# [seqfan] Re: Representation of real numbers by sequences

Giovanni Resta g.resta at iit.cnr.it
Mon May 18 14:48:35 CEST 2020

```Yes, I've seen the reddit post and the blog, it seems really interesting.

https://oeis.org/A301484 .

Giovanni

Il 18/05/2020 10:32, Oscar Cunningham ha scritto:

> I recently wrote a blog post
> (https://oscarcunningham.com/494/a-better-representation-for-real-numbers/)
> in which I described a method for representing real numbers as integer
> sequences.
>
> In brief, we define the function g by g(x) = x/(x+1), and then write a
> number in the form z+g(a_0+g(a_1+g(a_2+g(...)))), where z is an
> integer and a is a sequence of natural numbers.
>
> This representation is similar to the continued fraction, but improves
> on it in two ways.
>
> The first is that there is no nonuniqueness of representations. Each
> real number corresponds to a distinct sequence, and each sequence
> corresponds to a distinct number. The second is that the map is
> order-preserving (unlike continued fractions, in which increasing the
> terms in odd positions decreases the number). We still retain the nice
> properties of continued fractions of rational and quadratic roots.
>
> More formally we have:
>
> a) The representation defines an order-preserving bijection between
> the interval [0,1) and the set of functions from natural numbers to
> natural numbers (with the lexicographic order).
>
> b) The representation of x is eventually all 0s if and only if x is
> rational.
>
> c) The representation of x is eventually periodic if and only if x is
> the root of a quadratic.
>
> There has also been some interesting discussion of this blog post on
> reddit
> in which someone pointed out that the sequence 0,1,2,3,... corresponds
> under this representation to the real number J_0(2)/J_1(2) where J is
> the Bessel function.
>
> Best regards,
>
> Oscar Cunningham
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
```