# PEP 3141 – A Type Hierarchy for Numbers

PEP
3141
Title
A Type Hierarchy for Numbers
Author
Jeffrey Yasskin <jyasskin at google.com>
Status
Final
Type
Standards Track
Created
23-Apr-2007
Post-History
25-Apr-2007, 16-May-2007, 02-Aug-2007

Contents

## Abstract

This proposal defines a hierarchy of Abstract Base Classes (ABCs) (PEP 3119) to represent number-like classes. It proposes a hierarchy of `Number :> Complex :> Real :> Rational :> Integral` where `A :> B` means “A is a supertype of B”. The hierarchy is inspired by Scheme’s numeric tower .

## Rationale

Functions that take numbers as arguments should be able to determine the properties of those numbers, and if and when overloading based on types is added to the language, should be overloadable based on the types of the arguments. For example, slicing requires its arguments to be `Integrals`, and the functions in the `math` module require their arguments to be `Real`.

## Specification

This PEP specifies a set of Abstract Base Classes, and suggests a general strategy for implementing some of the methods. It uses terminology from PEP 3119, but the hierarchy is intended to be meaningful for any systematic method of defining sets of classes.

The type checks in the standard library should use these classes instead of the concrete built-ins.

### Numeric Classes

We begin with a Number class to make it easy for people to be fuzzy about what kind of number they expect. This class only helps with overloading; it doesn’t provide any operations.

```class Number(metaclass=ABCMeta): pass
```

Most implementations of complex numbers will be hashable, but if you need to rely on that, you’ll have to check it explicitly: mutable numbers are supported by this hierarchy.

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

In short, those are: conversion to complex, bool(), .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.
"""

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

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

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

This should subclass Real.
"""
raise NotImplementedError

@abstractproperty
def imag(self):
"""Retrieve the real component of this number.

This should subclass Real.
"""
raise NotImplementedError

@abstractmethod
raise NotImplementedError

@abstractmethod
raise NotImplementedError

@abstractmethod
def __neg__(self):
raise NotImplementedError

def __pos__(self):
"""Coerces self to whatever class defines the method."""
raise NotImplementedError

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

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

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

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

@abstractmethod
def __div__(self, other):
"""a/b; should promote to float or complex when necessary."""
raise NotImplementedError

@abstractmethod
def __rdiv__(self, other):
raise NotImplementedError

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

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

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

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

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

# __ne__ is inherited from object and negates whatever __eq__ does.
```

The `Real` ABC indicates that the value is on the real line, and supports the operations of the `float` builtin. Real numbers are totally ordered except for NaNs (which this PEP basically ignores).

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

In short, those are: conversion to float, trunc(), math.floor(),
math.ceil(), round(), divmod(), //, %, <, <=, >, and >=.

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

# XXX What to do about the __int__ implementation that's
# currently present on float?  Get rid of it?

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

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

If ndigits is omitted or None, returns an Integral,
otherwise returns a Real, preferably of the same type as
self. Types may choose which direction to round half. For
example, float rounds half toward even.

"""
raise NotImplementedError

def __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):
"""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):
"""The floor() of self/other. Integral."""
raise NotImplementedError

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

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

See
https://mail.python.org/pipermail/python-3000/2006-May/001735.html
and consider using "self/other - trunc(self/other)"
instead if you're worried about round-off errors.
"""
raise NotImplementedError

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

@abstractmethod
def __lt__(self, other):
"""< on Reals defines a total ordering, except perhaps for NaN."""
raise NotImplementedError

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

# __gt__ and __ge__ are automatically done by reversing the arguments.
# (But __le__ is not computed as the opposite of __gt__!)

# Concrete implementations of Complex abstract methods.
# Subclasses may override these, but don't have to.

def __complex__(self):
return complex(float(self))

@property
def real(self):
return +self

@property
def imag(self):
return 0

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

We should clean up Demo/classes/Rat.py and promote it into rational.py in the standard library. Then it will implement the Rational ABC.

```class Rational(Real, Exact):
""".numerator and .denominator should be in lowest terms."""

@abstractproperty
def numerator(self):
raise NotImplementedError

@abstractproperty
def denominator(self):
raise NotImplementedError

# Concrete implementation of Real's conversion to float.
# (This invokes Integer.__div__().)

def __float__(self):
return self.numerator / self.denominator
```

And finally integers:

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

@abstractmethod
def __int__(self):
raise NotImplementedError

def __index__(self):
"""__index__() exists because float has __int__()."""
return int(self)

def __lshift__(self, other):
return int(self) << int(other)

def __rlshift__(self, other):
return int(other) << int(self)

def __rshift__(self, other):
return int(self) >> int(other)

def __rrshift__(self, other):
return int(other) >> int(self)

def __and__(self, other):
return int(self) & int(other)

def __rand__(self, other):
return int(other) & int(self)

def __xor__(self, other):
return int(self) ^ int(other)

def __rxor__(self, other):
return int(other) ^ int(self)

def __or__(self, other):
return int(self) | int(other)

def __ror__(self, other):
return int(other) | int(self)

def __invert__(self):
return ~int(self)

# 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
```

### Changes to operations and __magic__ methods

To support more precise narrowing from float to int (and more generally, from Real to Integral), we propose the following new __magic__ methods, to be called from the corresponding library functions. All of these return Integrals rather than Reals.

1. `__trunc__(self)`, called from a new builtin `trunc(x)`, which returns the Integral closest to `x` between 0 and `x`.
2. `__floor__(self)`, called from `math.floor(x)`, which returns the greatest Integral `<= x`.
3. `__ceil__(self)`, called from `math.ceil(x)`, which returns the least Integral `>= x`.
4. `__round__(self)`, called from `round(x)`, which returns the Integral closest to `x`, rounding half as the type chooses. `float` will change in 3.0 to round half toward even. There is also a 2-argument version, `__round__(self, ndigits)`, called from `round(x, ndigits)`, which should return a Real.

In 2.6, `math.floor`, `math.ceil`, and `round` will continue to return floats.

The `int()` conversion implemented by `float` is equivalent to `trunc()`. In general, the `int()` conversion should try `__int__()` first and if it is not found, try `__trunc__()`.

`complex.__{divmod,mod,floordiv,int,float}__` also go away. It would be nice to provide a nice error message to help confused porters, but not appearing in `help(complex)` is more important.

### 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, a complex type could reasonably implement hash() as follows:

```def __hash__(self):
return hash(complex(self))
```

but should be careful of any values that fall outside of the built in complex’s range or precision.

#### 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)
```

#### 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):

if isinstance(other, MyIntegral):
else:
return NotImplemented

if isinstance(other, MyIntegral):
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`:

1. If A defines an __add__ which accepts b, all is well.
2. 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.)
3. Then B’s __radd__ gets a chance. If it accepts a, all is well.
4. If it falls back to the boilerplate, there are no more possible methods to try, so this is where the default implementation should live.
5. 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`.

## Rejected Alternatives

The initial version of this PEP defined an algebraic hierarchy inspired by a Haskell Numeric Prelude  including MonoidUnderPlus, AdditiveGroup, Ring, and Field, and mentioned several other possible algebraic types before getting to the numbers. We had expected this to be useful to people using vectors and matrices, but the NumPy community really wasn’t interested, and we ran into the issue that even if `x` is an instance of `X <: MonoidUnderPlus` and `y` is an instance of `Y <: MonoidUnderPlus`, `x + y` may still not make sense.

Then we gave the numbers a much more branching structure to include things like the Gaussian Integers and Z/nZ, which could be Complex but wouldn’t necessarily support things like division. The community decided that this was too much complication for Python, so I’ve now scaled back the proposal to resemble the Scheme numeric tower much more closely.

## The Decimal Type

After consultation with its authors it has been decided that the `Decimal` type should not at this time be made part of the numeric tower.

## Acknowledgements

Thanks to Neal Norwitz for encouraging me to write this PEP in the first place, to Travis Oliphant for pointing out that the numpy people didn’t really care about the algebraic concepts, to Alan Isaac for reminding me that Scheme had already done this, and to Guido van Rossum and lots of other people on the mailing list for refining the concept.