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Re: infinity notations



> And I'll concede my perceived necessity to denote an ambiguously signed
> infinity in exchange for the prevention of incorrectly signed infinities,
> which means that the region about 0 must be considered correspondingly
> invalid, (i.e. both are considered NaN or 0/0). yielding:
> 
> 
>           /  NaN  \         or equivalently:          /  0/0  \
>          /    |    \                                 /    |    \
>         -Inf  |  +Inf                               -1/0  |  +1/0
>         ------+------- (reciprocal projection axis) ------+------
>         -0.0  |   0.0                               -0.0  |  +0.0
>          \    |    /                                 \    |    /
>           \  NaN  /                                   \  0/0  /
>               |                                           |
>               0                                           0
>   (negative projection axis)                  (negative projection axis)
> 
> (where NaN and +-Inf may be thought of as symbols defined as 0/0 and +-1/0)
> 
> Which helps eliminates the ordering concern, although it's likely still a
> good idea to define (= -0.0 0 +0.0) => #t, and (< -0.0 0 +0.0) => #t, etc.
> 
> However then 0/0 denotes all ambiguities in either sign or value, even those
> which may be very small, then Therefore:
> 
> (+ +0.0 -0.0) => 0/0 [aka NaN]
> 
> as otherwise:
> 
>  (/ (+ +0.0 -0.0 +0.0)) :: (/ (+ 0 +0.0)) :: (/ +0.0) => +Inf
> 
> [which would be incorrect]
> 
> Thereby one can argue that this is actually good, as then the iterative sum
> of alternating infinitely small value about 0 is considered ambiguous, which
> would typically be the case. and correspondingly yield 0/0 for all
> ambiguities in either sign or significant magnitude.
> 
> (tan pi/2) => 0/0
> (/ 0.0 0.0) => 0/0

and as it may not be obvious, the difference between any two equivalently
valued inexact value is an exact 0. I.e.:

(- 1.5 1.5) => 0

as there is no inexact 0, as that would imply a value about 0 with an
ambiguous sign, which would both have a value range which overlaps +0.0,
0, and +0.0; and who's reciprocal was not self consistent. (or if one
chooses, an exact 0 is equivalent to an inexact 0, both mean absolute 0.)