`!numbers` --- Numeric abstract base classes

numbers
Source code: 
Lib/numbers.py
The 
!numbers module (
3141) defines a hierarchy of numeric 
abstract base classes <abstract base class> which progressively define more operations. None of the types defined in this module are intended to be instantiated.
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).
The numeric tower
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).
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, 
~Complex.real, 
~Complex.imag, and 
~Complex.conjugate.
Subtypes 
Real and adds 
~Rational.numerator and 
~Rational.denominator properties. It also provides a default for 
float.
The 
~Rational.numerator and 
~Rational.denominator values should be instances of 
Integral and should be in lowest terms with 
~Rational.denominator positive.
numerator
Abstract. The numerator of this rational number.
denominator
Abstract. The denominator of this rational number.
Subtypes 
Rational and adds a conversion to 
int. Provides defaults for 
float, 
~Rational.numerator, and 
~Rational.denominator. Adds abstract methods for 
pow with modulus and bit-string operations: 
<<, 
>>, 
&, 
^, 
|, 
~.
Notes for type implementers
Implementers 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. See also 
numeric-hash.
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)
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 
~object.__add__ and 
~object.__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:
If 
A defines an 
~object.__add__ which accepts 
b, all is well.
If 
A falls back to the boilerplate code, and it were to return a value from 
~object.__add__, we'd miss the possibility that 
B defines a more intelligent 
~object.__radd__, so the boilerplate should return 
NotImplemented from 
!__add__. (Or 
A may not implement 
!__add__ at all.)
Then 
B's 
~object.__radd__ gets a chance. If it accepts 
a, all is well.
If it falls back to the boilerplate, there are no more possible methods to try, so this is where the default implementation should live.
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 
~object.__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, 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, Real):
            return fallback_operator(float(a), float(b))
        elif isinstance(a, 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)
# ...