# Copyright 2007 Google, Inc. All Rights Reserved. # 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.""" ############ Maintenance notes ######################################### # # ABCs are different from other standard library modules in that they # specify compliance tests. In general, once an ABC has been published, # new methods (either abstract or concrete) cannot be added. # # Though classes that inherit from an ABC would automatically receive a # new mixin method, registered classes would become non-compliant and # violate the contract promised by ``isinstance(someobj, SomeABC)``. # # Though irritating, the correct procedure for adding new abstract or # mixin methods is to create a new ABC as a subclass of the previous # ABC. # # Because they are so hard to change, new ABCs should have their APIs # carefully thought through prior to publication. # # Since ABCMeta only checks for the presence of methods, it is possible # to alter the signature of a method by adding optional arguments # or changing parameter names. This is still a bit dubious but at # least it won't cause isinstance() to return an incorrect result. # # ####################################################################### from abc import ABCMeta, abstractmethod __all__ = ["Number", "Complex", "Real", "Rational", "Integral"] 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). 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 def __add__(self, other): """self + other""" raise NotImplementedError @abstractmethod def __radd__(self, other): """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 @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) 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 (other // self, other % self). 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) 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 int(self.numerator) / int(self.denominator) class Integral(Rational): """Integral adds methods that work on integral numbers. In short, these are conversion to int, pow with modulus, 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)