Abstract

    Today, Python's numerical model is similar to the C numeric model:
    there are several unrelated numerical types, and when operations
    between numerical types are requested, coercions happen.  While
    the C rationale for the numerical model is that it is very similar
    to what happens at the hardware level, that rationale does not
    apply to Python.  So, while it is acceptable to C programmers that
    2/3 == 0, it is surprising to many Python programmers.

    NOTE: in the light of recent discussions in the newsgroup, the
    motivation in this PEP (and details) need to be extended.

Rationale

    In usability studies, one of the least usable aspect of Python was
    the fact that integer division returns the floor of the division.
    This makes it hard to program correctly, requiring casts to
    float() in various parts through the code.  Python's numerical
    model stems from C, while an model that might be easier to work with 
    can be based on the mathematical understanding of numbers.

Other Numerical Models

    Perl's numerical model is that there is one type of numbers --
    floating point numbers.  While it is consistent and superficially
    non-surprising, it tends to have subtle gotchas.  One of these is
    that printing numbers is very tricky, and requires correct
    rounding.  In Perl, there is also a mode where all numbers are
    integers.  This mode also has its share of problems, which arise
    from the fact that there is not even an approximate way of
    dividing numbers and getting meaningful answers.

Suggested Interface For Python's Numerical Model

    While coercion rules will remain for add-on types and classes, the
    built in type system will have exactly one Python type -- a
    number.  There are several things which can be considered "number
    methods":

    1. isnatural()
    2. isintegral()
    3. isrational()
    4. isreal()
    5. iscomplex()

    a. isexact()

    Obviously, a number which answers true to a question from 1 to 5, will
    also answer true to any following question. If "isexact()" is not true, 
    then any answer might be wrong.
    (But not horribly wrong: it's close to the truth.)

    Now, there is two thing the models promises for the field operations
    (+, -, /, *): 

    - If both operands satisfy isexact(), the result satisfies
      isexact().

    - All field rules are true, except that for not-isexact() numbers,
      they might be only approximately true.

    One consequence of these two rules is that all exact calcutions
    are done as (complex) rationals: since the field laws must hold,
    then

        (a/b)*b == a

    must hold.

    There is built-in function, inexact() which takes a number
    and returns an inexact number which is a good approximation.
    Inexact numbers must be as least as accurate as if they were
    using IEEE-754.

    Several of the classical Python functions will return exact numbers
    even when given inexact numbers: e.g, int().

Coercion

    The number type does not define nb_coerce
    Any numeric operation slot, when receiving something other then PyNumber,
    refuses to implement it.

Inexact Operations

    The functions in the "math" module will be allowed to return
    inexact results for exact values.  However, they will never return
    a non-real number.  The functions in the "cmath" module are also
    allowed to return an inexact result for an exact argument, and are
    furthermore allowed to return a complex result for a real
    argument.

Numerical Python Issues

    People who use Numerical Python do so for high-performance vector
    operations.  Therefore, NumPy should keep its hardware based
    numeric model.

Unresolved Issues

    Which number literals will be exact, and which inexact?

    How do we deal with IEEE 754 operations? (probably, isnan/isinf should
    be methods)

    On 64-bit machines, comparisons between ints and floats may be
    broken when the comparison involves conversion to float.  Ditto
    for comparisons between longs and floats.  This can be dealt with
    by avoiding the conversion to float.  (Due to Andrew Koenig.)

Copyright

    This document has been placed in the public domain.