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).

learn more… | top users | synonyms

4
votes
3answers
79 views

Challenging $\lim_{x \rightarrow 10} \frac{1}{\lfloor x \rfloor} = \frac{1}{10}$ for $\epsilon=\frac{1}{2}$.

Consider the (incorrect) claim that $$\lim_{x \rightarrow 10} \frac{1}{\lfloor x \rfloor} = \frac{1}{10}.$$ How might I find the largest $\delta$ such that I can challenge $\epsilon = 1/2$? Clearly ...
3
votes
2answers
52 views

Maximum and minimum of a sum involving floor functions of rational numbers (contest question)

This question originates from the 1996 Canada National Olympiad. Let $r_1, r_2, \dots, r_m$ be a given set of $m$ positive rational numbers such that $\sum\limits^{m}_{k=1}{r_k} = 1 \tag{1}$ ...
0
votes
2answers
42 views

Left and Right limits of $\lfloor\lfloor x\rfloor\rfloor$ at $x = 0$

I'm self-studying calculus from Larson's Calculus 8E and on page 102 and I don't understand why $$\lim_{x\to 0^-} \frac{f(x) - f(0)}{x-0} = \frac{\lfloor\lfloor x\rfloor\rfloor - 0}{x} = \infty $$ ...
2
votes
1answer
32 views

Solving recurrence equation with floor and ceil functions

I have a recurrence equation that would be very easy to solve (without ceil and floor functions) but I can't solve them exactly including floor and ceil. \begin{align} k (1) &= 0\\ k(n) &= ...
0
votes
1answer
19 views

Possible value of $x$ so that fractions are in simplest form.

Which of the following could be the possible value of $x$ for which each of the fractions is in its simplest form, where $\lfloor{x\rfloor}$ stands for greatest integer less than or equal to ...
3
votes
2answers
26 views

Prove the following floor function identity

The identity is this: $\lfloor 2x \rfloor + \lfloor 2y \rfloor \geq \lfloor x \rfloor + \lfloor y \rfloor + \lfloor x+y \rfloor$ I am truly stumped. Help is appreciated. Thank you!
1
vote
3answers
121 views

What is $\left\lfloor0.\overline{9}\right\rfloor$? [duplicate]

We know that $0.\overline{9} = 1$ but then what is $\left\lfloor0.\overline{9}\right\rfloor$? My thought process went: $0.\overline{9} = 1$ so therefore $\left\lfloor0.\overline{9}\right\rfloor = ...
1
vote
2answers
40 views

Proving basic floor function inequality: $-1 \lt \lfloor 2x \rfloor - 2 \lfloor x \rfloor \lt 2$

As a direct consequence of the definition of $\lfloor x \rfloor $ I know that $$2x-1 \lt \lfloor 2x \rfloor \le 2x$$ and $$2x-2 \lt 2\lfloor x\rfloor \le 2x$$ How can I use these to show that $-1 \lt ...
1
vote
1answer
42 views

floor function problems

There are two parts of my problem. Given $n$ and $x$, $\lfloor \frac nx \rfloor = q$, what is the maximum possible value for $x$ such that we obtain the same floor value? i.e. $\lfloor \frac nx ...
4
votes
1answer
24 views

Arrangement of any number of objects from $n$ objects

Prove that the total number of arrangements of objects by taking any number of objects from $n$ different objects is $\lfloor e \times n! - 1 \rfloor$, where $e$ is the natural base. I tried it ...
0
votes
1answer
24 views

How many coins do we need to get $k$ amount

In the far away land of coinsville, they use $4$ different coins as currency, $\{1,10,100,200\}$ What is the computational class of the amount of coins (minimal!!) we need to get $k$ amount? Well, ...
4
votes
1answer
108 views

how to sum the floors of ratios n/k when prime factorization of n is known

According to @harald-hanche-olsen the sum of the floors of ratios of $n/k$ is approximately: $$n(\ln n-1-\ln2)<\sum_{k=2}^{n-1}\Bigl\lfloor \frac nk\Bigr\rfloor<n\ln n.$$ If the prime ...
2
votes
3answers
55 views

Proof for $\left\lfloor\frac 1j\left\lfloor\frac nk\right\rfloor\right\rfloor=\left\lfloor\frac n{jk}\right\rfloor$

Problem: For positive integers $n,j,k$, prove that the following holds: $$\left\lfloor\frac 1j\left\lfloor\frac nk\right\rfloor\right\rfloor=\left\lfloor\frac n{jk}\right\rfloor$$ I simply ...
0
votes
1answer
43 views

Show $ \left\lfloor\frac{2n}{p} \right\rfloor - 2 \left\lfloor \frac{n}{p} \right\rfloor \in \{ 0, 1 \}$

I conjecture that $$ \left\lfloor\frac{2n}{p} \right\rfloor - 2 \left\lfloor \frac{n}{p} \right\rfloor \in \{ 0, 1 \}. $$ I know that it is always nonnegative, and equals $1$ for $n < p \le 2n$, ...
3
votes
3answers
140 views

why is limit $\lim_{x\rightarrow 0}⌊\frac{\sin x}{x}⌋ = 0$?

I was evaluating the limit $$f(x) = \lim_{x\rightarrow 0} \left\lfloor\frac{\sin x}{x}\right\rfloor$$ and I substituted the equivalent infinitesimal $\sin(x) \sim x$, obtaining $f(x) = 1$. But on ...
0
votes
1answer
54 views

Does the series: $\sum_{n=1}^\infty (-1)^n \lbrack {\sqrt\frac{n}{2}} \rbrack$ Converge?

Does the series: $\sum_{n=1}^\infty (-1)^n \lbrack {\sqrt\frac{n}{2}} \rbrack$ Converge ? Note: by the brackets I mean the floor function. I tried to substitute numbers and look at the members of ...
2
votes
1answer
55 views

Check my proof of a property of the greatest integer function?

Prove that $\forall n \in \mathbb{Z}, \lfloor x + n \rfloor = \lfloor x \rfloor + n $. Proof: Let $K = \{\ k\ |\ k\in\mathbb{Z},\ k \leq x+n\}$. Then, by definition, $$ \lfloor x + n \rfloor = j ...
3
votes
3answers
63 views

$f(x)$=$\left\lfloor { x }^{ 2 } \right\rfloor -\left\lfloor x \right\rfloor ^{ 2 }$ is discontinuous for all integer values of x except only at x=1

How to prove that $f(x)$=$\left\lfloor { x }^{ 2 } \right\rfloor -\left\lfloor x \right\rfloor ^{ 2 }$ is discontinuous for all integer values of x except only at $x=1$ ? Ya,even I used intuition at ...
1
vote
0answers
60 views

Characterizing the cosets of a cycle of a finite abelian group with a linear combination of floor functions

Prelude Cconsider the finite abelian group $\mathcal G = \prod_{i=1}^A \mathbb Z_{a_i}$ and let $\mathbf s \in \mathcal G$. Let $\mathcal H = \operatorname{grp}({\mathbf s})$ be the subgroup of ...
3
votes
1answer
49 views

Find this closed form $\sum_{k=1}^{n}\left(\lfloor a_{k}\rfloor +\lfloor a_{k}+\frac{1}{2}\rfloor \right)$

Let $\dfrac{1}{a_{k}}=\dfrac{1}{k^2}+\dfrac{1}{k^2+1}+\cdots+\dfrac{1}{(k+1)^2-1}$ I need some ideas to exploit for finding the closed form of $$\sum_{k=1}^{n}\left(\lfloor a_{k}\rfloor +\lfloor ...
7
votes
3answers
68 views

Closed form for the sum $\sum_{k=0}^n a^k \left\lfloor\frac{k}{p}\right\rfloor$

I want a closed form for the sum $$S=\sum_{k=0}^n a^k \left\lfloor\frac{k}{p}\right\rfloor$$ where: $a\ne 1<p<n;\quad p\in\mathbb Z$ I know a related identity, $$\quad\displaystyle ...
4
votes
1answer
68 views

If $\lfloor x^i\rfloor =i,i=1,2,3,\cdots,n$ find the maximum of $n$

Find the maximum $n$ for which there exist a real number $x$ such that $$\lfloor x^i\rfloor =i,\quad i=1,2,3,\ldots,n.$$ $\lfloor x\rfloor =1$,then $1<x<2$, $\lfloor x^2\rfloor =2$ then ...
0
votes
2answers
40 views

Indefinite Integral of Floor Function Integration by Substitution

From knowing the anti-derivative of floor function to be x*floor(x), is it possible to find the derivative of a function contained within a floor function? The ...
0
votes
2answers
68 views

$\mathcal{I}=\int\limits_0^0 \{x\}^{\lfloor x\rfloor}\,\mathrm dx=0\textrm{ or undefined ?}$

Consider the following integral: $$\mathcal{I}=\int\limits_0^0 \{x\}^{\lfloor x\rfloor}\,\mathrm dx$$ Now, my concern is that at $x=0$, the value of the integrand is $0^0$ which is undefined. It's ...
0
votes
4answers
54 views

Prove that $n! \geq {\lfloor n/2 \rfloor}^{\lfloor n/2 \rfloor}$

I am trying to see why it holds that, for $n \in \mathbb{N},$ $$n! \geq {\lfloor n/2 \rfloor}^{\lfloor n/2 \rfloor}.$$ I would appreciate help to see this.
4
votes
3answers
73 views

Solving $7[x]+23\{x\}=191$

For every real number $x$, $[x]$ denotes the largest integer less than or equal to $x$ and $\{x\}=x-[x]$. The number of real solutions of $$7[x]+23\{x\}=191$$ is (a) 0 $\quad$ ...
2
votes
2answers
38 views

Concerning Roots of the cubic equation $f(x)=x^3+x^2-5x-1$ and the Greatest Integer (or Floor) function

The Question I got into a rather tight corner with this question. It says: Let $\alpha, \beta, \gamma$ be the roots of $f(x)=0$, where $f(x)=x^3+x^2-5x-1$. Then, the value of ...
2
votes
1answer
29 views

Are the following expression equivalent?

I need to check wether this expressions are equal but I haven't yet learned enough about the floor function to tell (I also $$ \frac 1 4 \left (5-\cos \left(n \frac \pi 2 \right)-2 (-1)^n \left(1+ ...
0
votes
2answers
36 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
46 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 ...
6
votes
4answers
221 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
25 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
34 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
votes
4answers
85 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 $m$. ...
0
votes
2answers
33 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} ...
1
vote
2answers
51 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
89 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
54 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
238 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
votes
2answers
64 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
110 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
48 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
103 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
68 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
62 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
41 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
104 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 ...
1
vote
2answers
49 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
173 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 ...