>> For *every* one, there are some real numbers that do not have a finite
>> representation. That's what makes them irrational.
>
>    "Some" is quite an understatement.
>
>    The set of real numbers with a finite representation is countable, while
> the set of real numbers without a finite representation is uncountable,
> making the latter set much "larger".

On the other hand, the distinction is typically irrelevant. Usually you 
don't actually "need" the precise value of (say) Sqrt(2); some rational 
approximation is sufficient.

Perhaps for similar reasons, you don't often come across real numbers 
that aren't compactly representable...

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