. It is clear that it has not a ...
0
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2answers
50 views

How to calculate with $\lceil \; \;\; \rceil$

I have a problem calculating with ceils. So If I have $\frac{\lceil \frac{n}{2} \rceil}{np}$, this is not the same as $\frac{\lceil \frac{1}{2} \rceil}{p}$. So do you have some rules how to ...
0
votes
1answer
49 views

Calculate floor of a non-negative value, without using ceiling, round, modulo, abs or if

I want to calculate floor of a non-negative value, without using ceiling, round, modulo, abs or if. I can calculate ceiling of a non-negative value, without using floor, round, modulo, abs or if: ...
2
votes
1answer
51 views

How prove this$\lfloor \sqrt{2x-\lfloor\sqrt{2x}\rfloor}\rfloor=\lfloor\frac{\sqrt{8x+1}-1}{2}\rfloor$

Question: let $x\ge 0$, show that $$\lfloor \sqrt{2x-\lfloor\sqrt{2x}\rfloor}\rfloor=\lfloor\dfrac{\sqrt{8x+1}-1}{2}\rfloor$$ My idea: let $\lfloor \sqrt{2x}\rfloor =m$ then ...
7
votes
1answer
142 views

A formula for $\lfloor n\rfloor+\left\lfloor \frac n2\right\rfloor+ \left\lfloor \frac n3\right\rfloor+\ldots+\left\lfloor \frac nk\right\rfloor$?

Is there any formula to calculate: $$\lfloor n\rfloor+\left\lfloor \frac n2\right\rfloor+ \left\lfloor \frac n3\right\rfloor+\ldots+\left\lfloor \frac nk\right\rfloor$$ with $n$ and $k$ positive ...
0
votes
3answers
64 views

Finding the fourier series of floor function

Find the fourier series for $f(x)=\cases{x-[x]\quad x\in\mathbb{R\setminus Z} \\ \frac 1 2\quad x\in\mathbb{Z}}$ on $[-\pi,\pi]$ and its values for $x=1.5,3,5$. In order to find the series I need ...
8
votes
3answers
120 views

Is it true that $\left\lfloor\sum_{s=1}^n\operatorname{Li}_s\left(\frac 1k \right)\right\rfloor\stackrel{?}{=}\left\lfloor\frac nk \right\rfloor$

While studying polylogarithms I observed the following. Let $n>0$ and $k>1$ be integers. Is the following statement true? $$\left\lfloor \sum_{s=1}^n \operatorname{Li}_s\left( \frac{1}{k} ...
0
votes
1answer
67 views

Measurability of the floor function

Let $u(x)=⌊x⌋$, i.e the largest integer not greater than $x$ . Determine $\{u≥a\}$ for all $a\in \mathbb{R}$. Show that $u$ is Borel-measurable. Can anyone help me with this problem?
0
votes
2answers
41 views

Proving that $x-\left\lfloor \frac{x}{2} \right\rfloor\; =\; \left\lfloor \frac{x+1}{2} \right\rfloor$

How would you prove that if $x$ is an integer, then $$x-\left\lfloor \frac{x}{2} \right\rfloor\; =\; \left\lfloor \frac{x+1}{2} \right\rfloor$$ I tried to start by saying that if $x$ is an even ...
1
vote
1answer
149 views

For every $x\in\mathbb R$ and $\varepsilon$ > 0 , there exist $\,q,q'\in\mathbb Q$, such that $q<x<q'$ and $\left |q-q' \right |< \varepsilon$

I'm asked to prove that for every $\varepsilon$ > 0 , there exists two rational numbers $q$ and $q'$ such that $q<x<q'$ and $\left |q-q' \right |<\varepsilon$ where $x$ is a real number. ...
1
vote
2answers
84 views

Proving the general formula [nx] where [.] is the floor function.

I've been trying to solve a exercise that asks me to prove the following generalization for the floor function: $$\lfloor nx\rfloor = \sum_{k=0}^{n-1} {\lfloor x + \frac kn \rfloor}$$ I've already ...
0
votes
3answers
37 views

Limit, Greatest Integer function?

Q. Find $\lim _{x\to 0}\left(1-x+\left[x-1\right]+\left[1-x\right]\right)$ where $\left[y\right]$ denotes the greatest integer function not exceeding 'y'.