For questions related to the fractional part of a number.

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-1
votes
2answers
80 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
79 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 ...
1
vote
1answer
23 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
51 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 ...
2
votes
2answers
35 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
29 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 ...
0
votes
1answer
65 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
34 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
122 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
88 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
75 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
80 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 $\quad$ ...
4
votes
3answers
206 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
82 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
53 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
38 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)= ...
5
votes
1answer
79 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
62 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 ...
1
vote
1answer
27 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
64 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
59 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
55 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
250 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 ...
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
126 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 ...
2
votes
1answer
75 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
478 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
643 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
371 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
119 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}) + ...
2
votes
1answer
218 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
56 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
122 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 ...
1
vote
0answers
63 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 - ...
1
vote
1answer
70 views

Find any sequence in fractional part of $e^x$?

For any infinite sequence of digits $s$, does an integer number $x$ always exist, such that the fractional part of the solution for $e^x = s$?
1
vote
0answers
67 views

Order of summation of Moebius function with summations of fractional parts as coefficients

I want to show that $\displaystyle\sum_{i=0}^n\left(\mu(i)\sum_{j=1}^{\lfloor\frac{n}{i}\rfloor}\{jx\}\right)=O(n)$ for $x\in (0,1)$. I have tried to use the result that ...
0
votes
1answer
63 views

Inequality involving fractional parts

Find the greatest real number c such that $\{c\sqrt2\}\ge\frac{c}{n}$ for all positive integers n.
1
vote
1answer
95 views

How do evaluate an inequality that involves a fractional part?

I am stuck on how to evaluate whether the following condition is true: Let $\{k\}$ be the fractional part of a real number such that $\{k\} = k - \lfloor{k}\rfloor$. if $\{\frac{x}{2}\} < ...
0
votes
1answer
235 views

Fractional part of Median always .5 or .0

If we find the mid value of two integer number,it's decimal part would always contain .5 or .0 exactly For Example: (5+10)/2=7 .5 (6+2)/2=4 .0 But,in some coding challenge they asked to calculate ...
0
votes
4answers
2k views

Simple math: how to extract the fractional portion from a decimal

Mathematically how do I get the cents from a dollar value (ex: $21.99$)? As a programmer, I would simply convert to a string and grab everything after the decimal... but I would think this would be ...
4
votes
1answer
177 views

Functional Prime Sums

Let $ f: \mathbb{N} \to \mathbb{N} $ be a number-theoretic function satisfying $ f(xy) = f(x) + f(y) $ whenever $ \gcd(x,y) = 1 $. How can I prove that $$ \sum_{\substack{p ~ \text{prime}; \\ p \leq ...
3
votes
2answers
1k views

How do I get the integer part of a number by using basic arithmetic?

While it is trivial to simply remove the fractional part of an irrational or rational number, and in programming I could just use the floor() or ...
0
votes
4answers
145 views

Fractional Parts Proof

OK, here are two questions out of Nathanon's Additive Number Theory from the section on fractional parts ($\S$4.4). I think I'm missing something. I don't understand what there is to prove? Let ...
0
votes
1answer
80 views

A combinatorics problem refer to this problem?

If i define $f(m,n)=$ $$\sum_{1\leq k\leq mn}\left\{ \frac{k}{m}\right\} \left\{ \frac{k}{n}\right\} .$$ Then prove $$f(m+n,n) - f(m,n) =\frac{n^2-n}{4}$$ for all $m$ and $n$. This question came ...
18
votes
3answers
567 views

A sum of fractional parts.

I am looking to evaluate the sum $$\sum_{1\leq k\leq mn}\left\{ \frac{k}{m}\right\} \left\{ \frac{k}{n}\right\} .$$ Using matlab, and experimenting around, it seems to be $\frac{(m-1)(n-1)}{4}$ when ...