# A Density in [0; 1]

1. May 13, 2016

### Calabi

Hello, let be $x \in \mathbb{R} - \mathbb{Q}$, do we have de density of $\{nx - \lfloor{nx} / n \in \mathbb{N}\}$ in $[0; 1]$ please?

I think yes but it's just an intuition : if I take a and b in $[0; 1]$ with a < b, I have an irrationnal between them let call it c but I don't know hat to do with it.

Thank you in advance and have a nice afternoon.

2. May 13, 2016

### Staff: Mentor

In principle you already have said what you need. How is a dense subset formally defined? It might be a bit of work to formally prove that there are always irrational numbers as close to a given rational one as wanted. But given this fact, it's basically the definition of density itself.
E.g.: https://en.wikipedia.org/wiki/Dense_set

3. May 13, 2016

### Calabi

No that's not what I mean : the irrationnal number are dense in $\mathbb{R}$ so in $[0; 1]$ too (1)
And I wanted to use that to show that
$\{nx - \lfloor nx \rfloor / n \in \mathbb{N}\}$ in [0; 1], where x is an irrationnal number, by using (1).

4. May 13, 2016

### micromass

Staff Emeritus
5. May 13, 2016