# Re: infinities reformulated [was Re: My ideas about infinity in Scheme (revised)]

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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

```

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