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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34 views

Floor inequality with prime

If $a$ and $b$ are positive integers and $a\ge b$ and $b$ is an odd prime, show that: $$\left\lfloor \frac{6a-1}{b}\right\rfloor+\left\lfloor\frac{a}{b}\right\rfloor\ge \left\lfloor ...
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1answer
26 views

Floor inequality: $\lfloor \frac{6a-1}{b}\rfloor+\lfloor\frac{a}{b}\rfloor\ge \lfloor \frac{2a}{b}\rfloor+\lfloor \frac{3a-1}{b}\rfloor+\cdots$

If $a$ and $b$ are positive integers and $a\ge b$, show that: $$\left\lfloor \frac{6a-1}{b}\right\rfloor+\left\lfloor\frac{a}{b}\right\rfloor\ge \left\lfloor \frac{2a}{b}\right\rfloor+\left\lfloor ...
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1answer
31 views

Floor inequality: $\lfloor x+y\rfloor\ge \lfloor x\rfloor+\lfloor y\rfloor$

I remember seeing the inequality $\lfloor x+y\rfloor\ge \lfloor x\rfloor+\lfloor y\rfloor$ somewhere which is true for all reals. So I was wondering what's wrong with this proof? For all reals $a,b$ ...
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1answer
90 views

Solve the system $ x \lfloor y \rfloor = 7 $ and $ y \lfloor x \rfloor = 8 $.

Solve the following system for $ x,y \in \mathbb{R} $: \begin{align} x \lfloor y \rfloor & = 7, \\ y \lfloor x \rfloor & = 8. \end{align} It could be reducing to one variable, but it is ...
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2answers
35 views

Prove that every natural number is a limit of a subsequence

I need to prove that every natural number is a partial limit (limit of a subsequence) to the sequence ${{a}_{n}}=n-{{\left\lfloor \sqrt{n} \right\rfloor }^{2}}$ . I already found that 0 is a partial ...
5
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1answer
140 views

Prove a relationship involving floor functions

I am trying to prove that a particular expression is a lower bound for a very unusually-behaved function. The whole proof will be complete if I can just nail down the details of one technical lemma ...
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1answer
13 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
28 views

Finding the sum of many square root values (greatest integer function)?

I am not really sure how to go about this mathematically, but I used my programming skills to find the answer quite quickly: ...
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0answers
87 views
+100

Find a function so whenever it is near a lattice point $\lim_{x \rightarrow [x_0]}f(x)=[y_0]$

A function $f:\mathbb{R} \rightarrow \mathbb{R}$ is an "easy estimator" if any point on f,$(x_0,y_0)$ is near a lattice point $([x_0],[y_0])$ then $\lim_{x \rightarrow [x_0]}f(x)=[y_0]$. In other ...
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1answer
72 views

Evaluate: $\int_{0}^{0.9} \lfloor x-2\lfloor x\rfloor\rfloor$ [closed]

Evaluate the following integral: $$\int\limits_{0}^{0.9} \big\lfloor x-2\left\lfloor x\right\rfloor\big\rfloor \mathrm dx $$ where $\lfloor \cdot\rfloor$ denotes the greatest integer function (floor ...
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18 views

Integrating a function over the space of all `triangles' given a condition on their geometry (involving the floor function)

Without writing the limits, I need $$ \int\int\int\int a^{\left\lfloor \sqrt{x_i^2+y_i^2} \right\rfloor + \left\lfloor \sqrt{x_j^2+y_j^2} \right\rfloor} \, \textrm{d}x_i \, \textrm{d}x_j \, ...
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1answer
32 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
48 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
20 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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1answer
34 views

Trouble proving floor function is onto?

I'm trying to do a proof of a floor function being onto, but I'm not sure where to go from here. I don't want to ask the question outright because I want to figure it out myself, but I know that if ...
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1answer
81 views

How can I prove the following sequence is convergence [closed]

We know the floor function $[x]$ such that $[2.4]=2$ , $[-2.4]=-3$ , $[2]=2$ . How can I prove the following sequence $$x_n=\frac{[2^n \sqrt2 ]}{2^n}$$ converges to $\sqrt2$ . And in general the ...
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0answers
14 views

Prove that the number of words in a positive integer a in base Beta is 1 + ilog(a)/BPW where Beta = pow(2, BPW)

The number of digits (words) in a base Beta of any positive integer a is 1 + floor(log(a)) where the log is in base Beta. When the base Beta = 2 this is just the number of bits. That is 1 + lg(a) is ...
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0answers
8 views

Minimization of multivariable function which contains floor

Let $f$ be a function such that: $$f(r,h,n) = 4 \pi (r+h)^2 - \big ( \big \lfloor 2 \frac h {r+h} \big \rfloor + n \big ) 2 \pi h (r+h) $$ where $r,h > 0$ and $n \in \mathbb N^*$ as well as ...
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1answer
38 views

Sum of floor functions

For a positive integer $n$, let $$\def\fl#1{\left\lfloor\frac n{#1}\right\rfloor}f(n)=\fl 1+\fl2+\fl3+\cdots+\fl n.$$ Find $f(1,000,000)−f(999,999)$. I know the changes in the floors will take place ...
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58 views

Number Theory : Show that $\sum_{i=0}^\infty$ $ [\frac{n}{2^i}+\frac{1}{2}]$ $=$ $2n$

I was doing some basic Number Theory problems and came across this problem : Show that for any integer $n$ $\geq$ $1$ ; $$\sum_{i=0}^\infty [\frac{n}{2^i}+\frac{1}{2}] = 2n$$ ; where $[x]$ ...
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21 views

How to calculate the Integer portion of a fraction using only +, -, $\div$ and *?

I made something in excel that calculates the days left until a given date, and from that how many weeks were left. I had it so that 9 days displayed as 1.2 using this formula: ...
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1answer
137 views

Prove this Floor function indentity $\sum_{k=0}^{n-1} \bigl\lfloor \frac{ak+b}{c} \bigr\rfloor$

Assmue $a,b,c$ be postive integers. Show that: $$\sum_{k=0}^{n-1} \left\lfloor \frac{ak+b}{c} \right\rfloor = \sum_{k=0}^{\left\lfloor \dfrac{an+b}{c}\right\rfloor} ...
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1answer
17 views

How can we prove that $\left(\frac{1}{2^n}\lfloor 2^nX\rfloor\right)_{n\in\mathbb{N}}\uparrow X$? [closed]

Let $(\Omega,\mathcal{A})$ be a measurable space $X$ be $\mathcal{A}$-$\mathcal{B}(\overline{\mathbb{R}})$-measurable Moreover, let $$X_n:=\frac{1}{2^n}\lfloor 2^nX\rfloor$$ How can we prove that ...
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1answer
50 views

Inequalities with floor function

How large should $n$ be in order for the following inequality to hold? $$\left\lfloor \frac{n}{m} \right\rfloor \leq 2 \left\lfloor \frac{n}{2m} \right\rfloor$$ Thanks.
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6answers
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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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1answer
65 views

Floor function of a factorial

Compute $$\left\lfloor \frac{1000!}{1!+2!+\cdots+999!} \right\rfloor.$$ How can I start with the problem? I thought of dividing by some number, but then I thought that some small numbers when added ...
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1answer
55 views

If $ f(x) = \left\{\frac{3x}{2}\right\}\;,$ Where $\{x\}=x-\lfloor x \rfloor$. Then real solution of $f(f(x))=0$

If $\displaystyle f(x) = \left\{\frac{3x}{2}\right\}\;,$ Where $\{x\}=x-\lfloor x \rfloor$. Then no. of solution of the equation $f(f(x))=0$ and $f(f(f(x)))=0$ and $f(f(f(f(x))))=0$. ...
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3answers
39 views

Quadratic equation involving floor function.

If equations $x^2-3x+4=0$ and $ 4x^2-2\lfloor3a+b\rfloor x+b=0\space (a,b\space\epsilon\space R) $ have a common root then the complete set of values of $a$ is ? I have not yet been able to develop ...
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1answer
56 views

Prove that if $\lim_{x\to\infty}f(x)=L$ exists and finite, and $\lim_{x\to\infty}\lfloor f(x)\rfloor$ doesn't exist then L is an integer

Let $f$ be a continuous function on $(0, \infty)$ s.t $\lim \limits_{x \to \infty}f(x) = L$ exists and finite, but $\lim \limits_{x\to \infty} \lfloor f(x) \rfloor$ doesn't exist. Prove that L ...
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1answer
48 views

Combining Floor Functions

Hey fellow math enthusiasts, $$\left\lfloor \frac{(n+1)^2}{c} \right\rfloor - \left\lfloor \frac{n^2}{c} \right\rfloor = \left\lfloor \frac{(n+1)^2-n^2}{c} \right\rfloor + f(c,n)$$ where $c, n \in ...
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1answer
25 views

Floor function right hand limit

Show $$\lim_{x\to 0^+}\frac{x}a\cdot\left\lfloor\frac{b}x\right\rfloor=\frac{b}a\;.$$ I think I should use boundedness of $\left\lfloor\dfrac{b}x\right\rfloor$, ...
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1answer
55 views

Restating a floor function as a finite sum

It seems to me that a floor function can be expressed as a finite sum that is open to the Möbius function. Does it follow for all nonnegative integers $a$ that: ...
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1answer
24 views

Finding an $f(x,y,n)$ such that $round[f(x,y,n)] = \lfloor\frac xn \rfloor + \lfloor\frac yn \rfloor$

Problem: I have an equation: $$\left\lfloor\frac xn\right\rfloor + \left\lfloor\frac yn\right\rfloor$$ I need to find an equation that does NOT use the floor function, but will take those same two ...
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1answer
17 views

How do I solve $N = \lfloor\dfrac{A-B}{B}\rfloor$ for $A$

So I'm not too familiar with the floor function and I was wondering if there was a simple way of solving $N = \lfloor\dfrac{A-B}{B}\rfloor$ with $A$ alone on the LHS?
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2answers
72 views

Calculate $\lim_{x \to 0^+} x\lfloor 1/x\rfloor$

I need to find the following limit: $$ \lim\limits_{x \to 0^+} x \left\lfloor\frac{1}{x}\right\rfloor $$ Thanks!
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1answer
28 views

limit of a floor function

I came across the following limit in a math book $\lim_{x\to \infty}\frac{(x^x)}{E(x)^{E(x)}} $ where $E(x)$ represent the floor function, and the question was to prove that this limit doesn't ...
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49 views

Geometric progression of terms involving floor function

Which positive real number has the property that $x ,\lfloor x\rfloor,$ and $x-\lfloor x\rfloor$ form a geometric progression (in that order)?
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2answers
61 views

How do I find a closed form for this sequence?

I want to find a closed form for $$\sum_{i=0,1,...}{\left\lfloor\frac{n}{2^i}\right\rfloor}$$ e.g. ...
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2answers
112 views

Integral with floor function

Find the integral $$\int_1^{1000}\frac{dx}{x+⌊\log_{10}(x)⌋}$$ The logarithm is creating some problems along with floor function.
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2answers
366 views

Find $f'(8.23)$ where $f(x)=23|x|−37\lfloor x\rfloor+58\{x\}+88\arccos(\sin x)−40\max(x,0)$

Let $$f(x)=23|x|−37\lfloor x\rfloor+58\{x\}+88\arccos(\sin x)−40\max(x,0).$$ Find $f^\prime(8.23)$. Note: For a real number $x$, $\{x\}=x−\lfloor x\rfloor$ denotes the fractional part of x. I don't ...
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1answer
23 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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Floor function of scale of stopping time with translation is non-increasing

Oké, so this question was one we had with a course of Stochastic Integration, it is however part of bigger proof, but I'll formulate the part I am uncertain about. The question is as follows: $T$ is ...
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3answers
62 views

How to prove this floor function equation?

How can I prove the following equation? $$ \lfloor nx \rfloor = \lfloor x \rfloor + \Big\lfloor x + \frac{1}{n} \Big\rfloor + \Big\lfloor x + \frac{2}{n} \Big\rfloor + \Big\lfloor x + ...
4
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3answers
65 views

$\int_{0}^{n^{2}} \lfloor \sqrt{t} \rfloor\ dt =$?

To find the integral $\int_{0}^{n^{2}} \lfloor \sqrt{t} \rfloor\ dt$ if $n \geq 1$ is an integer and $\lfloor \cdot \rfloor$ is the floor function, I began with the observation that $$\int_{0}^{n^{2}} ...
0
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1answer
27 views

Prove that $-\lfloor x+1 \rfloor = \lfloor -x \rfloor$ for all real $x$ not an integer

Let $\lfloor x \rfloor$ be the greatest integer lower bound of any real number $x$. Prove that if a real $x$ is not an integer, then $$-\lfloor x+1 \rfloor = \lfloor -x \rfloor.$$
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1answer
42 views

Sum of $\lfloor k^{1/3} \rfloor$

I am faced with the following sum: $$\sum_{k=0}^m \lfloor k^{1/3} \rfloor$$ Where $m$ is a positive integer. I have determined a formula for the last couple of terms such that $\lfloor n^{1/3} ...
4
votes
1answer
64 views

Prove or disprove that $\sin\lfloor x\rfloor$ is periodic.

The title says it all. I was plotting random functions on my phone and noticed this graph. I don't think this function is periodic (WA also agrees). Is there a way to prove if a function like this is ...
0
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1answer
17 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 ...
2
votes
2answers
46 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 ...
0
votes
1answer
45 views

What can the floor of a square root be rewritten as?

For example: Let $n$ be natural (positive) number. $\lfloor\sqrt{n+1}\rfloor$ Floor: round down to nearest integer. What can I rewrite this as? Can I break it down or apart?