# Tagged Questions

For questions related to the fractional part of a number.

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### $\lfloor x\rfloor \cdot \lfloor x^2\rfloor = \lfloor x^3\rfloor$ means that $x$ is close to an integer

Suppose $x>30$ is a number satisfying $\lfloor x\rfloor \cdot \lfloor x^2\rfloor = \lfloor x^3\rfloor$. Prove that $\{x\}<\frac{1}{2700}$, where $\{x\}$ is the fractional part of $x$. My ...
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### Prove that $4m + 1$ is a perfect square if $\{ \sqrt {n + \sqrt n}\} = \{\sqrt m\}$

Let $n,m \in \mathbb{N}-\{0\}$ so that $\{ \sqrt {n + \sqrt n}\} = \{\sqrt m\} \tag1$ Prove that $4m + 1$ is a perfect square. ($\{x\}$ is the fractional part of $x$) No idea how to start. ...
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### How do I the fractional approximation of a fraction?

I am studying a Computer Organization course, but in the slides it is mentioned that I have to convert fraction numbers to IEEE floating point representation. To do that first, I have to convert the ...
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### Can BCD-formatted numbers have fractional parts?

Can BCD-formatted numbers contain fractional parts in them? For instance, is there some way to represent a number like: 123.45 In BCD format?
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### 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 ...
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### 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 ...
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### 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,...
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### 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 ...
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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 ...
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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 ...
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 ...
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 ?