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======= 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 > 3.14159265358979323846 > 3.1415926535897932384626433 > 3.141592653589793238462643383279 > 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 > x: > > (= -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. > = = = = = = = = = = = = = = = = = = = = Chongkai Zhu mathematica@xxxxxxxxx 2005-05-25