Questions tagged [floor-function]

The floor function, also known as the greatest integer function, maps a real number $x$ to the greatest integer less than or equal to $x$ (often denoted $\lfloor x \rfloor$). See also (ceiling-function).

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1answer
66 views

Are there ways to solve miscellaneous equations such as $\sin x=\log [x]$ without drawing the graphs?

Consider the example $$\sin x=\log [x]$$ where $[\,·\,]$ represents Greatest Integer Function. It is a miscellaneous equation, and I have been told that the only way to solve it is to draw the graphs ...
-1
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1answer
32 views

Which of the following relations are true?Here $[x]$ denotes the greatest integer less then or equal to $x$ (as in option b) symbol

Which of the following relations are true? $1)$ $(-1)^{\frac{n(n-1)}{2}} = (-1)^{\frac{n(n+1)}{2}}$ $2)$ $(-1)^{\frac{n(n-1)}{2}} = (-1)^{[\frac{n}{2}]}$ $3)$ $(-1)^{\frac{n(n-1)}{2}} = (-1)^{n^2}$ ...
2
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3answers
46 views

Prove an equality with floor function.

Let $p\in \Bbb N \ne 0$ and $x\in \Bbb R$. prove that $$\left\lfloor \frac {\lfloor px \rfloor}{p} \right\rfloor=\lfloor x\rfloor$$ I tried using the double inequality $$\lfloor px\rfloor \le px<...
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1answer
69 views

Continuity of $f(x)=\left\lfloor \frac12 x -1\right\rfloor$ [closed]

How can you define the domain of a floor function, i.e. in interval notation, for which it is continuous? For example, for the following function: $$f(x)=\left\lfloor \frac12 x -1\right\rfloor$$
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1answer
25 views

Is it possible to solve $n=\text{floor}\left(\frac{L-1}{k}\right)$ for $L$?

Is it possible to solve $$n=\text{floor}\left(\frac{L-1}{k}\right), n,k,L \in \mathbb{Z}^+$$ for $L$?
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1answer
40 views

Inequality with floor function supremum infimum

Let $A = \{ x \in \mathbb{R}^+ : x \lfloor \frac{1}{x} \rfloor \leq \frac{3}{4} \}$, with $\lfloor x \rfloor=\max \{d \in \mathbb{Z}: d\leq x  \}$. Find, $\sup{A}$ and $\inf{A}$. If $x>1$, $\...
4
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1answer
64 views

Evaluate $\lfloor n / \lfloor n / \lfloor \sqrt n \rfloor \rfloor \rfloor$ for positive integers $n$.

I'm considering: $$ \left\lfloor {n \over \lfloor n / \lfloor \sqrt n \rfloor\rfloor} \right\rfloor \:\:\:\: \forall n \in \mathbf N^+ $$ which seems to be $\lfloor \sqrt n \rfloor$. Is there any ...
3
votes
2answers
172 views

Calculate $ \biggr\lfloor \frac{1}{4^{\frac{1}{3}}} + \frac{1}{5^{\frac{1}{3}}} + … + \frac{1}{1000000^{\frac{1}{3}}} \biggr\rfloor$

Calculate $ \biggr\lfloor \frac{1}{4^{\frac{1}{3}}} + \frac{1}{5^{\frac{1}{3}}} + \frac{1}{6^{\frac{1}{3}}} + ... + \frac{1}{1000000^{\frac{1}{3}}} \biggr\rfloor$ I am just clueless. I just ...
3
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3answers
150 views

How to calculate the limit $\lim_{x\to1}\lfloor\sin^{-1}(x)\rfloor$?

The question is about finding $$\lim_{x\to1} f(x)$$ where $$f(x) = \lfloor\sin^{-1}(x)\rfloor$$ The function takes the value $1$ at $x = 1$ but while approaching $1$ from the left side, it takes the ...
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1answer
182 views

Proving equivalence of two conditions

$a,b,c,d,p,q$ are integers such that $$c=b-a\quad,\quad 0\leq d\quad,\quad 1\leq a\leq b\leq d+1\quad,\quad 1\leq p\quad,\quad 1\leq q$$ How to argue that the cumbersome condition $$\left\lfloor \...
3
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4answers
93 views

The priority of limits

How are the two expressions different? $$\lim_{x\to0}\bigg\lfloor\frac{\sin{x}}{x}\bigg\rfloor$$ and $$\bigg\lfloor\lim_{x\to0}\frac{\sin{x}}{x}\bigg\rfloor$$ If limit is inside the floor function, ...
1
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3answers
45 views

Solve for $x$ in the equation containing ${\lfloor{x}\rfloor}$ and $\{x\}$

Calculate all possible values of $x$ satisfying, $$\frac{\lfloor{x}\rfloor}{\lfloor{x-2}\rfloor}-\frac{\lfloor{x-2}\rfloor}{\lfloor{x}\rfloor}=\frac{8\{x\}+12}{\lfloor{x}\rfloor \lfloor{x-2}\...
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2answers
49 views

Properties of floors, ceilings and modulus

I'm trying to reduce one calculation in an iterative Successive Over-Relaxation procedure for a program I'm writing. The code that works does this calculation: $$ s = \left\lfloor\frac{b + \left\...
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0answers
46 views

Find the cardinality of the set $A_p$ defined as the following : [duplicate]

For any prime number $p$, $A_p$=the set of integers $d\in \{1,2,3,\dots, n\}$ such that the power of $p$ in the prime factorization of $d$ is odd. Then \begin{align*} A_p= & \lfloor\dfrac{n}{p}\...
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3answers
82 views

Prove $\forall x,y \in \mathbb{R} :\lfloor{x+y}\rfloor=\lfloor{x}\rfloor+\lfloor{y}\rfloor∨\lfloor{x+y}\rfloor=\lfloor{x}\rfloor+\lfloor{y}\rfloor+1$

Prove $∀x,y\ (x,y\in \mathbb{R}: \lfloor{x+y}\rfloor=\lfloor{x}\rfloor+\lfloor{y}\rfloor∨\lfloor{x+y}\rfloor=\lfloor{x}\rfloor+\lfloor{y}\rfloor+1)$ So, I let $\lfloor{x}\rfloor=m ≡ m≤x<m+1$ $...
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1answer
245 views

For any prime number $p$, let $A_p$ be the set of integers $d\in \{1,2,\dots, 999\}$ s. Then what is the cardinality of $A_p$?

For any prime number $p$, let $A_p$ be the set of integers $d\in \{1,2,\dots, 999\}$ such that the power of $p$ in the prime factorization of $d$ is odd. Then what is the cardinality of $A_p$? I ...
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1answer
50 views

Why isn't this true for $x<0$?

Prove that $$\lfloor{x}\rfloor=\bigg\lfloor{\frac{x+1}{2}}\bigg\rfloor+\bigg\lfloor{\frac{x+2}{2^2}}\bigg\rfloor+\bigg\lfloor{\frac{x+2^2}{2^3}}\bigg\rfloor+ \ldots$$ $x\geq{0}$ I asked this ...
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0answers
49 views

Prove that $\lfloor{x}\rfloor=\bigg\lfloor{\frac{x+1}{2}}\bigg\rfloor+\bigg\lfloor{\frac{x+2}{2^2}}\bigg\rfloor+\ldots$ [duplicate]

Prove that $$\lfloor{x}\rfloor=\bigg\lfloor{\frac{x+1}{2}}\bigg\rfloor+\bigg\lfloor{\frac{x+2}{2^2}}\bigg\rfloor+\bigg\lfloor{\frac{x+2^2}{2^3}}\bigg\rfloor+ \ldots$$ $x\geq{0}$ How should I ...
2
votes
1answer
46 views

Prove that, $\left\lfloor{\frac{x}{n}}\right\rfloor=\left\lfloor{\frac{\lfloor{x}\rfloor}{n}}\right\rfloor$ where $n \in{\mathbb{N}}$ [duplicate]

Prove that $$\left\lfloor{\frac{x}{n}}\right\rfloor=\left\lfloor{\frac{\lfloor{x}\rfloor}{n}}\right\rfloor,$$ where $n \in{\mathbb{N}}.$ My Attempt: Let $x=nt$. Then, I need to prove, $$\...
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1answer
46 views

prove that $x∉ℤ⇒\lfloor{-x}\rfloor=-\lfloor{x}\rfloor-1$

prove that $x∉ℤ⇒\lfloor{-x}\rfloor=-\lfloor{x}\rfloor-1$ So, I have that $(\lfloor{-x}\rfloor=n)⇔(n≤-x<n+1)$ And I also have that $(-\lfloor{x}\rfloor-1=n)⇔(\lfloor{x}\rfloor=-n-1)⇔(-n-1≤x<-n)$...
4
votes
2answers
248 views

Inverse floor function

studying a scientific article**, I ran into something I cannot explain: $$l := \left\lfloor{\frac{x+y}{2}}\right\rfloor ,\quad h := x - y \\ x = l + \left\lfloor{\frac{h+1}{2}}\right\rfloor, \quad y ...
1
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1answer
72 views

Convergence of stochastic process divided by floor.

Assume that $N(t)$ is a stochastic process with positive and integer values. We know that: $\frac{N(t)}{t} \xrightarrow{\text{P}} c, \hspace{0.5cm} t \rightarrow \infty$ where $c$ is positive and ...
4
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0answers
55 views

Alternating harmonic series containing floor function

Let $$S\left(a\right)=\sum_{n=0}^\infty \frac{\left(-1\right)^{\left[na\right]}}{\left[na\right]+1}$$ where $a\gt 0$, and $\left[\cdot\right]$ denotes the floor function. Consider $a\in \mathbb{Q}$, ...
4
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2answers
424 views

Given the equation $\sum_{k=11}^{99} \left[x + \frac{k}{100} \right] = 765$ find $[10 \, x]$.

Given the equation $\sum_{k=11}^{99} \left[x + \frac{k}{100} \right] = 765$ find $[10 \, x]$. I have tried this problem in many ways and I think it uses the identity $$[x]+\left[x+\frac{1}{n}\right]...
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2answers
42 views

Show that the number of multiples of $m$ in $[1, x]$ is $\lfloor \frac{x}{m} \rfloor$

I'm self-studying number theory using "A Computational Introduction to Number Theory and Algebra" by Victor Shoup, and this trivially-looking exercise throws me off: Let $m$ $\in$ $\mathbb{Z}$, $m$ ...
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3answers
45 views

Proving $\Big\lfloor{x}/({y}\cdot{z})\Big\rfloor=\Big\lfloor{\lfloor{x}/{y}\rfloor}/{z}\Big\rfloor$ for positive integers

I want to prove that the following identity is true for any positive integers $x,y,z$: $$\Big\lfloor{x}/({y}\cdot{z})\Big\rfloor=\Big\lfloor{\lfloor{x}/{y}\rfloor}/{z}\Big\rfloor$$ Here is a script ...
9
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1answer
674 views

Are ceiling and floor elementary functions?

According to the Wikipedia entry on elementary functions, the trigonometric functions and their inverses are elementary functions. It doesn't seem to me that the floor and ceiling functions should be ...
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2answers
53 views

Given some $n ∈ ℤ$ what conditions must $v$ satisfy for $n \left \lfloor {v} \right \rfloor $ = $\left \lfloor {n v} \right \rfloor $

I'm probably overthinking this. What constraints must you place on $v\in \mathbb R$ : $n \left \lfloor {v} \right \rfloor $ = $\left \lfloor {n v} \right \rfloor $ if $n$ is an arbitrary integer? I ...
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3answers
40 views

Find Range of $a$ if $ \lfloor \frac{1}{3}+2a \sin ^3 x \rfloor $ is an Even function

Find Range of $a$ if $$ f(x)= \lfloor \frac{1}{3}+2a \sin ^3 x \rfloor $$ is an Even function My try: we have $$f(-x)=f(x)$$ $\implies$ $$ \lfloor \frac{1}{3}+2a \sin ^3 x \rfloor=\lfloor \frac{1}{...
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2answers
58 views

Reference Request for solution of Ramanujan Identities

Where can I find the identities of Ramanujan concerning the Floor Function with its solution? Any site you can recomend to me?
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4answers
736 views

Finding the values of $x$ such that $[x^2]=[x]^2$ ($[\cdot]$ denotes the floor function)

I was trying to find the values of $x$ that makes the equation $[x^2]=[x]^2$ true where $[\cdot]$ denotes the floor function. I tried doing the following: Let $x=n+r$ where $n=[x]$ and $0\leq\ r<1$...
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1answer
43 views

Creation of infinite 'step' function

Suppose I have a simple function such as $(\sin \frac{\pi}{5} x)^2$, and I want to create a 'stepped' function that, for each integer $x$, jumps by $(\sin \frac{\pi}{5} x)^2$ - in effect, a stepped ...
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3answers
84 views

Calculate $\lim_{n \rightarrow \infty}$ ($n!e-[n!e]$)?

Calculate $$\lim_{n \to \infty} (n!e-[n!e])$$ I think that it will be $\infty $ as $\lim_{n \to \infty} (n!e-[n!e])= \infty - \infty = \infty $. Is it True/false ??
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2answers
58 views

How Can I Simplify An Inequality With A Floor Function?

I'm trying to convert between pagination by starting index $i$ & length $l$ and pagination by page number $p$ & page size $s$. I've gotten far enough to know that: for given $i$ and $l$ such ...
0
votes
1answer
52 views

Random variable transformation with floor function

Can someone help me with the following question: For each $n\geq 1$, let $X_{n}$ be a random variable following an exponential distribution with mean $n$. Determine $F_{n}$: the distribution function ...
0
votes
1answer
33 views

Find $a, b$ such that $f(x) = ax - \lfloor bx+c\rfloor$ is periodic and find its period, where $ab \ne 0$

Find $a, b$ such that $f(x) = ax - \lfloor bx+c\rfloor$ is periodic and find its period, where $ab \ne 0$ I've tried to do it the following way: $$ f(x) = f(x+T) \\ ax - \lfloor{bx + c}\rfloor = a(x+...
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votes
2answers
62 views

Given some $x ∈ ℝ$, find some $y ∈ ℝ: x < y < \left \lfloor {x+1} \right \rfloor $

Motivation: This Wolfram webpage suggests that you can represent a floor function analytically as: $$\left \lfloor x \right \rfloor := x + \frac{\tan^{-1}(\ \cot(\pi x) \ )}{\pi} - \frac{1}{2} \ \\ \...
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votes
2answers
579 views

Number theory question with floor function

Define $[a]$ as the largest integer not greater than $a$. For example, $\left[\frac{11}3\right]=3$. Given the function $$f(x)=\left[\frac x7\right]\left[\frac{37}x\right],$$ where $x$ is an ...
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2answers
70 views

Finding the Floor Function Explicit Formula for a Sequence

I have found come across two sequences that I know probably both have an explicit formula expressed in terms of the floor function, but I cannot quite figure it out. $0,1,1,2,4,4,5,5,6,8,8,9,9,10,12,...
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votes
2answers
77 views

$\lim_{n \to \infty} \sum_{k=1}^{\left\lfloor\frac{n}{2}\right\rfloor}{\binom{n-k}{k}\frac{1}{2^{n-k}}}$?

Consider the following limit: $$\lim_{n \to \infty} \sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor}{\binom{n-k}{k}\frac{1}{2^{n-k}}}.$$ I can find the limit numerically, but is it possible to ...
0
votes
2answers
60 views

Find the number of possible solution of |[x]-2x|=4

Find the number of possible values of x of the equation |[x]-2x|=4. |x| represent absolute value of x. [x] represent greatest integer lesser than x. I plotted the curve in desmos.com and got the ...
2
votes
2answers
209 views

How is addition and multiplication of step functions defined?

I'm going through "Calculus" by Tom Apostol. And I'm in this section: I think the book assumes that from the example I can extrapolate how the graph for any addition of step functions is done; ...
4
votes
3answers
98 views

Prove that $ \left\lfloor{\frac xn}\right\rfloor= \left\lfloor{\lfloor{x}\rfloor\over n}\right\rfloor$ where $n \ge 1, n \in \mathbb{N}$ [duplicate]

Prove that $ \left\lfloor{\frac xn}\right\rfloor= \left\lfloor{\lfloor{x}\rfloor\over n}\right\rfloor$ where $n \ge 1, n \in \mathbb{N}$ and $\lfloor{.}\rfloor$ represents Greatest Integer $\mathbf{\...
5
votes
2answers
103 views

What is the value of this product: $\prod_{n=1}^\infty \;\frac{3}{1+2 \cos(\frac{\pi}{3^n})} \;=\;? $

emphasized text$$\left[\, \prod_{n=1}^\infty \;\frac{3}{1+2 \cos\left(\frac{\pi}{3^n}\right)}\, \right]\, =\;? $$ Where $\;[\, .]\;$ denotes the integral part function. $\mathbf {My Attempt}$ I ...
0
votes
2answers
67 views

How to solve for x:$ \left \lfloor{x} \right \rfloor\ - n \cdot \left \lfloor{\frac{x}{n}} \right \rfloor\ = y$

Getting solutions to questions with floor functions, like: solve for x:$$ \left \lfloor{x} \right \rfloor\ - n \cdot \left \lfloor{\frac{x}{n}} \right \rfloor\ = y$$ $$st.: \ x ∈\{0, \mathbb{R^{+}} \}...
40
votes
3answers
1k views

Does this pattern continue $\lfloor\sqrt{44}\rfloor=6, \lfloor\sqrt{4444}\rfloor=66,\dots$?

By observing the following I have a feeling that the pattern continues. $$\lfloor \sqrt{44} \rfloor=6$$ $$\lfloor \sqrt{4444} \rfloor=66$$ $$\lfloor \sqrt{444444} \rfloor=666$$ $$\lfloor \sqrt{...
2
votes
0answers
27 views

Proof that a sequence of numbers must always contain a perfect square [duplicate]

Given $f(n) = n + \left \lfloor{\sqrt n}\right \rfloor$ Prove that for any natural number $n$, the sequence $n, f(n), f(f(n)),.......$ must always contain at least one perfect square. Now obviously ...
-1
votes
1answer
81 views

Find range of the function $f(x)=\frac{\left\{x\right\}}{1+(\lfloor x\rfloor)^2}$

Find range of the function $f : \mathbb{R} \to \mathbb{R}$ $$f(x)=\frac{\left\{x\right\}}{1+(\lfloor x\rfloor)^2}$$ My try: Obviously range contains zero, since for integers $\left\{x\right\}=0$ ...
4
votes
5answers
136 views

Number of solutions of $\left\{x\right\}+\left\{\frac{1}{x}\right\}=1$

Find the number of solutions of $$\left\{x\right\}+\left\{\frac{1}{x}\right\}=1,$$ where $\left\{\cdot\right\}$ denotes Fractional part of real number $x$. My try: When $x \gt 1$ we get $$\left\{x\...
1
vote
2answers
42 views

Continuity of a function containing infinite sum of floor function

How to find the points of discontinuity of the following function $$f(x) = \lim_{n\to \infty} \sum_{r=1}^n \frac{\lfloor2rx\rfloor}{n^2}$$