Density of Irrational Numbers in [0;1]

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In summary, the conversation discusses the density of a set in a given interval. The speaker suggests using irrationals to prove the density, and the other person provides a resource with a relevant theorem.
  • #1
Calabi
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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.

Could you help me please?

Thank you in advance and have a nice afternoon:oldbiggrin:.
 
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  • #2
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
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).
 

What is the definition of density of irrational numbers in [0;1]?

The density of irrational numbers in [0;1] refers to the frequency at which irrational numbers occur in the interval between 0 and 1. It is a measure of how closely packed the irrational numbers are in this range.

How is the density of irrational numbers in [0;1] calculated?

The density of irrational numbers in [0;1] is calculated by dividing the total number of irrational numbers in the interval by the total number of numbers in the interval. This ratio gives an approximation of the frequency at which irrational numbers occur in the range.

Why is the density of irrational numbers in [0;1] important?

The density of irrational numbers in [0;1] is important because it helps us understand the distribution of numbers in this interval. It also has applications in various mathematical theories and can provide insights into the nature of irrational numbers.

What is the relationship between the density of irrational numbers in [0;1] and the density of rational numbers in [0;1]?

The density of irrational numbers in [0;1] is equal to the complement of the density of rational numbers in [0;1]. In other words, the sum of the density of irrational numbers and the density of rational numbers in [0;1] is equal to 1.

Does the density of irrational numbers in [0;1] have a limit?

Yes, the density of irrational numbers in [0;1] has a limit of 1. This means that as the interval becomes smaller and smaller, the frequency of irrational numbers in the interval approaches 1. In other words, the interval contains almost entirely irrational numbers.

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