On 7/13/2013 3:16 PM, Warp wrote:
> Orchid Win7 v1 <voi### [at] devnull> wrote:
>> I was under the impression that if you allow infinitely long decimal
>> expansions then all reals are representable.
>
> The question would thus be: Can the set of real numbers be represented
> with digits chosen from a finite set?
>
> Or even: Does an infinite amount of digits (chosen from a finite set)
> become uncountably infinite? Doesn't that go contrary to the notion of
> countable sets?

This is why the reals are considered to have a higher countability than 
the integers.  Reals are allowed to have infinite precision (such as pi, 
for which any digital representation goes indefinitely without 
repeating), but integers are defined to have finite precision.

(Which brings up the question, why this imposition on the integers?)

Regards,
John

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