For questions related to the fractional part of a number.

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1answer
37 views

Grade 6 Math-Percentage homework help… Thank you. [on hold]

If 25% of the boys left the computer club, the ratio of the number of boys who remained to the number of girls would be 5:8. On Children's Day celebration, each girl received 4 sweets fewer than each ...
5
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2answers
75 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.
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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 ...
4
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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 $\quad$ ...
4
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3answers
117 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 ...
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4answers
62 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, ...
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5answers
51 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
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1answer
29 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
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0answers
43 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
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1answer
70 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
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2answers
56 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
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0answers
53 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
24 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
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3answers
60 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
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2answers
53 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
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2answers
32 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 ...
12
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1answer
208 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 ...
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3answers
76 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
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2answers
114 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 ...
14
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3answers
472 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 ...
12
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3answers
611 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 ...
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1answer
226 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
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1answer
65 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 ?
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1answer
50 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
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1answer
129 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 ...
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1answer
55 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
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5answers
117 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 ...
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0answers
52 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
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1answer
67 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 $e^x = s$?
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0answers
65 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
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1answer
61 views

Inequality involving fractional parts

Find the greatest real number c such that $\{c\sqrt2\}\ge\frac{c}{n}$ for all positive integers n.
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1answer
85 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
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1answer
159 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 ...
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4answers
1k 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
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1answer
174 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
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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
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4answers
135 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
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1answer
79 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 ...
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3answers
500 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 ...