> Yes. However, a real has at most two decimal expansions - one ending
> with a recurring 9, the other with a recurring 0. If you insert a rule
> that the latter is always the one to be chosen, the representation
> becomes unique.
>
> This allows you to unambiguously transform any 2D point into a 1D point.
> It is unclear to me whether it solves the reverse transformation...

If you take out all the numbers that could be represented with recurring 
0s or 9s from the 2D points, and the corresponding 1D representations of 
them, the remaining numbers have a 1-to-1 relation.

So the question comes down to is there a 1-to-1 relation between the 1D 
and 2D values you took out from the original sets.

The 2D coordinates taken out will be all possible pairs of numbers from 
this list:
0,1,0.1,0.2,0.3,...,0.9,0.01,0.02,0.03,...

The 1D numbers taken out will be the ones generated by interleaving the 
digits from all possible pairs of this list:
0.000...
0.999...
1.000...
0.099...
0.100...
0.199...
0.200...
...

It seems there is then an easy 1-to-1 mapping function for these 
remaining numbers, namely that if your 2D coordinate is the i'th and 
j'th element from the first list, then you use the i'th and j'th element 
from the 2nd list to generate your 1D coordinate.

So (0.3,0.4) is the 5th and 6th element from the first list, so you use 
the 5th and 6th from the second list (0.100...,0.199...), which gives 
your 1D value of 0.110909090909...

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