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Questions tagged [fractional-part]

For questions related to the fractional part of a number.

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Understanding why a fraction is the same as a division problem

I am trying to understand why fractions are another way to write division. I haven't seen a satisfying explanation for this yet. I know that division when we say for example a divided by b is like ...
Seeking_The_Truth's user avatar
0 votes
4 answers
130 views

Solve the equation $25\{x\}^2 −10x+ 1 = 0$

the problem We denote the fractional part of the number x by $\{x\}$. The sum of the solutions of the equation $25\{x\}^2 −10x+ 1 = 0$ is... my idea I noted $x=a+b, a=\{x\}, b=[x] \implies 25a^2-10a+...
IONELA BUCIU's user avatar
3 votes
2 answers
87 views

Solution-verification: Solve $3x^2-6x+4 = 6\{x\}\bigl(\lfloor x\rfloor - \{x\}\bigr)$

the problem Solve in the set of real numbers the following equation $$ 3x^2-6x+4 = 6\{x\}\bigl(\lfloor x\rfloor - \{x\}\bigr), $$ where $\lfloor x\rfloor$ and $\{x\}$ are the whole part and the ...
IONELA BUCIU's user avatar
2 votes
2 answers
77 views

Show that : $\sqrt{[x]\cdot \{x\}} +\sqrt{x \cdot \{x\}} + \sqrt{[x]\cdot x} \leq 2x$

Show that for any positive real number $x$ the inequality holds: $\sqrt{[x]\cdot \{x\}} +\sqrt{x \cdot \{x\}} + \sqrt{[x]\cdot x} \leq 2x$ where by $[a], \{a\}$ we mean the whole par and fractional ...
IONELA BUCIU's user avatar
0 votes
0 answers
42 views

How to show that $\sum_{j = 1}^{A \left({n}\right)} \left\{{\frac{n - {j}^{2}}{2\, j}}\right\}$ satisfies Weyl's criterion.

We can write this sum in terms of even ($j = 2k$) and odd ($j = 2k-1$) summation index as $$\sum_{j = 1}^{A \left({n}\right)} \left\{{\frac{n - {j}^{2}}{2\, j}}\right\} = \sum_{k = 1}^{\lfloor{A \left(...
Lorenz H Menke's user avatar
3 votes
1 answer
60 views

$\inf_{n\in \mathbb{N}^*} \left\{\frac{3^n}{2^n} \right\} > 0$?

The fractional part is defined of a positive real number $x$ is defined by $\{x\}:=x-\lfloor x\rfloor$. Is it true that $$\inf_{n\in \mathbb{N}^*} \left\{\frac{3^n}{2^n} \right\} > 0 \ ?$$ If so, ...
Nathan Portland's user avatar
2 votes
1 answer
66 views

Determine the set of values of the natural number $n\in N^*$, for which an equation admits nonzero real solutions

The problem Determine the set of values of the natural number $n\in N^*$, for which the equation $\{x\}+\{2x\}+\{3x\}=\lfloor nx\rfloor$ admits nonzero real solutions., where $\{a\}$ is the fractional ...
IONELA BUCIU's user avatar
4 votes
0 answers
172 views

Positive solutions to a linear Diophantine equation

Let $d,d',n\in \mathbb N$ be given. If you want, assume $(d,d')=1$. How many positive integer solutions does $$dx+d'x'=n$$ have? (Assuming $(d,d')=1$). I know there are $n/dd'+\mathcal O(1)$ solutions,...
tomos's user avatar
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1 vote
2 answers
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How to compute the following limit? $\lim\limits_{x\to\ 0^+} {\frac{x-\lfloor x \rfloor}{x+\lfloor x \rfloor}}$ [closed]

$$\lim\limits_{x\to\ 0^+} {\frac{x-\lfloor x \rfloor}{x+\lfloor x \rfloor}}$$ Here, $\lfloor x \rfloor$ represents the floor of $x$. I tried using a graphing calculator (desmos) to plot the function $...
Jesko's user avatar
  • 25
0 votes
1 answer
114 views

Find all $x \in \mathbb{R}$ such that $f: \mathbb{N} \rightarrow \mathbb{R}$, where $f(n) = \{2^n \cdot x\}$ is monotone on $\mathbb{N}$.

Find all $x \in \mathbb{R}$ such that $f: \mathbb{N} \rightarrow \mathbb{R}$, where $f(n) = \{2^n \cdot x\}$ (where $\{x\}$ denotes the fractional part of $x$), is increasing/decreasing on $\mathbb{N}$...
math.enthusiast9's user avatar
3 votes
1 answer
62 views

$F(x) = f([x]) \cdot f(\{x\})$ . Find $f$

Given a function $f: [0,\infty) \rightarrow (0,\infty)$ which admits primitives, and let $F$ be a primitive such that: $F(x) = f([x]) \cdot f(\{x\})$ for all x, and there exists $a>0$ s.t. $f(a)=F([...
RemWheel's user avatar
  • 167
6 votes
3 answers
253 views

Solve the equation $\{x\}+\frac{1}{\{x\}}=2+\left\{\frac{1}{[x]}\right\}$.

the question Solve the equation $\{x\}+\frac{1}{\{x\}}=2+\left\{\frac{1}{[x]}\right\}$, where $[a]$ and $\{ a\}$ represent the whole and fractional parts of the real number $a$ respectively. The idea ...
IONELA BUCIU's user avatar
1 vote
0 answers
48 views

For what sequences $a_n$ does $\lim\limits_{n\to\infty}\int\limits_{0}^{1}\{x+a_1\{x+a_2\{x+...a_n\}\}\}dx$ converge?

As a follow-up to the question Does $\lim\limits_{n\to\infty}\int\limits_{0}^{1}\{x+\frac{1}{2}\{x+\frac{1}{3}\{x+...\frac{1}{n}\}\}\}dx$ converge? where it was shown that $\lim\limits_{n\to\infty}\...
Dylan Levine's user avatar
  • 1,698
0 votes
2 answers
100 views

Solution-verification || Solve the equation $\{ \frac {3x + 5}{x + 2} \}+ \big \lfloor \frac{3x + 2}{x + 1} \big \rfloor = \frac{25}{9}$

Question: Solve the equation $$\left\{ \frac {3x + 5}{x + 2} \right\}+ \left\lfloor \frac{3x + 2}{x + 1} \right\rfloor = \frac{25}{9}$$ where $\lfloor a \rfloor$, $\{a\}$ represents the greatest ...
IONELA BUCIU's user avatar
18 votes
2 answers
545 views

Does $\lim\limits_{n\to\infty}\int\limits_{0}^{1}\{x+\frac{1}{2}\{x+\frac{1}{3}\{x+...\frac{1}{n}\}\}\}dx$ converge?

Where $\{x\}$ is the fractional part of $x$. According to Desmos, with $I_n=\int\limits_{0}^{1}\{x+\frac{1}{2}\{x+\frac{1}{3}\{x+...\frac{1}{n}\}\}\}dx$ $$\begin{array}{c|c} n & I_n \\ \hline 2 ...
Dylan Levine's user avatar
  • 1,698
2 votes
1 answer
123 views

Solutions to $\left\{x^n\right\}=\{x\}^n$

Let $x>1$ be a real number. For a (fixed) natural number $n>1$, is it possible to fully or partially classify all solutions to the equation $$\left\{x^n\right\}=\{x\}^n$$ where $\{\cdot\}$ ...
Dumbest person on earth's user avatar
2 votes
1 answer
140 views

Does $\displaystyle\lim\limits_{a\to\infty}\frac{1}{a}\int\limits_{-\infty}^{a}\{e^x\}dx=\frac{1}{2}$? [closed]

This summation is as far as I got with $L=\displaystyle\lim\limits_{a\to\infty}\frac{1}{a}\int\limits_{-\infty}^{a}\{e^x\}dx$: $$L=\lim\limits_{a\to\infty}\frac{\sum\limits_{n=1}^{a}\left(1-n\ln\left(\...
Dylan Levine's user avatar
  • 1,698
5 votes
1 answer
179 views

What is $\int\limits_0^1\left\{\frac{1}{x\left\{\frac{1}{x}\right\}}\right\}dx$?

$\{x\}$ is the fractional part of $x$. $\{x\}=x-\lfloor x\rfloor$ I ended up with this double summation: $$\lim\limits_{\substack{a\to\infty\\b\to\infty}}\sum_{m=1}^{a}\sum_{n=0}^{b}\left(\frac{1}{m}\...
Dylan Levine's user avatar
  • 1,698
0 votes
2 answers
55 views

Solution verification | Determine the positive rational number $x$ with the property that $x=3^{[\frac{3x-1}{2}-[\frac{3x}{2}]]-1}$

Question: Determine the positive rational number $x$ with the property that $x=3^{[\frac{3x-1}{2}-[\frac{3x}{2}]]-1}$, where $[a ]$ represents the whole part of the real number a. My idea Can you ...
IONELA BUCIU's user avatar
10 votes
4 answers
324 views

Distribution of the fractional parts of $n/1, n/2,\ldots, n/n$ as $n$ tends to infinity

Motivation: I felt excited by the answer of X-Rui in this thread. So, I tried to generalize his answer. I tried to obtain the distribution in $[0,1]$ of the fractional parts of the numbers $n/1, n/2, ....
MikeTeX's user avatar
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15 votes
3 answers
443 views

A curious property of the fractional part

I've found a curious property of the fractional part. Namely, let be given a number $n$. Then among the numbers $ n/1, n/2, n/3, .... n/n, $ the proportion of elements whose fractional part is $\geq 1/...
MikeTeX's user avatar
  • 2,088
2 votes
1 answer
204 views

Solving $\lfloor 2022 x \rfloor+\{2023 x\}=2023$

question Solve in the set of real numbers the equation $\lfloor 2022 x \rfloor+\{2023 x\}=2023$, where (where $\lfloor a\rfloor$ and $\{a\}$ represent the whole part, respectively the fractional part ...
IONELA BUCIU's user avatar
2 votes
1 answer
54 views

On an identity involving fractional parts

Let $p$ and $q$ be two positive coprime integers. I found in a research paper the following identity $$\sum_{m=1}^{q-1}\left\{\frac{mp}{q}\right\}\cot\left(\frac{\pi m}{q}\right)=\sum_{m=1}^{q-1}\frac{...
user45783's user avatar
0 votes
1 answer
136 views

Solution Verification Solve the equation $\frac{38-11x}{6}=\{\frac{2x+1}{6}\}+\{\frac{x+17}{3}\}$ in the set of real numbers

question Solve the equation $\frac{38-11x}{6}=\{\frac{2x+1}{6}\}+\{\frac{x+17}{3}\}$ in the set of real numbers. my idea Because ${a}$ is between $1$ and $0$ we can say that $0\leq\frac{38-11x}{6}<...
IONELA BUCIU's user avatar
4 votes
0 answers
107 views

A characterization of the floor function

Our recent research has arrived at probable uniqueness conditions for the floor (integer part) function. Is it true that: A function $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies the conditions: (a) $...
M.H.Hooshmand's user avatar
1 vote
2 answers
161 views

Prove or disprove: the sequence $a_n = \{ \alpha n \}$ (fractional part) converges if and only if $\alpha \in \mathbb{Z}$

Prove or disprove: the sequence $a_n = \{ \alpha n \}$ (fractional part) converges if and only if $\alpha \in \mathbb{Z}$ I'm quite sure that this statement is true, but I'm having a little difficulty ...
Saelee's user avatar
  • 133
1 vote
2 answers
125 views

Evaluating $\int_0^{100}{e^{\{x\}}}dx$ on Desmos

I was watching this video, which shows the following result: $$\int_0^{100}{e^{\{x\}}}dx=100(e-1)$$ I thought to check the result on Desmos, but it's giving me a different answer! *Note: I am using $\{...
Andrew Sotomayor's user avatar
2 votes
0 answers
52 views

Sum of fractional parts greater not equal to $1$

I was trying to solve Putnam's B6 from 1995: "For any $a>0$,set $S(a)=\{\lfloor{na}\rfloor|n\in \mathbb{N}\}$. Show that there are no three positive reals $a,b,c$ such that $$S(a)\cap S(b)=S(b)...
João Rafael Silva de Azeredo's user avatar
-2 votes
1 answer
99 views

Understanding a strange question [closed]

This question is bizarre to me. I simply don't understand it. Plot the graph of the function $x\longmapsto \{x\}$ Any ideas what this means? What is $\{x\}$, and how can you draw a transformation ...
Superunknown's user avatar
  • 2,973
1 vote
0 answers
64 views

Evaluation of sum and asymptotic expansion of $\sum_{k=1}^{\lfloor{m/2}\rfloor} \left\{{\sqrt{k^2+n}}\right\}$

Find the sum if possible and the asymptotic expansion to high order of $$W \left({m, n}\right) = \sum_{k=1}^{\lfloor{m/2}\rfloor} \left\{{\sqrt{k^2+n}}\right\}$$ where $\left\{{...}\right\}$ is the ...
Lorenz H Menke's user avatar
24 votes
3 answers
673 views

Real number $x$ such that $\{ x^n\}$ is constant for all $n\in S$

The golden ratio satisfies the property that $$\{\phi^{-1}\}=\{\phi\}=\{\phi^2\} = 0.618\cdots$$ where $\{x\}$ is the fractional part of $x$, equal to $x-\lfloor x\rfloor$. Inspired by that, I was ...
Varun Vejalla's user avatar
4 votes
1 answer
110 views

Consider the set $A_N$ of all fractions $\left(\frac{3}{2}\right)^n \pmod{1}$ for $n\le N.$ Prove that $\min(A_N)→0$ as $N→∞.$

This is related to the well-known unsolved problem in number theory that concerns the distribution of $(3/2)^n \pmod{1}$. This sequence is believed to be uniformly distributed. Has this simpler ...
user965964's user avatar
2 votes
0 answers
54 views

If $x > 1$ then does the set $ \lbrace{ \lbrace{ x^n\rbrace}: n\in\mathbb{N} \rbrace}\ $ always contain either $0$ or $1$ as a limit point?

This question is somewhat related to my previous question here. Here, $\lbrace{ \cdot \rbrace}$ means fractional part. Is it true that if $x > 1\ $ then either $0$ or $1$ (or both) are accumulation ...
Adam Rubinson's user avatar
4 votes
2 answers
87 views

Find the domain of this function involving fractional part and greatest integer function

Consider a function $$y=\log_{10}\{\log_{10}\lfloor\log_{10}(\log_{10}x)\rfloor\}$$ where $\{x\}$ denotes the fractional part of $x$ and $\lfloor x\rfloor$ denotes the greatest integer less than or ...
MathStackexchangeIsMarvellous's user avatar
3 votes
1 answer
60 views

Integration by parts of the fractional part function between n and n+1?

If $n \in \mathbb{N}$, then, on $]n;n+1[$, the fractional part function is differentiable and its derivative equals $1$ ? Then, is it possible to integrate by parts this :$\displaystyle \int \limits_{...
someone's user avatar
  • 63
6 votes
1 answer
114 views

Solve the equation: $\frac{x^2 - 2x + 3}{\lbrace x\rbrace} + \frac{2023}{2}{\lbrace x\rbrace} = 2\lfloor x \rfloor + 88$

Problem Solve the equation $\frac{x^2 - 2x + 3}{\lbrace x\rbrace} + \frac{2023}{2}{\lbrace x\rbrace} = 2\lfloor x \rfloor + 88$, where $\lfloor x \rfloor$ and $\{x\}$ are the greatest integer less ...
JHumpdos's user avatar
  • 167
2 votes
1 answer
82 views

Why is $\{x/2\}$ always either $\{x\}/2$ or $\{x\}/2 + 1/2$?

I was working on a problem involving floor functions and fractional parts. The question is to prove that $\lfloor x \rfloor - 2 \cdot \lfloor \frac{x}{2} \rfloor$ is either $0$ or $1$. In one of the ...
hikaru jakafura's user avatar
1 vote
1 answer
49 views

Question regarding finding domain .. want to know is there any better method

We need to find the domain of $$\dfrac{1}{[x]^2-x-2\{x\}}$$ I tried to find where the denominator is becoming $0$, then I will remove those value of real numbers and I will have my domain. Simplifying ...
Nandini's user avatar
  • 159
0 votes
1 answer
92 views

Inequality involving the distance to the nearest integer

i've been searching for several days trying to prove this inequality below. Let $\alpha$ an irrationnal number. First let's write for $n,j>0$ some integers, $\sigma_j^{+}(n)=1$ if $0<\{n\alpha\}&...
OdeurAtroce's user avatar
2 votes
0 answers
63 views

Right Hand Limit involving fractional part

The value of $$L=\lim _{x \rightarrow[a]^{+}} \frac{\mathrm{e}^{\{x\}}-\{\mathrm{x}\}-1}{\{\mathrm{x}\}^{2}}$$ can be: (A) $[a]$ (B) $2$ (C) $\{a\}$ (D)$\frac12$ where where [.] and {⋅} denotes ...
Wolgwang's user avatar
  • 1,563
4 votes
1 answer
461 views

Fractional part of $(3+\sqrt{2})^n$

For $r\in\mathbb{R}$, define $\Vert{r}\Vert:=\inf_{n\in\mathbb{Z}}\vert r-n\vert$. By Question 4208947 and Question 1536761, we know that $\lim_{n\rightarrow \infty}\Vert(3+2\sqrt{2})^n\Vert=0$ and $\...
Cofield's user avatar
  • 123
1 vote
1 answer
55 views

Upper limit of sequence involving the fractional part

In the survey "The Problems Submitted by Ramanujan to the Journal of the Indian Mathematical Society", the lower limit of the sequences $n\{n\sqrt 2\}, n\{n\sqrt 3\}$, where $\{\cdot\} $ ...
user1150324's user avatar
0 votes
1 answer
60 views

What is the measure of this set?

I was wondering what is the 2 dimensional Lebesgue measure of this set $\Gamma^2 := \{ (\{at\},\{bt\}) \; : \; t \in \mathbb{R} \} \subset \mathbb{R}^2$ Where $\{x\}$ denotes the fractional part of ,$...
Paul's user avatar
  • 1,374
3 votes
1 answer
110 views

Find $\sum_{i=1}^{200}f(i)$

$$f(x)=\left\{ \begin{array}{ c l } \left[\frac{1}{\{\sqrt{x}\}}\right] & \quad \textrm{if $x≠k^2$} \\ 0 & \quad \textrm{otherwise} \end{array} \right.$$ Where $x,k\in \mathbb{N}$ and [.] ...
Leibniz-Z's user avatar
  • 1,019
1 vote
0 answers
70 views

Lebesgue integration of fractional part in on $L^1[0,1]$ [duplicate]

I was trying to solve the following problem: Suppose $h\colon [0,1]\to \mathbb{R}$ is a continuous function such that $\int_0^1 h(x)dx=0$. For $x\ge 0$, let $\{ x \}$ represent the fractional part of $...
user1136316's user avatar
1 vote
2 answers
106 views

Find the real numbers $x$ such that $\{x\} =\frac {x-3}{2\lfloor x\rfloor-5}.$

Determine the real numbers that verify the relation $$\{x\} =\frac {x-3}{2\lfloor x\rfloor-5}$$ where $\{x\}$ represents the fractional part, respectively $\lfloor x\rfloor$ represents the whole part ...
user avatar
0 votes
1 answer
194 views

Is the set $\{\textrm{fractional part of } n\sqrt{2}|n \in \mathbb{N}\}$ dense in $[0,1]$ [duplicate]

Let $m,n \in \mathbb{N}$ such that $m$ is unequal to $n$ then the fractional part of $m\sqrt{2}$ is unequal to the fractional part of $n\sqrt{2}$. So my questions is: is it true that the set $\{\...
user avatar
0 votes
0 answers
25 views

Let $\iota(z)\in[0,1)$ denote the fractional part of $z\in\mathbb R$. Can we derive a formula for the toroidal distance of $\iota(z_1)-\iota(z_2)$?

Let $$\left|x\right|_{(-1,\:1)}:=\left.\begin{cases}|x|&\text{, if }|x|\le\frac12;\\1-|x|&\text{, otherwise}\end{cases}\right\}\;\;\;\text{for }x\in(-1,1).$$ Now, let $$\iota:\mathbb R\to[0,1)\...
0xbadf00d's user avatar
  • 13.9k
2 votes
0 answers
45 views

Benford and logarithms: are the fractional parts of the logarithm well-distributed on average?

What, if anything, can be said about the distribution of $$ X_n = \sum_{k=1}^n \{\alpha\log k\} $$ where $\alpha$ is a nonzero real constant and $\{x\}=x-\lfloor x\rfloor$ is the fractional part of $x$...
Charles's user avatar
  • 32.3k
-2 votes
1 answer
62 views

An interesting integral about γ [closed]

It is well known that $$\gamma = \lim_{n \to +\infty} (H_n − \ln(n)) = 0.5772(...)$$ where $H_n$ is the sum of reciprocals of all integers from $1$ to $n$. Prove that $$\int_1^\infty \frac{\{ x\}}{x^...
Pulkit Sabharwal's user avatar

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