For questions related to the fractional part of a number.

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0
votes
1answer
57 views

Integral of Fractional Part $\int_{0}^{1} \{ \frac{1}{x} \}dx$

Does the integral exist? $\displaystyle\int_{0}^{1}\{\frac{1}{x}\}dx,\quad$ where {x} is the fractional part. I have broken it into $$\displaystyle\int_{0}^{1}\frac{1}{x}-\lfloor \frac{1}{x} \rfloor ...
4
votes
1answer
68 views

Calculate a limit $\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \Big\{\frac{k}{\sqrt{3}}\Big\} $

The problem is to calculate a limit $$ \lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \Big\{\frac{k}{\sqrt{3}}\Big\} $$ where {$\cdot$} is a fractional part. I believe that this limit is equal to $\...
7
votes
3answers
173 views

The equation $\{x^2\} + \{x\}=1$ has no solution over positive rationals

Prove there is no positive rational $x$ so that $$\{x^2\} + \{x\}=1 \tag1 $$ Let $x=\frac p q$ and $p=qc+r, p, q, c, r \in \mathbb{N}, 0 \le r \lt q$ From (1) $\{ 2c \frac r q + (\frac r q)^2\} +...
3
votes
2answers
95 views

The graph of the function $f(x)= \left\{ \frac{1}{2 x} \right\}- \frac{1}{2}\left\{ \frac{1}{x} \right\} $ for $0<x<1$

Let for reals $$\{x\}=\text{Frac}(x)$$ the fractional part function, take for example the more common definition, the first (there is a different definition as you see in this MathWorld's Page, ...
1
vote
1answer
44 views

Interval for the solutions of $\{x+1\}<x^2-2x$ where $\{x\}$ is the fractional part of $x$.

Find the interval(s) which contain solutions of $$\{x+1\}<x^2-2x$$ where $\{x\}$ is the fractional part of $x$. I was told that one way of solving this would be graphically. However I generally ...
1
vote
2answers
55 views

Proving the fractional equation: $\{2^{n-1}\sqrt{3}\}=0.b_nb_{n+1}\ldots_{(2)}$

Prove that $$\{2^{n-1}\sqrt{3}\}=0.b_nb_{n+1}\ldots_{(2)}$$ where $\sqrt 3 = 1.b_1b_2b_3 \dots _{(2)}$. (Note: $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$.) I am not sure ...
2
votes
3answers
358 views

WolframAlpha function that returns the 'decimal' part of a number [closed]

Is there a function or command in Wolfram Alpha for getting only the decimal part of a number? Something like this: DecimalPart(3.4231) = 0.4231 I will be using ...
7
votes
1answer
1k views

A curious property of $\operatorname{frac}(e\cdot k)$

Let $\alpha > 0$ be a real number and let us consider the set $S(\alpha)$ of those natural numbers $n$ such that the fractional part of $\alpha \cdot n$ "begins" with the representation of $n$ (in ...
12
votes
3answers
197 views

Evaluate $\int_0^1\int_0^1 \left\{ \frac{e^x}{e^y} \right\}dxdy$

I want compute this integral $$\int_0^1\int_0^1 \left\{ \frac{e^x}{e^y} \right\}dxdy, $$ where $ \left\{ x \right\} $ is the fractional part function. Following PROBLEMA 171, Prueba de a), last ...
2
votes
2answers
67 views

If $x$, $\{x\}$, $\lfloor x\rfloor$ are in G.P, find $x$.

If $x$, $\{x\}$, $\lfloor x\rfloor$ are in Geometric Progression, find $x$; $x \neq 0$. Here, $\{x\}=x-\lfloor x\rfloor$ Some properties are pretty evident: $$0\leq \{x\} < 1 \tag{1}$$ $$\...
0
votes
1answer
23 views

If $l>10$ and $\{a\}>\frac{1}{l}$ then there exists $k \in \{1,\dots, l\}$ such that $\{ka\} \in [1/10,2/10]$

It is probably a simple question, but I wasn't able to solve it. Let $n=10$ and $l>n$ an integer. Let $a>0$ a real number such that $\{a\}>\frac{1}{l}$ (where $\{ \cdot \}$ denotes the ...
0
votes
1answer
34 views

Simplifiying (or getting the exponent thing outside) the expression $\{({\rm complicated\, stuff})^{1/n}\}$ [closed]

So i have like a very complicated expression involving many variables, and i want to simplify it, the only problem comes from the fact that this complicated expression is itself inside a fractional ...
-1
votes
2answers
91 views

Prove that limit of the fractional part of $\sqrt{n^2+n}$ is $\frac{1}{2}$ [duplicate]

Prove that $$\operatorname{frac}(\sqrt{n^2 + n}) \to \frac{1}{2}$$ ($n \in \mathbb{N}$, $\operatorname{frac}$ is fractional part of number) I think I should use just definition of limit and find $N$ ...
4
votes
1answer
87 views

Summation of fractional parts $\frac{m}{n}$, where $2 \leq n < m$ (amateur)

I am looking for the result of the sum of the fractional part of the following number: $$f(m):=\sum_{n=2}^{m-1}Frac\left(\frac{m}{n}\right)$$ After some research I have found $2$ possible solutions:...
1
vote
1answer
29 views

fractional chromatic number

A fractional chromatic number of a graph $G=(V,E)$, is $min~\sum_{I}y_{I}$, and for every vertex $v\in V$ we have $\sum_{\{I:v \in I\}}y_{I} \ge1$(the condition). I'm just a little confused on ...
0
votes
1answer
61 views

Find $\lim_{x\to 0^{\pm}}f(x)=\frac{\arcsin(1-\left\{x\right\})\times\arccos(1-\left\{x\right\})}{\sqrt{2\left\{x\right\}}\times(1-\left\{x\right\})}$

If $f(x)=\frac{\arcsin(1-\left\{x\right\})\times\arccos(1-\left\{x\right\})}{\sqrt{2\left\{x\right\}}\times(1-\left\{x\right\})}$ Find $\lim_{x\to 0^+}f(x)$ and $\lim_{x\to 0^-}f(x)$ where $\left\{x\...
2
votes
2answers
39 views

What is a field with bounded whole and unbounded fractional part called?

Given numbers of the form: $W + \frac n d$ where $d \gt n \ge 0, W \ge 0$, and all are integers, when defining addition and multiplication on these numbers, I want $W$ to be bound by a positive ...
0
votes
0answers
32 views

Solve system with complex fractional equations

I need to maximize a function with equality constraint, so I made the Lagrangian function and I found the partial derivatives, which are these: $$ \frac{|h_1|^2}{\sigma^2(\frac{x b}{\sigma^2+wd}+\...
0
votes
1answer
68 views

What is the closest fraction (that isn't something like 31415…/1000…) that gets you pretty close to pi?

I'm just wondering what is the closest fraction (question for math nerds and geniuses) that isn't like pi/length-of-pi that gets you relatively close (like accurate to the 20th place) to pi? For ...
1
vote
1answer
35 views

What is $\operatorname{frac(x)}$ or $\{x\}$?

I understand this is an opinion kind of a question...but still: Well I know that $\operatorname{frac(x)}$ or $\{x\}$ stands for the fractional part of $x$ but how is it exactly defined? Quoting ...
5
votes
1answer
130 views

$\sin x$ as a sum involving fractional parts

Does there exist a formula giving a sense to the formal equation $$ \sin x=-\pi\sum_{n=1}^{+\infty}\frac{\mu(n)}{n}\left\{\frac{nx}{2\pi}\right\}, $$ where $\mu$ is the Möbius function, $\{\cdot\}$ ...
5
votes
2answers
92 views

What is $x$ if $\{x\}+\{\frac{1}{x} \}=1$ ? ({} - fractional part)?

What is $x$ if $$\large\{x\}+\left\{\dfrac{1}{x}\right\}=1$$($\{\}$ - fractional part)? I need a direction, or a proof, please answer descriptively. Thank you very much.
0
votes
2answers
76 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 ...
4
votes
3answers
88 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 $\quad$ (b) 1 ...
4
votes
3answers
272 views

Calculate fractional part of square root without taking square root

Let's say I have a number $x>0$ and I need to calculate the fractional part of its square root: $$f(x) = \sqrt x-\lfloor\sqrt x\rfloor$$ If I have $\lfloor\sqrt x\rfloor$ available, is there a ...
1
vote
4answers
90 views

How find the fractional part of $5^{200}$ divided by $8$?

Finding the fractional part of $\frac{5^{200}}{8}$. I've had this problem given to me (we're learning the Binomial Theorem and all.) So obviously I thought I'd apply the binomial theorem to it, ...
1
vote
5answers
54 views

Irrational number multiplied by its fractional part becomes rational (SOLVED)

Here's a Korean middle school midterm problem I've been struggling for quite some time now. "$X$ is an irrational number such that $X>0$, and $Y$ is fractional part of $X$. If $$X^2+Y^2=27$$, find ...
3
votes
1answer
42 views

Addition of Fractional Part Function

my question is simple. For some reason I can't seem to deduce whether the statement: {x} + {y} = {x+y} Is true, where $x,y \in \mathbb{Q} $ and {x} denotes the fractional part of x. This really is ...
0
votes
0answers
45 views

solving the partial diffential ODE $ y^{s} (x)=y(x) $

how could i solve the differential equation with frational derivatives $$ \frac{d^{s}}{dx^{s}}y(x) =y(x)$$ here 's' is a real number my idea is to make the ansatz with the series $ y(x)= \sum_{n=0}^...
5
votes
1answer
85 views

prove the inequality with fractional parts

$$ \frac{n^k-n}{2} \leq \left\{\sqrt[k]{1}\right\} + \left\{\sqrt[k]{2}\right\} + \dots + \left\{\sqrt[k]{n^k}\right\} \leq \frac{n^k-1}{2} $$ how it can be proven?
3
votes
2answers
69 views

Numbers $a$ that are the sum of the fractional parts $\{x^2\} + \{x\}$ for some $x$

Are there infinitely many rational numbers $a\in\mathbb{Q}$ with the following property: $\{x^2\}+\{x\}=a$ for infinitely many $x\in\mathbb{Q}^+$
0
votes
0answers
54 views

Conversions of real numbers

Given functions $f:\Bbb Z_+\to \Bbb Z_n $ and $g:Z_+\to \Bbb Z_m$ and suppose $$\displaystyle\sum_{k=1}^\infty f(k)\cdot n^{-k}=\sum_{k=1}^\infty g(k)\cdot m^{-k}$$ Is there a method to express $f(k)...
1
vote
1answer
30 views

Partition of fractional parts where each sum of them has to be at least 1

Let $ a_1,\ldots,a_t \in \mathbb{Q} \setminus \mathbb{Z} $ be with $ \sum_{i=1}^t \lbrace a_i \rbrace \in \left[k,k+1\right) $ for some $ k \in \mathbb{N} $ with $ k \ge 4 $. Here $ \lbrace x \rbrace $...
0
votes
3answers
68 views

What is the value of $\left\{\frac{3^{1001}}{82}\right\}$

Let $$x=\left\{\frac{3^{1001}}{82}\right\}$$ where $\{\}$ denotes fractional part. What is the value of $x$? First I noticed that $x=\frac y{82}$ for some $y\in\mathbb{Z}$ and $0\le y\le81$. But what ...
0
votes
2answers
68 views

fractional part of the square of natural number

How can if prove that the sequence :$$a_n\:=\left\{\sqrt{n}\right\}\left(fractional\:part\:of\:\sqrt{n}\right)\:=\:\:\sqrt{n}\:-\:\left[\sqrt{n}\right]$$ is bounded from above by 1? So far i try ...
0
votes
2answers
73 views

Expressing n mod m in terms of floor values?

I'm trying to prove the expression: $$\left\lceil\frac{n}m\right\rceil = \left\lfloor n+m-\frac1m\right\rfloor\;,$$ where $n,m$ are integers` So I've come across this article (PDF) which gives a ...
13
votes
1answer
268 views

A fractional part integral giving $\frac{F_{n-1}}{F_n}-\frac{(-1)^n}{F_n^2}\ln\left(\!\frac{F_{n+2}-F_n\gamma}{F_{n+1}-F_n\gamma}\right)$

I've been asked to elaborate on the following evaluation: $$ \begin{align}\\ \displaystyle {\large\int_0^{1}} \!\cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {\ddots + \cfrac 1 { 1 + \psi (\left\{1/x\...
0
votes
3answers
82 views

Is the derivative of $\{x\}$ on $(0,1)$ always equal to $1$?

Define the function $f:(0,1)\to[0,1]$ where $f(x)=\{x\}$ is the fractional part of $x$. Am I correct in thinking that $f'(x)=1$ for $x\in(0,1)$? I'm asking because I think what I say is correct, but ...
0
votes
2answers
138 views

Evaluating $\displaystyle\sum_{x=a}^b \left\lfloor {\frac{k}{x}} \right\rfloor$

I'm trying to find a nicer form to evaluate this sum, but the floor function is throwing me off. $$\sum_{x=a}^b \left\lfloor {\frac{k}{x}} \right\rfloor$$ This is the most I've been able to do so far,...
2
votes
1answer
83 views

Evaluating an Indefinite Integral involving $\{x\} = x - \lfloor x \rfloor$

This is probably a pretty simple question, but I just want to check something that I'm not completely sure about. I want to evaluate $\int{\{x\}}^ndx$, with $\{x\} = x - \lfloor x \rfloor$. Instead of ...
14
votes
3answers
487 views

Equality of sums with fractional parts of the form $\sum_{k=1}^{n}k\{\frac{mk}{n}\}$

I recently encountered the following equality ($\{\}$ denotes fractional part): $$\sum_{k=1}^{65}k\left\{\frac{8k}{65}\right\}=\sum_{k=1}^{65}k\left\{\frac{18k}{65}\right\}$$ and found it very ...
14
votes
3answers
676 views

Another integral for $\pi$

Here is a new integral for $\pi$. $$\int_{0}^{1}\sqrt{\frac{\left\{1/x\right\}}{1-\left\{1/x\right\}}}\, \frac{\mathrm{d}x}{1-x} = \pi $$ where $\left\{x\right\}$ denotes the fractional part of ...
1
vote
1answer
523 views

How to find out the period of fractional part of x

I came across this solved example in a book, it says - Find the period of the function : $f(x)=\sin(4\pi x)+\{3x\}$, where $\{x\}$ denotes the fractional part of $x$. Now I know that if $f(x)$ is ...
2
votes
1answer
180 views

limit involving sine of fractional part

What can you say about the following limit : $$ \lim_{x\rightarrow 1} \dfrac{x\sin\lbrace x\rbrace}{x-1} $$ where $\lbrace x\rbrace$ is the fractional part of x Whether this limit exists ?
0
votes
1answer
53 views

Verify my proof on elementary number theory

I've tried to prove this theorem, which is very simple, but is a kind of practice for me. Let $a,b$ be two positive integers. Therefore, if $a+b$ is a composite number, $frac(\frac{a}{l}) + frac(\...
2
votes
1answer
293 views

Irrational fractional parts - patterns

Some fractional-part list plots are: $\text{listplot of }|[\pi x]-\pi x|\text{, for }x \in \mathbb{Z} \text{ and } \text{listplot of }|[ex]-ex|\text{, for }x \in \mathbb{Z}$ $\text{listplot of ...
1
vote
1answer
59 views

Relations between $\sqrt x$ and $\sqrt{x+n}$

Is there any relation between $\sqrt x$ and $\sqrt{x+n}$? I am interested in the fractional part mostly. n and x are both positive integers, n is much greater than x.
0
votes
5answers
123 views

Prove that $\left\lfloor \lfloor x/2\rfloor/2 \right\rfloor=\lfloor x/4\rfloor$ for all $x$. [duplicate]

This I approached the problem. I let $x = n + e$ where $n$ is an integer and $e$ is a decimal less than $1$ but not less than $0$. I substituted that into the equation to get $\left\lfloor \lfloor (n+...
1
vote
0answers
75 views

Solving a system of equations with fractional parts and a system with round parts

I have the following two systems of equations: $a = x_{11} - \{x_{11} + \frac{4 - \sqrt{2}}{7}b + \frac{4 - \sqrt{2}}{7}c + \frac{2\sqrt{2} - 1}{7}d\}$ $b = x_{12} - \{x_{12} + \frac{4 - \sqrt{2}}{7}...
1
vote
1answer
79 views

Finding limit points of {$\sqrt n$}

How can I find limit points of {$\sqrt n$}, where {.} represents the fractional part of a number. Intuitively it should be $[0,1]$, but what is a rigorous argument?