Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [fractional-part]

For questions related to the fractional part of a number.

If $r$ is a real number, we can define the floor function $$\lfloor r \rfloor = \max \{n \in \mathbb{Z} : n \le r\}$$

to be the greatest integer which is not larger than $r$. The fractional part of $r$, frequently written $\{r\}$ or $r \bmod 1$, is then defined to be $$\{r\} = r - \lfloor r \rfloor$$

The fractional part of any number is thus a non-negative real number which is strictly less than $1$. The fractional part of a number is rational if and only if the number itself is rational.

Fractional parts satisfy the inequality $$\{x+y\}\leq\{x\}+\{y\},$$ with equality iff the RHS is less than $1$. Another useful fact is that, for $x\in\mathbb R\setminus\mathbb Q$, the sequence $$a_n=\{nx\}$$ is equidistributed (and in particular, dense) in $[0,1]$.

Questions involving fractional parts can often be tagged with as well.

Reference: Fractional part.

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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 wondering for what subsets $S$ of…
Varun Vejalla
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For $x\in\mathbb R\setminus\mathbb Q$, the set $\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$ is dense on $[0,1)$

Let $x\in \mathbb{R}$ an irrational number. Define $X=\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$. Prove that $X$ is dense on $[0,1)$. Can anyone give some hint to solve this problem? I tried contradiction but could not reach a proof. I spend part…
EQJ
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Minimum values of the sequence $\{n\sqrt{2}\}$

I have been studying the sequence $$\{n\sqrt{2}\}$$ where $\{x\}:= x-\lfloor x\rfloor$ is the "fractional part" function. I am particularly interested in the values of $n$ for which $\{n\sqrt{2}\}$ has an extremely small value - that is, when…
Franklin Pezzuti Dyer
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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 $x$. Do you have any proof?
Olivier Oloa
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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 $m,n$ are relatively prime. How can we prove…
Eric Naslund
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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\right\}+1)}}}} \:\mathrm{d}x & =…
Olivier Oloa
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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), as detailed in the last paragraph of page 109 and the…
user243301
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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 & \frac{1}{2} \\ \hline 3 & \frac{19}{36} \\…
Dylan Levine
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Is $1$ a limit point of the fractional part of $1.5^n$?

It is an open problem whether the fractional part of $\left(\dfrac32\right)^n$ is dense in $[0...1]$. The problem is: is $1$ a limit point of the above sequence? An equivalent formulation is: $\forall \epsilon > 0: \exists n \in \Bbb N: 1 -…
DHMO
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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 $\geq 1/2$ tends to $2.588\ldots$ as $n$ tends to…
MikeTeX
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Sum of $\{n\sqrt{2}\}$

I would like to prove (rigorously, not intuitively) that $$\sum_{n=1}^N \{n\sqrt{2}\}=\frac{N}{2}+\mathcal{O}(\sqrt{N})$$ where $\{\}$ is the "fractional part" function. I understand intuitively why this is true, and that's how I came up with this…
Franklin Pezzuti Dyer
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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 interesting as most of the individual summands on one…
ruadath
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The fractional part of $n\log(n)$

When I was thinking about my other question on the sequence $$p(n)=\min_a\left\{a+b,\ \left\lfloor\frac {2^a}{3^b}\right\rfloor=n\right\}$$ I found an interesting link with the sequence $$q(n)=\{n\log(n)\}=n\log(n)-[n\log(n)]$$ the fractional part…
E. Joseph
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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 $n/1, n/2, .... n/n$ as $n$ tends to $\infty$. There may be…
MikeTeX
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What is the behaviour of the first $n$ digits of ${\underbrace{99\dots99}_{n\text{ nines}}}^{\overbrace{99\dots99}^{n\text{ nines}}}$ as $n\to\infty$

For a natural number $n$, let $f(n)$ denote the first $n$ digits of the decimal expansion of $${\underbrace{99\dots99}_{n\text{ nines}}}^{\overbrace{99\dots99}^{n\text{ nines}}}=(10^n-1)^{10^n-1}.$$ So we have $10^{n-1}\le f(n)<10^n$. Then what can…
Maximilian Janisch
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