For questions related to the fractional part of a number.

learn more… | top users | synonyms

1
vote
4answers
49 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
43 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
25 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
42 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
67 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
52 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
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
20 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
56 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
3answers
47 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
21 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 ...
11
votes
1answer
182 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
70 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
100 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 ...
15
votes
3answers
467 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
votes
3answers
563 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
121 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
56 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
49 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
113 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
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
votes
5answers
115 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
47 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
63 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
63 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
60 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
81 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
123 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
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
votes
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
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
129 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
78 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
474 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 ...