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Re: infinities reformulated [was Re: My ideas about infinity in Scheme (revised)]
======= At 2005-05-25, 01:19:41 Aubrey Jaffer wrote: =======
>I have come up with a clearer formulation:
> 6.2.2x Inexactness
> In an implementation which represents inexact real numbers with
> IEEE-754 64-bit flonums:
> (= 3.141592653589793
> 3.14159265358979323846264338327950288) ==> #t
> Thus an inexact real number represents not a single value, but a
> neighborhood of (mathematical) real numbers. The inaccuracies of
> inexact calculations are due to misalignment of functional
> projection of a given neighborhood onto the real line neighborhoods.
Where does the length of the "neighborhood" come from? How will it
be stored and passed?
> The interpretation of real infinities is that 1/0 represents real
> numbers greater than can be encoded by finite inexacts in the
> implementation (> 179.76931348623158e306 for IEEE-754 64-bit
> flonums) and that -1/0 represents numbers less than can be encoded
> by finite inexacts in the implementation (< -179.76931348623158e306
> for IEEE-754 64-bit flonums).
What if an implementation supports arbitrary big real numbers?
> This preserves the total ordering of
> the (mathematical) real numbers and extends Scheme's representation
> to cover the entire real line. Note that no numerical infinity,
> with its attendant theoretical problems, is constructed; 1/0 and
> -1/0 represent the half-lines beyond either end of the
> implementation's inexact rational range. For any finite real number
> (= -1/0 x)) ==> #f
> (= 1/0 x)) ==> #f
> (< -1/0 x 1/0)) ==> #t
> (> 1/0 x -1/0)) ==> #t
> Implementations of Scheme which provide inexact real numbers shall
> implement positive infinity and negative infinity as unique inexact
> real numbers.
> An optional third infinity, which is not real, may be returned by a
> numerical function when no inexact neighborhood (including
> infinities) contains the correct answer. An implementation may
> report a violation of an implementation restriction in any
> calculation for which the result would be an unreal infinity.
>I believe these semantics avoid the problems caused by introduction of
>new elements to the field.
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