`numbers`

— Numeric abstract base classes**Source code:** Lib/numbers.py

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)`

.

*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.

*abstractmethod*`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:`<<`

,`>>`

,`&`

,`^`

,`|`

,`~`

.

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))
```

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)
```

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`

:

- 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, 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)
# ...
```