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Re: infinity notations



> Some new possibilities have come to light.  Here are all the possible
> Scheme infinity notations that I know of:
> Notations C and G use a trailing `.' to indicate inexactness as K
> does.  This requires a small extension to R5RS number syntax, as
> rational notation (`1/0') does not currently allow a trailing period.

- however may not be necessary if +Inf is symbolically defined as +1/0.

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