# Source code for numbers

```
# Licensed to PSF under a Contributor Agreement.

"""Abstract Base Classes (ABCs) for numbers, according to PEP 3141.

TODO: Fill out more detailed documentation on the operators."""

from abc import ABCMeta, abstractmethod

__all__ = ["Number", "Complex", "Real", "Rational", "Integral"]

[docs]class Number(metaclass=ABCMeta):
"""All numbers inherit from this class.

If you just want to check if an argument x is a number, without
caring what kind, use isinstance(x, Number).
"""
__slots__ = ()

# Concrete numeric types must provide their own hash implementation
__hash__ = None

## Notes on Decimal
## ----------------
## Decimal has all of the methods specified by the Real abc, but it should
## not be registered as a Real because decimals do not interoperate with
## binary floats (i.e.  Decimal('3.14') + 2.71828 is undefined).  But,
## abstract reals are expected to interoperate (i.e. R1 + R2 should be
## expected to work if R1 and R2 are both Reals).

[docs]class Complex(Number):
"""Complex defines the operations that work on the builtin complex type.

In short, those are: a conversion to complex, .real, .imag, +, -,
*, /, abs(), .conjugate, ==, and !=.

If it is given heterogeneous arguments, and doesn't have special
knowledge about them, it should fall back to the builtin complex
type as described below.
"""

__slots__ = ()

@abstractmethod
def __complex__(self):
"""Return a builtin complex instance. Called for complex(self)."""

def __bool__(self):
"""True if self != 0. Called for bool(self)."""
return self != 0

@property
@abstractmethod
def real(self):
"""Retrieve the real component of this number.

This should subclass Real.
"""
raise NotImplementedError

@property
@abstractmethod
def imag(self):
"""Retrieve the imaginary component of this number.

This should subclass Real.
"""
raise NotImplementedError

@abstractmethod
"""self + other"""
raise NotImplementedError

@abstractmethod
"""other + self"""
raise NotImplementedError

@abstractmethod
def __neg__(self):
"""-self"""
raise NotImplementedError

@abstractmethod
def __pos__(self):
"""+self"""
raise NotImplementedError

def __sub__(self, other):
"""self - other"""
return self + -other

def __rsub__(self, other):
"""other - self"""
return -self + other

@abstractmethod
def __mul__(self, other):
"""self * other"""
raise NotImplementedError

@abstractmethod
def __rmul__(self, other):
"""other * self"""
raise NotImplementedError

@abstractmethod
def __truediv__(self, other):
"""self / other: Should promote to float when necessary."""
raise NotImplementedError

@abstractmethod
def __rtruediv__(self, other):
"""other / self"""
raise NotImplementedError

@abstractmethod
def __pow__(self, exponent):
"""self**exponent; should promote to float or complex when necessary."""
raise NotImplementedError

@abstractmethod
def __rpow__(self, base):
"""base ** self"""
raise NotImplementedError

@abstractmethod
def __abs__(self):
"""Returns the Real distance from 0. Called for abs(self)."""
raise NotImplementedError

[docs]    @abstractmethod
def conjugate(self):
"""(x+y*i).conjugate() returns (x-y*i)."""
raise NotImplementedError

@abstractmethod
def __eq__(self, other):
"""self == other"""
raise NotImplementedError

Complex.register(complex)

[docs]class Real(Complex):
"""To Complex, Real adds the operations that work on real numbers.

In short, those are: a conversion to float, trunc(), divmod,
%, , and >=.

Real also provides defaults for the derived operations.
"""

__slots__ = ()

@abstractmethod
def __float__(self):
"""Any Real can be converted to a native float object.

Called for float(self)."""
raise NotImplementedError

@abstractmethod
def __trunc__(self):
"""trunc(self): Truncates self to an Integral.

Returns an Integral i such that:
* i>0 iff self>0;
* abs(i) <= abs(self);
* for any Integral j satisfying the first two conditions,
abs(i) >= abs(j) [i.e. i has "maximal" abs among those].
i.e. "truncate towards 0".
"""
raise NotImplementedError

@abstractmethod
def __floor__(self):
"""Finds the greatest Integral <= self."""
raise NotImplementedError

@abstractmethod
def __ceil__(self):
"""Finds the least Integral >= self."""
raise NotImplementedError

@abstractmethod
def __round__(self, ndigits=None):
"""Rounds self to ndigits decimal places, defaulting to 0.

If ndigits is omitted or None, returns an Integral, otherwise
returns a Real. Rounds half toward even.
"""
raise NotImplementedError

def __divmod__(self, other):
"""divmod(self, other): The pair (self // other, self % other).

Sometimes this can be computed faster than the pair of
operations.
"""
return (self // other, self % other)

def __rdivmod__(self, other):
"""divmod(other, self): The pair (self // other, self % other).

Sometimes this can be computed faster than the pair of
operations.
"""
return (other // self, other % self)

@abstractmethod
def __floordiv__(self, other):
"""self // other: The floor() of self/other."""
raise NotImplementedError

@abstractmethod
def __rfloordiv__(self, other):
"""other // self: The floor() of other/self."""
raise NotImplementedError

@abstractmethod
def __mod__(self, other):
"""self % other"""
raise NotImplementedError

@abstractmethod
def __rmod__(self, other):
"""other % self"""
raise NotImplementedError

@abstractmethod
def __lt__(self, other):
"""self < other

< on Reals defines a total ordering, except perhaps for NaN."""
raise NotImplementedError

@abstractmethod
def __le__(self, other):
"""self <= other"""
raise NotImplementedError

# Concrete implementations of Complex abstract methods.
def __complex__(self):
"""complex(self) == complex(float(self), 0)"""
return complex(float(self))

@property
def real(self):
"""Real numbers are their real component."""
return +self

@property
def imag(self):
"""Real numbers have no imaginary component."""
return 0

def conjugate(self):
"""Conjugate is a no-op for Reals."""
return +self

Real.register(float)

[docs]class Rational(Real):
""".numerator and .denominator should be in lowest terms."""

__slots__ = ()

@property
@abstractmethod
def numerator(self):
raise NotImplementedError

@property
@abstractmethod
def denominator(self):
raise NotImplementedError

# Concrete implementation of Real's conversion to float.
def __float__(self):
"""float(self) = self.numerator / self.denominator

It's important that this conversion use the integer's "true"
division rather than casting one side to float before dividing
so that ratios of huge integers convert without overflowing.

"""
return self.numerator / self.denominator

[docs]class Integral(Rational):
"""Integral adds a conversion to int and the bit-string operations."""

__slots__ = ()

@abstractmethod
def __int__(self):
"""int(self)"""
raise NotImplementedError

def __index__(self):
"""Called whenever an index is needed, such as in slicing"""
return int(self)

@abstractmethod
def __pow__(self, exponent, modulus=None):
"""self ** exponent % modulus, but maybe faster.

Accept the modulus argument if you want to support the
3-argument version of pow(). Raise a TypeError if exponent < 0
or any argument isn't Integral. Otherwise, just implement the
2-argument version described in Complex.
"""
raise NotImplementedError

@abstractmethod
def __lshift__(self, other):
"""self << other"""
raise NotImplementedError

@abstractmethod
def __rlshift__(self, other):
"""other << self"""
raise NotImplementedError

@abstractmethod
def __rshift__(self, other):
"""self >> other"""
raise NotImplementedError

@abstractmethod
def __rrshift__(self, other):
"""other >> self"""
raise NotImplementedError

@abstractmethod
def __and__(self, other):
"""self & other"""
raise NotImplementedError

@abstractmethod
def __rand__(self, other):
"""other & self"""
raise NotImplementedError

@abstractmethod
def __xor__(self, other):
"""self ^ other"""
raise NotImplementedError

@abstractmethod
def __rxor__(self, other):
"""other ^ self"""
raise NotImplementedError

@abstractmethod
def __or__(self, other):
"""self | other"""
raise NotImplementedError

@abstractmethod
def __ror__(self, other):
"""other | self"""
raise NotImplementedError

@abstractmethod
def __invert__(self):
"""~self"""
raise NotImplementedError

# Concrete implementations of Rational and Real abstract methods.
def __float__(self):
"""float(self) == float(int(self))"""
return float(int(self))

@property
def numerator(self):
"""Integers are their own numerators."""
return +self

@property
def denominator(self):
"""Integers have a denominator of 1."""
return 1

Integral.register(int)

```