"numbers" --- 數值的抽象基底類別
********************************

**原始碼：**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 are intended to be
instantiated.

class numbers.Number

   數值階層結構的基礎。如果你只想確認引數 *x* 是不是數值、並不關心其型
   別，請使用 "isinstance(x, Number)"。


數值的階層
==========

class numbers.Complex

   這個型別的子類別描述了複數並包含適用於內建 "complex" 型別的操作。這
   些操作有："complex" 和 "bool" 的轉換、"real"、"imag"、"+"、"-"、"*"
   、"/"、"**"、"abs()"、"conjugate()"、"==" 以及 "!="。除 "-" 和 "!="
   之外所有操作都是抽象的。

   real

      為抽象的。取得該數值的實數部分。

   imag

      為抽象的。取得該數值的虛數部分。

   abstractmethod conjugate()

      為抽象的。回傳共軛複數，例如 "(1+3j).conjugate() == (1-3j)"。

class numbers.Real

   To "Complex", "Real" adds the operations that work on real numbers.

   簡單的說，有 "float" 的轉換、"math.trunc()"、"round()"、
   "math.floor()"、"math.ceil()"、"divmod()"、"//"、"%"、 "<"、"<="、
   ">"、和 ">="。

   實數同樣提供 "complex()"、"real"、"imag" 和 "conjugate()" 的預設值
   。

class numbers.Rational

   "Real" 的子型別，並增加了 "numerator" 和 "denominator" 這兩種特性。
   它也會提供 "float()" 的預設值。

   "numerator" 和 "denominator" 的值必須是 "Integral" 的實例且
   "denominator" 要是正數。

   numerator

      為抽象的。

   denominator

      為抽象的。

class numbers.Integral

   "Rational" 的子型別，並增加了 "int" 的轉換操作。為 "float()"、
   "numerator" 和 "denominator" 提供了預設值。為 "pow()" 方法增加了求
   餘 (modulus) 和位元字串運算 (bit-string operations) 的抽象方法：
   "<<"、">>"、"&"、"^"、"|"、"~"。


給型別實作者的註記
==================

實作者需注意，相等的數值除了大小相等外，還必須擁有同樣的雜湊值。當使用
兩個不同的實數擴充時，這可能是很微妙的。例如，"fractions.Fraction" 底
下的 "hash()" 實作如下：

   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))


加入更多數值 ABC
----------------

當然，還有更多用於數值的 ABC，如果不加入它們就不會有健全的階層。你可以
在 "Complex" 和 "Real" 中加入 "MyFoo"，像是：

   class MyFoo(Complex): ...
   MyFoo.register(Real)


實作算術操作
------------

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

"Complex" 的子類別有 5 種不同的混合型別操作。我將上面提到所有不涉及
"MyIntegral" 和 "OtherTypeIKnowAbout" 的程式碼稱作「模板 (boilerplate)
」。"a" 是 "Complex" 之子型別 "A" 的實例 ("a : A <: Complex")，同時 "b
: B <: Complex"。我將要計算 "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. 如果沒有成功回退到模板，就沒有更多的方法可以去嘗試，因此這裡將使用
   預設的實作。

5. 如果 "B <: A"，Python 會在 "A.__add__" 之前嘗試 "B.__radd__"。這是
   可行的，因為它是透過對 "A" 的理解而實作的，所以這可以在交給
   "Complex" 之前處理好這些實例。

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".

由於大部分對任意給定類型的操作都十分相似的，定義一個為任意給定運算子生
成向前 (forward) 與向後 (reverse) 實例的輔助函式可能會非常有用。例如，
"fractions.Fraction" 使用了：

   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, Real):
               return fallback_operator(float(a), float(b))
           elif isinstance(a, 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)

   # ...
