[seqfan] Representation of real numbers by sequences

[Oscar Cunningham]

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 
(https://old.reddit.com/r/math/comments/glhu6f/a_better_representation_for_real_numbers/), 
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