8.1. numbers
— Numeric abstract base classes¶
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)
.
8.1.1. The numeric tower¶

class
numbers.
Complex
¶ Subclasses of this type describe complex numbers and include the operations that work on the builtin
complex
type. These are: conversions tocomplex
andbool
,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() == (13j)
.


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
, andconjugate()
.
8.1.2. 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))
8.1.2.1. 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)
8.1.2.2. Implementing the arithmetic operations¶
We want to implement the arithmetic operations so that mixedmode
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 mixedtype 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__add__()
which acceptsb
, all is well. If
A
falls back to the boilerplate code, and it were to return a value from__add__()
, we’d miss the possibility thatB
defines a more intelligent__radd__()
, so the boilerplate should returnNotImplemented
from__add__()
. (OrA
may not implement__add__()
at all.) Then
B
‘s__radd__()
gets a chance. If it acceptsa
, 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 triesB.__radd__
beforeA.__add__
. This is ok, because it was implemented with knowledge ofA
, so it can handle those instances before delegating toComplex
.
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, 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)
# ...