# 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`

• `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` :

Note

• If `A` defines an add() which accepts `b` , 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 that `B` defines a more intelligent radd(), so the boilerplate should return NotImplemented from add(). (Or `A` may not implement add() at all.)

• Then `B` ’s 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 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)

# ...
``````