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
18 views

Approximate summation of flooring function

I have this summation: $\sum_{k=0}^x \lfloor{\frac{k}{c}}\rfloor$ Do you have any ideas on any general expressions that can approximate this? P.S I know I can approximate it with a Fourier ...
-3
votes
1answer
37 views

The integer part of $x+1$ is the integer part of $x$ plus $1$ [closed]

How do you solve the proof: If $x$ is a real number, then: $[x+1] = [x] + 1$. For my proof, I tried to describe the interior of the argument inside the parentheses, but I was unsuccessful. Please ...
-1
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1answer
37 views

How to prove this has no real solution?

How to prove that $\left\lfloor x \right\rfloor +\left\lfloor 2x \right\rfloor +\left\lfloor 4x \right\rfloor +\left\lfloor 8x \right\rfloor +\left\lfloor 16x \right\rfloor +\left\lfloor 32x ...
5
votes
4answers
188 views

Floor function to the base 2

I'm not a math guy, so I'm kinda confused about this. I have a program that needs to calculate the floor base $2$ number of a float. Let say a number $4$, that base $2$ floor would be $4$. Other ...
2
votes
1answer
23 views

Multiplying a floor function to a number

Is it correct to write: $\cfrac{\left\lfloor{\cfrac{\pi y^2}{3\sqrt{3}x^2}}\right\rfloor}{n} \times\sqrt{3}x =\left\lfloor\cfrac{\pi y^2}{3xn}\right\rfloor$ ?
1
vote
1answer
29 views

Integer division through multiplication by reciprocal

Please help me to understand (prove) why the following statement is true. For any natural number $w > 0$ and divisor $b \in \left[ 1, 2^w \right)$, if we define a natural number $inv(b)$ such that ...
0
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4answers
83 views

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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2answers
30 views

Solving an inequality involving a floor

Increasing the integer $k$, I can make the floor of $L/k$ smaller than $r$: $$\left\lfloor \frac{L}{k} \right\rfloor \lt r$$ where $L, k, r$ are positive integers, $k\leq \lfloor \frac{L}{2} ...
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2answers
44 views

Solving equations including floor function.

I got a little trouble solving equations that involve floor function in an efficient way. For example : $$ \left\lfloor\frac{x+3}{2}\right\rfloor = \frac{4x+5}{3} $$ In the one above, I get that ...
4
votes
2answers
80 views

For $n>2, n\in\mathbb{Z}$, why is this true: $\left\lfloor 1/\left(\frac{1}{n^2}+\frac{1}{(n+1)^2}+\cdots+\frac{1}{(n+n)^2}\right)\right\rfloor=2n-3$

Let $n>2$ be a positive integer, prove that $$\left\lfloor \dfrac{1}{\dfrac{1}{n^2}+\dfrac{1}{(n+1)^2}+\cdots+\dfrac{1}{(n+n)^2}}\right\rfloor=2n-3?$$ before I use hand Calculation $n=2,3,4$,maybe ...
2
votes
1answer
44 views

find the limit of a floor function

The function f is defined $f(x)=\frac{\lfloor x^2\rfloor}{x^2}$ I need to find the limit of the function at an arbitrary point. For the continuous parts it was fine, and also for right sided limit at ...
10
votes
1answer
220 views

Compare $\sum_{k=1}^n \left\lfloor \frac{k}{\varphi}\right\rfloor$ …

Given two integer sequences \begin{equation*} \displaystyle A_n=\sum_{k=1}^n \left\lfloor \frac{k}{\varphi}\right\rfloor$ \end{equation*} \begin{equation*} ...
0
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2answers
47 views

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

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.
2
votes
1answer
83 views

Find the upper and lower sum of an integral with a floor

I'm having some trouble and looking for some help with a problem i'm trying to solve. Without the floor function it would be easy but the floor has made it a bit trickier: Find the upper and lower ...
0
votes
2answers
43 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, ...
4
votes
5answers
82 views

Prove that $[a/b]+[2a/b]+[3a/b]+…+[(b-1)a/b]=(a-1)(b-1)/2$

If a and b are positive integers with no common factor how to show that $[a/b]+[2a/b]+[3a/b]+...+[(b-1)a/b]=(a-1)(b-1)/2)$,where [.] denotes the greatest integer function? Im not able to understand ...
2
votes
0answers
54 views

Sum of Floor of Square Root: $S = \sum_{k=1}^{n} \lfloor \sqrt{k}\rfloor$

$$ S = \sum_{k=1}^{n} \lfloor \sqrt{k}\rfloor. $$ Hello, I´m trying to solve this summation. I was able to get $$ a_n = 2a_{(n-1)} - a_{(n-2)} $$ for non perfect square numbers and $$ a_n = ...
1
vote
1answer
59 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 ...
1
vote
1answer
27 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 ...
0
votes
1answer
38 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 $a,b$ ...
0
votes
1answer
103 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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vote
2answers
46 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
votes
1answer
159 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 ...
0
votes
1answer
15 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 ...
0
votes
1answer
33 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: ...
-1
votes
1answer
132 views

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 ...
0
votes
1answer
79 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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0answers
20 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 \, ...
1
vote
1answer
38 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, ...
0
votes
1answer
52 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> ...
0
votes
1answer
26 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) > ... > ...
1
vote
1answer
36 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 ...
0
votes
1answer
86 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 ...
0
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0answers
15 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 ...
0
votes
0answers
9 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 ...
0
votes
1answer
40 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 ...
3
votes
2answers
61 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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0answers
25 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: ...
3
votes
1answer
139 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} ...
0
votes
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 ...
0
votes
1answer
55 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
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
votes
1answer
68 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 ...
3
votes
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 $f(f(f(f(x))))=0$. ...
1
vote
3answers
45 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 ...
0
votes
1answer
57 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 ...
2
votes
1answer
51 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 ...
1
vote
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 $\left\lfloor\dfrac{b}x\right\rfloor$, ...
1
vote
1answer
60 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: ...
1
vote
1answer
26 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 ...