Zara wrote:
> I am stubborn on this: a finite representation of *any* real number is
> impossible.
> LEDA REAL represents correctly the algebraic part of the real numbers.
> Thus, it fails to represent pi or e.
> Boots rational represents correctly rational numbers. Thus, it fauls
> to represnet pi or e.
> GMP allows us to extend tha preccission odf the mantissa of tha
> floating point representation of a rela number. Thus, it fails to
> represent pi, e or 0.1
>
> As I noted on other message, the infinite quantityof real numberts is
> too much of an infinity.

I think everybody around here agrees with you. What you say is so true
as being trite, and therefore not worth mentioning. Cantor would be
either very satisfied or very annoyed to hear that :o).

A good point is to stop abusing terminology: "infinite precision" does
not really describe what's going on, maybe "unbounded precision" or the
consecrated "arbitrary precision" would fare closer.


Andrei

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Arne Schmitz wrote:
> Zara wrote:
>
>> mpf_t is simply a floating point with a huge mantissa. But you are
>> unable to represent 0.1 exactly with it, so the problem remains the
>> same.
>
> Interesting thought... What if one uses two arbitrary precision integers to
> represent the integer and fractional part of a real number? That should be
> possible.

A rational number can be exactly represented with its numerator and its
denominator, but what about irrational numbers? Can you represent pi, e,
or sqrt(2) with two arbitrary precision integers for its integer and
fractional parts? The answer is no.

To be able to exactly represent real numbers would be to be able to
handle expressions without evaluating them numerically; e.g. represent

    (1 + sqrt(5)) / 2

by an expression tree such as:

              ( / )
                |
          ,-----+-----.
          |           |
        ( + )         2
          |
    ,-----+-----.
    |           |
    1        ( sqrt )
                |
                5

This is certainly possible. Mathematica does it. :)

--
Seungbeom Kim

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Zara wrote:
> On Tue,  7 Nov 2006 23:15:49 CST, "blwy10" <blwy10v@gmail.com> wrote:
>
>
>>This is my first time posting here so pardon if I lack any etiquette.
>>Anyway, I would like to ask if there was any paper submitted to the C++
>>Standards Committee with regards to an infinite precision floating
>>point type or class, or if there are any plans to do so. A brief check
>>on the papers available on
>>http://www.open-std.org/jtc1/sc22/wg21/docs/papers/ did not reveal
>>anything, though I may have accidentally overlooked some items. I
>>understand that there is already a proposal submitted with regards to
>>infinite precision integers ("Proposal for an Infinite Precision
>>Integer for Library Technical Report 2, Revision 1" N2020 2006-06-22),
>>so I was just wondering if there was a similar paper for its floating
>>point counterpart.
>>
>
>
> An infinite precision floating point class is impossible.

LEDA REAL

Louis.

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blwy10 ha scritto:
> Hmm...I never thought of it that way before.
>
> But then in that case, what do you call types such as mpf_t found in
> GMP? And has there ever been a proposal for the inclusion of such a
> type/class into the C++ Standard?
>

The GMP doc itself uses the term "arbitrary precision" on page 16 which
sounds quite a good term to me. Maybe "user-controlled precision" could
be even better.

Ganesh

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