# Definition talk:Real Number

I've never seen the approach of constructing the real numbers in the interval $[0..1]$.

Why would you do that? The interval isn't closed under $+$. Unless you're suggesting working with $\Z \times [0..1)$? I'm not sure what you mean.

I always knew of the Cauchy, Dedekind and Stevin construction of $\R$ but looking at Wikipedia's sources there are many other approaches as well.

IMHO, I think PW should somehow show a recognition that these approaches exist (to be comprehensive) and then come back to it in the future. Unless of course anyone wants to do it. --Jshflynn (talk) 10:24, 16 January 2013 (UTC)

- What page does this comment relate to? --prime mover (talk) 12:00, 16 January 2013 (UTC)

- No matter, I understand - this was the result of a faulty template. --prime mover (talk) 12:08, 16 January 2013 (UTC)

Jshflynn, yes, $\Z \times [0,1)$. Thanks to some recent work by I don't remember whom, some of these approaches are now or will shortly be available. --Dfeuer (talk) 15:07, 16 January 2013 (UTC)

- Another option is to build the strictly positive reals, the strictly negative reals, and take the disjoint union of those with $0$. That justifies the usual notion of a real number as either $0$ or an optionally signed infinite decimal value. I'll have to dig up the book I have that uses one of these approaches. I think it's a number theory book. --Dfeuer (talk) 16:54, 16 January 2013 (UTC)

## Circular problems with the definition using Cauchy Sequences

Many articles about sequences assume we have defined real numbers. Examples: Definition:Convergent Sequence/Note on Domain of N, Definition:Metric Space.

If we don't want to invalidate the construction of $\R$ by Cauchy-sequences, we have to make sure that all definitions and theorems used in its construction do not use $\R$ (better: don't even mention it). This means: when defining an equivalence relation on Cauchy Sequences, this *has* to be done separately for rational numbers. The theorem for general metric spaces can not be used in the construction of $\R$. --barto (talk) 07:49, 28 January 2017 (EST)

- This might be a serious issue. We will have to address this carefully. — Lord_Farin (talk) 04:37, 29 January 2017 (EST)