9.1. "numbers" — Numeric abstract base classes
**********************************************

New in version 2.6.

The "numbers" module (**PEP 3141**) defines a hierarchy of numeric
*abstract base classes* which progressively define more operations.
None of the types defined in this module can be instantiated.

class numbers.Number

   The root of the numeric hierarchy. If you just want to check if an
   argument *x* is a number, without caring what kind, use
   "isinstance(x, Number)".


9.1.1. The numeric tower
========================

class numbers.Complex

   Subclasses of this type describe complex numbers and include the
   operations that work on the built-in "complex" type. These are:
   conversions to "complex" and "bool", "real", "imag", "+", "-", "*",
   "/", "abs()", "conjugate()", "==", and "!=". All except "-" and
   "!=" are abstract.

   real

      Abstract. Retrieves the real component of this number.

   imag

      Abstract. Retrieves the imaginary component of this number.

   conjugate()

      Abstract. Returns the complex conjugate. For example,
      "(1+3j).conjugate() == (1-3j)".

class numbers.Real

   To "Complex", "Real" adds the operations that work on real numbers.

   In short, those are: a conversion to "float", "math.trunc()",
   "round()", "math.floor()", "math.ceil()", "divmod()", "//", "%",
   "<", "<=", ">", and ">=".

   Real also provides defaults for "complex()", "real", "imag", and
   "conjugate()".

class numbers.Rational

   Subtypes "Real" and adds "numerator" and "denominator" properties,
   which should be in lowest terms. With these, it provides a default
   for "float()".

   numerator

      Abstract.

   denominator

      Abstract.

class numbers.Integral

   Subtypes "Rational" and adds a conversion to "int".  Provides
   defaults for "float()", "numerator", and "denominator".  Adds
   abstract methods for "**" and bit-string operations: "<<", ">>",
   "&", "^", "|", "~".


9.1.2. Notes for type implementors
==================================

Implementors should be careful to make equal numbers equal and hash
them to the same values. This may be subtle if there are two different
extensions of the real numbers. For example, "fractions.Fraction"
implements "hash()" as follows:

   def __hash__(self):
       if self.denominator == 1:
           # Get integers right.
           return hash(self.numerator)
       # Expensive check, but definitely correct.
       if self == float(self):
           return hash(float(self))
       else:
           # Use tuple's hash to avoid a high collision rate on
           # simple fractions.
           return hash((self.numerator, self.denominator))


9.1.2.1. Adding More Numeric ABCs
---------------------------------

There are, of course, more possible ABCs for numbers, and this would
be a poor hierarchy if it precluded the possibility of adding those.
You can add "MyFoo" between "Complex" and "Real" with:

   class MyFoo(Complex): ...
   MyFoo.register(Real)


9.1.2.2. Implementing the arithmetic operations
-----------------------------------------------

We want to implement the arithmetic operations so that mixed-mode
operations either call an implementation whose author knew about the
types of both arguments, or convert both to the nearest built in type
and do the operation there. For subtypes of "Integral", this means
that "__add__()" and "__radd__()" should be defined as:

   class MyIntegral(Integral):

       def __add__(self, other):
           if isinstance(other, MyIntegral):
               return do_my_adding_stuff(self, other)
           elif isinstance(other, OtherTypeIKnowAbout):
               return do_my_other_adding_stuff(self, other)
           else:
               return NotImplemented

       def __radd__(self, other):
           if isinstance(other, MyIntegral):
               return do_my_adding_stuff(other, self)
           elif isinstance(other, OtherTypeIKnowAbout):
               return do_my_other_adding_stuff(other, self)
           elif isinstance(other, Integral):
               return int(other) + int(self)
           elif isinstance(other, Real):
               return float(other) + float(self)
           elif isinstance(other, Complex):
               return complex(other) + complex(self)
           else:
               return NotImplemented

There are 5 different cases for a mixed-type operation on subclasses
of "Complex". I’ll refer to all of the above code that doesn’t refer
to "MyIntegral" and "OtherTypeIKnowAbout" as “boilerplate”. "a" will
be an instance of "A", which is a subtype of "Complex" ("a : A <:
Complex"), and "b : B <: Complex". I’ll consider "a + b":

   1. If "A" defines an "__add__()" which accepts "b", all is well.

   2. If "A" falls back to the boilerplate code, and it were to
      return a value from "__add__()", we’d miss the possibility that
      "B" defines a more intelligent "__radd__()", so the boilerplate
      should return "NotImplemented" from "__add__()". (Or "A" may not
      implement "__add__()" at all.)

   3. Then "B"’s "__radd__()" gets a chance. If it accepts "a", all
      is well.

   4. If it falls back to the boilerplate, there are no more
      possible methods to try, so this is where the default
      implementation should live.

   5. If "B <: A", Python tries "B.__radd__" before "A.__add__".
      This is ok, because it was implemented with knowledge of "A", so
      it can handle those instances before delegating to "Complex".

If "A <: Complex" and "B <: Real" without sharing any other knowledge,
then the appropriate shared operation is the one involving the built
in "complex", and both "__radd__()" s land there, so "a+b == b+a".

Because most of the operations on any given type will be very similar,
it can be useful to define a helper function which generates the
forward and reverse instances of any given operator. For example,
"fractions.Fraction" uses:

   def _operator_fallbacks(monomorphic_operator, fallback_operator):
       def forward(a, b):
           if isinstance(b, (int, long, Fraction)):
               return monomorphic_operator(a, b)
           elif isinstance(b, float):
               return fallback_operator(float(a), b)
           elif isinstance(b, complex):
               return fallback_operator(complex(a), b)
           else:
               return NotImplemented
       forward.__name__ = '__' + fallback_operator.__name__ + '__'
       forward.__doc__ = monomorphic_operator.__doc__

       def reverse(b, a):
           if isinstance(a, Rational):
               # Includes ints.
               return monomorphic_operator(a, b)
           elif isinstance(a, numbers.Real):
               return fallback_operator(float(a), float(b))
           elif isinstance(a, numbers.Complex):
               return fallback_operator(complex(a), complex(b))
           else:
               return NotImplemented
       reverse.__name__ = '__r' + fallback_operator.__name__ + '__'
       reverse.__doc__ = monomorphic_operator.__doc__

       return forward, reverse

   def _add(a, b):
       """a + b"""
       return Fraction(a.numerator * b.denominator +
                       b.numerator * a.denominator,
                       a.denominator * b.denominator)

   __add__, __radd__ = _operator_fallbacks(_add, operator.add)

   # ...
