Add rational.Rational as an implementation of numbers.Rational with infinite

precision. This has been discussed at http://bugs.python.org/issue1682. It's
useful primarily for teaching, but it also demonstrates how to implement a
member of the numeric tower, including fallbacks for mixed-mode arithmetic.

I expect to write a couple more patches in this area:
 * Rational.from_decimal()
 * Rational.trim/approximate() (maybe with different names)
 * Maybe remove the parentheses from Rational.__str__()
 * Maybe rename one of the Rational classes
 * Maybe make Rational('3/2') work.

3 files changed, 711 insertions, 3 deletions

diff --git a/Lib/numbers.py b/Lib/numbers.py
index 3c13290..8e02203 100644
--- a/Lib/numbers.py
+++ b/Lib/numbers.py
@@ -5,6 +5,7 @@
 
 TODO: Fill out more detailed documentation on the operators."""
 
+from __future__ import division
 from abc import ABCMeta, abstractmethod, abstractproperty
 
 __all__ = ["Number", "Exact", "Inexact",
@@ -63,7 +64,8 @@ class Complex(Number):
     def __complex__(self):
         """Return a builtin complex instance. Called for complex(self)."""
 
-    def __bool__(self):
+    # Will be __bool__ in 3.0.
+    def __nonzero__(self):
         """True if self != 0. Called for bool(self)."""
         return self != 0
 
@@ -98,6 +100,7 @@ class Complex(Number):
         """-self"""
         raise NotImplementedError
 
+    @abstractmethod
     def __pos__(self):
         """+self"""
         raise NotImplementedError
@@ -122,12 +125,28 @@ class Complex(Number):
 
     @abstractmethod
     def __div__(self, other):
-        """self / other; should promote to float or complex when necessary."""
+        """self / other without __future__ division
+
+        May promote to float.
+        """
         raise NotImplementedError
 
     @abstractmethod
     def __rdiv__(self, other):
-        """other / self"""
+        """other / self without __future__ division"""
+        raise NotImplementedError
+
+    @abstractmethod
+    def __truediv__(self, other):
+        """self / other with __future__ division.
+
+        Should promote to float when necessary.
+        """
+        raise NotImplementedError
+
+    @abstractmethod
+    def __rtruediv__(self, other):
+        """other / self with __future__ division"""
         raise NotImplementedError
 
     @abstractmethod
diff --git a/Lib/rational.py b/Lib/rational.py
new file mode 100755
index 0000000..d455dc6
--- /dev/null
+++ b/Lib/rational.py
@@ -0,0 +1,410 @@
+# Originally contributed by Sjoerd Mullender.
+# Significantly modified by Jeffrey Yasskin <jyasskin at gmail.com>.
+
+"""Rational, infinite-precision, real numbers."""
+
+from __future__ import division
+import math
+import numbers
+import operator
+
+__all__ = ["Rational"]
+
+RationalAbc = numbers.Rational
+
+
+def _gcd(a, b):
+    """Calculate the Greatest Common Divisor.
+
+    Unless b==0, the result will have the same sign as b (so that when
+    b is divided by it, the result comes out positive).
+    """
+    while b:
+        a, b = b, a%b
+    return a
+
+
+def _binary_float_to_ratio(x):
+    """x -> (top, bot), a pair of ints s.t. x = top/bot.
+
+    The conversion is done exactly, without rounding.
+    bot > 0 guaranteed.
+    Some form of binary fp is assumed.
+    Pass NaNs or infinities at your own risk.
+
+    >>> _binary_float_to_ratio(10.0)
+    (10, 1)
+    >>> _binary_float_to_ratio(0.0)
+    (0, 1)
+    >>> _binary_float_to_ratio(-.25)
+    (-1, 4)
+    """
+
+    if x == 0:
+        return 0, 1
+    f, e = math.frexp(x)
+    signbit = 1
+    if f < 0:
+        f = -f
+        signbit = -1
+    assert 0.5 <= f < 1.0
+    # x = signbit * f * 2**e exactly
+
+    # Suck up CHUNK bits at a time; 28 is enough so that we suck
+    # up all bits in 2 iterations for all known binary double-
+    # precision formats, and small enough to fit in an int.
+    CHUNK = 28
+    top = 0
+    # invariant: x = signbit * (top + f) * 2**e exactly
+    while f:
+        f = math.ldexp(f, CHUNK)
+        digit = trunc(f)
+        assert digit >> CHUNK == 0
+        top = (top << CHUNK) | digit
+        f = f - digit
+        assert 0.0 <= f < 1.0
+        e = e - CHUNK
+    assert top
+
+    # Add in the sign bit.
+    top = signbit * top
+
+    # now x = top * 2**e exactly; fold in 2**e
+    if e>0:
+        return (top * 2**e, 1)
+    else:
+        return (top, 2 ** -e)
+
+
+class Rational(RationalAbc):
+    """This class implements rational numbers.
+
+    Rational(8, 6) will produce a rational number equivalent to
+    4/3. Both arguments must be Integral. The numerator defaults to 0
+    and the denominator defaults to 1 so that Rational(3) == 3 and
+    Rational() == 0.
+
+    """
+
+    __slots__ = ('_numerator', '_denominator')
+
+    def __init__(self, numerator=0, denominator=1):
+        if (not isinstance(numerator, numbers.Integral) and
+            isinstance(numerator, RationalAbc) and
+            denominator == 1):
+            # Handle copies from other rationals.
+            other_rational = numerator
+            numerator = other_rational.numerator
+            denominator = other_rational.denominator
+
+        if (not isinstance(numerator, numbers.Integral) or
+            not isinstance(denominator, numbers.Integral)):
+            raise TypeError("Rational(%(numerator)s, %(denominator)s):"
+                            " Both arguments must be integral." % locals())
+
+        if denominator == 0:
+            raise ZeroDivisionError('Rational(%s, 0)' % numerator)
+
+        g = _gcd(numerator, denominator)
+        self._numerator = int(numerator // g)
+        self._denominator = int(denominator // g)
+
+    @classmethod
+    def from_float(cls, f):
+        """Converts a float to a rational number, exactly."""
+        if not isinstance(f, float):
+            raise TypeError("%s.from_float() only takes floats, not %r (%s)" %
+                            (cls.__name__, f, type(f).__name__))
+        if math.isnan(f) or math.isinf(f):
+            raise TypeError("Cannot convert %r to %s." % (f, cls.__name__))
+        return cls(*_binary_float_to_ratio(f))
+
+    @property
+    def numerator(a):
+        return a._numerator
+
+    @property
+    def denominator(a):
+        return a._denominator
+
+    def __repr__(self):
+        """repr(self)"""
+        return ('rational.Rational(%r,%r)' %
+                (self.numerator, self.denominator))
+
+    def __str__(self):
+        """str(self)"""
+        if self.denominator == 1:
+            return str(self.numerator)
+        else:
+            return '(%s/%s)' % (self.numerator, self.denominator)
+
+    def _operator_fallbacks(monomorphic_operator, fallback_operator):
+        """Generates forward and reverse operators given a purely-rational
+        operator and a function from the operator module.
+
+        Use this like:
+        __op__, __rop__ = _operator_fallbacks(just_rational_op, operator.op)
+
+        """
+        def forward(a, b):
+            if isinstance(b, RationalAbc):
+                # Includes ints.
+                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, RationalAbc):
+                # 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 Rational(a.numerator * b.denominator +
+                        b.numerator * a.denominator,
+                        a.denominator * b.denominator)
+
+    __add__, __radd__ = _operator_fallbacks(_add, operator.add)
+
+    def _sub(a, b):
+        """a - b"""
+        return Rational(a.numerator * b.denominator -
+                        b.numerator * a.denominator,
+                        a.denominator * b.denominator)
+
+    __sub__, __rsub__ = _operator_fallbacks(_sub, operator.sub)
+
+    def _mul(a, b):
+        """a * b"""
+        return Rational(a.numerator * b.numerator, a.denominator * b.denominator)
+
+    __mul__, __rmul__ = _operator_fallbacks(_mul, operator.mul)
+
+    def _div(a, b):
+        """a / b"""
+        return Rational(a.numerator * b.denominator,
+                        a.denominator * b.numerator)
+
+    __truediv__, __rtruediv__ = _operator_fallbacks(_div, operator.truediv)
+    __div__, __rdiv__ = _operator_fallbacks(_div, operator.div)
+
+    @classmethod
+    def _floordiv(cls, a, b):
+        div = a / b
+        if isinstance(div, RationalAbc):
+            # trunc(math.floor(div)) doesn't work if the rational is
+            # more precise than a float because the intermediate
+            # rounding may cross an integer boundary.
+            return div.numerator // div.denominator
+        else:
+            return math.floor(div)
+
+    def __floordiv__(a, b):
+        """a // b"""
+        # Will be math.floor(a / b) in 3.0.
+        return a._floordiv(a, b)
+
+    def __rfloordiv__(b, a):
+        """a // b"""
+        # Will be math.floor(a / b) in 3.0.
+        return b._floordiv(a, b)
+
+    @classmethod
+    def _mod(cls, a, b):
+        div = a // b
+        return a - b * div
+
+    def __mod__(a, b):
+        """a % b"""
+        return a._mod(a, b)
+
+    def __rmod__(b, a):
+        """a % b"""
+        return b._mod(a, b)
+
+    def __pow__(a, b):
+        """a ** b
+
+        If b is not an integer, the result will be a float or complex
+        since roots are generally irrational. If b is an integer, the
+        result will be rational.
+
+        """
+        if isinstance(b, RationalAbc):
+            if b.denominator == 1:
+                power = b.numerator
+                if power >= 0:
+                    return Rational(a.numerator ** power,
+                                    a.denominator ** power)
+                else:
+                    return Rational(a.denominator ** -power,
+                                    a.numerator ** -power)
+            else:
+                # A fractional power will generally produce an
+                # irrational number.
+                return float(a) ** float(b)
+        else:
+            return float(a) ** b
+
+    def __rpow__(b, a):
+        """a ** b"""
+        if b.denominator == 1 and b.numerator >= 0:
+            # If a is an int, keep it that way if possible.
+            return a ** b.numerator
+
+        if isinstance(a, RationalAbc):
+            return Rational(a.numerator, a.denominator) ** b
+
+        if b.denominator == 1:
+            return a ** b.numerator
+
+        return a ** float(b)
+
+    def __pos__(a):
+        """+a: Coerces a subclass instance to Rational"""
+        return Rational(a.numerator, a.denominator)
+
+    def __neg__(a):
+        """-a"""
+        return Rational(-a.numerator, a.denominator)
+
+    def __abs__(a):
+        """abs(a)"""
+        return Rational(abs(a.numerator), a.denominator)
+
+    def __trunc__(a):
+        """trunc(a)"""
+        if a.numerator < 0:
+            return -(-a.numerator // a.denominator)
+        else:
+            return a.numerator // a.denominator
+
+    def __floor__(a):
+        """Will be math.floor(a) in 3.0."""
+        return a.numerator // a.denominator
+
+    def __ceil__(a):
+        """Will be math.ceil(a) in 3.0."""
+        # The negations cleverly convince floordiv to return the ceiling.
+        return -(-a.numerator // a.denominator)
+
+    def __round__(self, ndigits=None):
+        """Will be round(self, ndigits) in 3.0.
+
+        Rounds half toward even.
+        """
+        if ndigits is None:
+            floor, remainder = divmod(self.numerator, self.denominator)
+            if remainder * 2 < self.denominator:
+                return floor
+            elif remainder * 2 > self.denominator:
+                return floor + 1
+            # Deal with the half case:
+            elif floor % 2 == 0:
+                return floor
+            else:
+                return floor + 1
+        shift = 10**abs(ndigits)
+        # See _operator_fallbacks.forward to check that the results of
+        # these operations will always be Rational and therefore have
+        # __round__().
+        if ndigits > 0:
+            return Rational((self * shift).__round__(), shift)
+        else:
+            return Rational((self / shift).__round__() * shift)
+
+    def __hash__(self):
+        """hash(self)
+
+        Tricky because values that are exactly representable as a
+        float must have the same hash as that float.
+
+        """
+        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))
+
+    def __eq__(a, b):
+        """a == b"""
+        if isinstance(b, RationalAbc):
+            return (a.numerator == b.numerator and
+                    a.denominator == b.denominator)
+        if isinstance(b, numbers.Complex) and b.imag == 0:
+            b = b.real
+        if isinstance(b, float):
+            return a == a.from_float(b)
+        else:
+            # XXX: If b.__eq__ is implemented like this method, it may
+            # give the wrong answer after float(a) changes a's
+            # value. Better ways of doing this are welcome.
+            return float(a) == b
+
+    def _subtractAndCompareToZero(a, b, op):
+        """Helper function for comparison operators.
+
+        Subtracts b from a, exactly if possible, and compares the
+        result with 0 using op, in such a way that the comparison
+        won't recurse. If the difference raises a TypeError, returns
+        NotImplemented instead.
+
+        """
+        if isinstance(b, numbers.Complex) and b.imag == 0:
+            b = b.real
+        if isinstance(b, float):
+            b = a.from_float(b)
+        try:
+            # XXX: If b <: Real but not <: RationalAbc, this is likely
+            # to fall back to a float. If the actual values differ by
+            # less than MIN_FLOAT, this could falsely call them equal,
+            # which would make <= inconsistent with ==. Better ways of
+            # doing this are welcome.
+            diff = a - b
+        except TypeError:
+            return NotImplemented
+        if isinstance(diff, RationalAbc):
+            return op(diff.numerator, 0)
+        return op(diff, 0)
+
+    def __lt__(a, b):
+        """a < b"""
+        return a._subtractAndCompareToZero(b, operator.lt)
+
+    def __gt__(a, b):
+        """a > b"""
+        return a._subtractAndCompareToZero(b, operator.gt)
+
+    def __le__(a, b):
+        """a <= b"""
+        return a._subtractAndCompareToZero(b, operator.le)
+
+    def __ge__(a, b):
+        """a >= b"""
+        return a._subtractAndCompareToZero(b, operator.ge)
+
+    def __nonzero__(a):
+        """a != 0"""
+        return a.numerator != 0
diff --git a/Lib/test/test_rational.py b/Lib/test/test_rational.py
new file mode 100644
index 0000000..654c46d
--- /dev/null
+++ b/Lib/test/test_rational.py
@@ -0,0 +1,279 @@
+"""Tests for Lib/rational.py."""
+
+from decimal import Decimal
+from test.test_support import run_unittest, verbose
+import math
+import operator
+import rational
+import unittest
+R = rational.Rational
+
+def _components(r):
+    return (r.numerator, r.denominator)
+
+class RationalTest(unittest.TestCase):
+
+    def assertTypedEquals(self, expected, actual):
+        """Asserts that both the types and values are the same."""
+        self.assertEquals(type(expected), type(actual))
+        self.assertEquals(expected, actual)
+
+    def assertRaisesMessage(self, exc_type, message,
+                            callable, *args, **kwargs):
+        """Asserts that callable(*args, **kwargs) raises exc_type(message)."""
+        try:
+            callable(*args, **kwargs)
+        except exc_type, e:
+            self.assertEquals(message, str(e))
+        else:
+            self.fail("%s not raised" % exc_type.__name__)
+
+    def testInit(self):
+        self.assertEquals((0, 1), _components(R()))
+        self.assertEquals((7, 1), _components(R(7)))
+        self.assertEquals((7, 3), _components(R(R(7, 3))))
+
+        self.assertEquals((-1, 1), _components(R(-1, 1)))
+        self.assertEquals((-1, 1), _components(R(1, -1)))
+        self.assertEquals((1, 1), _components(R(-2, -2)))
+        self.assertEquals((1, 2), _components(R(5, 10)))
+        self.assertEquals((7, 15), _components(R(7, 15)))
+        self.assertEquals((10**23, 1), _components(R(10**23)))
+
+        self.assertRaisesMessage(ZeroDivisionError, "Rational(12, 0)",
+                                 R, 12, 0)
+        self.assertRaises(TypeError, R, 1.5)
+        self.assertRaises(TypeError, R, 1.5 + 3j)
+
+    def testFromFloat(self):
+        self.assertRaisesMessage(
+            TypeError, "Rational.from_float() only takes floats, not 3 (int)",
+            R.from_float, 3)
+
+        self.assertEquals((0, 1), _components(R.from_float(-0.0)))
+        self.assertEquals((10, 1), _components(R.from_float(10.0)))
+        self.assertEquals((-5, 2), _components(R.from_float(-2.5)))
+        self.assertEquals((99999999999999991611392, 1),
+                          _components(R.from_float(1e23)))
+        self.assertEquals(float(10**23), float(R.from_float(1e23)))
+        self.assertEquals((3602879701896397, 1125899906842624),
+                          _components(R.from_float(3.2)))
+        self.assertEquals(3.2, float(R.from_float(3.2)))
+
+        inf = 1e1000
+        nan = inf - inf
+        self.assertRaisesMessage(
+            TypeError, "Cannot convert inf to Rational.",
+            R.from_float, inf)
+        self.assertRaisesMessage(
+            TypeError, "Cannot convert -inf to Rational.",
+            R.from_float, -inf)
+        self.assertRaisesMessage(
+            TypeError, "Cannot convert nan to Rational.",
+            R.from_float, nan)
+
+    def testConversions(self):
+        self.assertTypedEquals(-1, trunc(R(-11, 10)))
+        self.assertTypedEquals(-2, R(-11, 10).__floor__())
+        self.assertTypedEquals(-1, R(-11, 10).__ceil__())
+        self.assertTypedEquals(-1, R(-10, 10).__ceil__())
+
+        self.assertTypedEquals(0, R(-1, 10).__round__())
+        self.assertTypedEquals(0, R(-5, 10).__round__())
+        self.assertTypedEquals(-2, R(-15, 10).__round__())
+        self.assertTypedEquals(-1, R(-7, 10).__round__())
+
+        self.assertEquals(False, bool(R(0, 1)))
+        self.assertEquals(True, bool(R(3, 2)))
+        self.assertTypedEquals(0.1, float(R(1, 10)))
+
+        # Check that __float__ isn't implemented by converting the
+        # numerator and denominator to float before dividing.
+        self.assertRaises(OverflowError, float, long('2'*400+'7'))
+        self.assertAlmostEquals(2.0/3,
+                                float(R(long('2'*400+'7'), long('3'*400+'1'))))
+
+        self.assertTypedEquals(0.1+0j, complex(R(1,10)))
+
+    def testRound(self):
+        self.assertTypedEquals(R(-200), R(-150).__round__(-2))
+        self.assertTypedEquals(R(-200), R(-250).__round__(-2))
+        self.assertTypedEquals(R(30), R(26).__round__(-1))
+        self.assertTypedEquals(R(-2, 10), R(-15, 100).__round__(1))
+        self.assertTypedEquals(R(-2, 10), R(-25, 100).__round__(1))
+
+
+    def testArithmetic(self):
+        self.assertEquals(R(1, 2), R(1, 10) + R(2, 5))
+        self.assertEquals(R(-3, 10), R(1, 10) - R(2, 5))
+        self.assertEquals(R(1, 25), R(1, 10) * R(2, 5))
+        self.assertEquals(R(1, 4), R(1, 10) / R(2, 5))
+        self.assertTypedEquals(2, R(9, 10) // R(2, 5))
+        self.assertTypedEquals(10**23, R(10**23, 1) // R(1))
+        self.assertEquals(R(2, 3), R(-7, 3) % R(3, 2))
+        self.assertEquals(R(8, 27), R(2, 3) ** R(3))
+        self.assertEquals(R(27, 8), R(2, 3) ** R(-3))
+        self.assertTypedEquals(2.0, R(4) ** R(1, 2))
+        # Will return 1j in 3.0:
+        self.assertRaises(ValueError, pow, R(-1), R(1, 2))
+
+    def testMixedArithmetic(self):
+        self.assertTypedEquals(R(11, 10), R(1, 10) + 1)
+        self.assertTypedEquals(1.1, R(1, 10) + 1.0)
+        self.assertTypedEquals(1.1 + 0j, R(1, 10) + (1.0 + 0j))
+        self.assertTypedEquals(R(11, 10), 1 + R(1, 10))
+        self.assertTypedEquals(1.1, 1.0 + R(1, 10))
+        self.assertTypedEquals(1.1 + 0j, (1.0 + 0j) + R(1, 10))
+
+        self.assertTypedEquals(R(-9, 10), R(1, 10) - 1)
+        self.assertTypedEquals(-0.9, R(1, 10) - 1.0)
+        self.assertTypedEquals(-0.9 + 0j, R(1, 10) - (1.0 + 0j))
+        self.assertTypedEquals(R(9, 10), 1 - R(1, 10))
+        self.assertTypedEquals(0.9, 1.0 - R(1, 10))
+        self.assertTypedEquals(0.9 + 0j, (1.0 + 0j) - R(1, 10))
+
+        self.assertTypedEquals(R(1, 10), R(1, 10) * 1)
+        self.assertTypedEquals(0.1, R(1, 10) * 1.0)
+        self.assertTypedEquals(0.1 + 0j, R(1, 10) * (1.0 + 0j))
+        self.assertTypedEquals(R(1, 10), 1 * R(1, 10))
+        self.assertTypedEquals(0.1, 1.0 * R(1, 10))
+        self.assertTypedEquals(0.1 + 0j, (1.0 + 0j) * R(1, 10))
+
+        self.assertTypedEquals(R(1, 10), R(1, 10) / 1)
+        self.assertTypedEquals(0.1, R(1, 10) / 1.0)
+        self.assertTypedEquals(0.1 + 0j, R(1, 10) / (1.0 + 0j))
+        self.assertTypedEquals(R(10, 1), 1 / R(1, 10))
+        self.assertTypedEquals(10.0, 1.0 / R(1, 10))
+        self.assertTypedEquals(10.0 + 0j, (1.0 + 0j) / R(1, 10))
+
+        self.assertTypedEquals(0, R(1, 10) // 1)
+        self.assertTypedEquals(0.0, R(1, 10) // 1.0)
+        self.assertTypedEquals(10, 1 // R(1, 10))
+        self.assertTypedEquals(10**23, 10**22 // R(1, 10))
+        self.assertTypedEquals(10.0, 1.0 // R(1, 10))
+
+        self.assertTypedEquals(R(1, 10), R(1, 10) % 1)
+        self.assertTypedEquals(0.1, R(1, 10) % 1.0)
+        self.assertTypedEquals(R(0, 1), 1 % R(1, 10))
+        self.assertTypedEquals(0.0, 1.0 % R(1, 10))
+
+        # No need for divmod since we don't override it.
+
+        # ** has more interesting conversion rules.
+        self.assertTypedEquals(R(100, 1), R(1, 10) ** -2)
+        self.assertTypedEquals(R(100, 1), R(10, 1) ** 2)
+        self.assertTypedEquals(0.1, R(1, 10) ** 1.0)
+        self.assertTypedEquals(0.1 + 0j, R(1, 10) ** (1.0 + 0j))
+        self.assertTypedEquals(4 , 2 ** R(2, 1))
+        # Will return 1j in 3.0:
+        self.assertRaises(ValueError, pow, (-1), R(1, 2))
+        self.assertTypedEquals(R(1, 4) , 2 ** R(-2, 1))
+        self.assertTypedEquals(2.0 , 4 ** R(1, 2))
+        self.assertTypedEquals(0.25, 2.0 ** R(-2, 1))
+        self.assertTypedEquals(1.0 + 0j, (1.0 + 0j) ** R(1, 10))
+
+    def testMixingWithDecimal(self):
+        """Decimal refuses mixed comparisons."""
+        self.assertRaisesMessage(
+            TypeError,
+            "unsupported operand type(s) for +: 'Rational' and 'Decimal'",
+            operator.add, R(3,11), Decimal('3.1415926'))
+        self.assertNotEquals(R(5, 2), Decimal('2.5'))
+
+    def testComparisons(self):
+        self.assertTrue(R(1, 2) < R(2, 3))
+        self.assertFalse(R(1, 2) < R(1, 2))
+        self.assertTrue(R(1, 2) <= R(2, 3))
+        self.assertTrue(R(1, 2) <= R(1, 2))
+        self.assertFalse(R(2, 3) <= R(1, 2))
+        self.assertTrue(R(1, 2) == R(1, 2))
+        self.assertFalse(R(1, 2) == R(1, 3))
+
+    def testMixedLess(self):
+        self.assertTrue(2 < R(5, 2))
+        self.assertFalse(2 < R(4, 2))
+        self.assertTrue(R(5, 2) < 3)
+        self.assertFalse(R(4, 2) < 2)
+
+        self.assertTrue(R(1, 2) < 0.6)
+        self.assertFalse(R(1, 2) < 0.4)
+        self.assertTrue(0.4 < R(1, 2))
+        self.assertFalse(0.5 < R(1, 2))
+
+    def testMixedLessEqual(self):
+        self.assertTrue(0.5 <= R(1, 2))
+        self.assertFalse(0.6 <= R(1, 2))
+        self.assertTrue(R(1, 2) <= 0.5)
+        self.assertFalse(R(1, 2) <= 0.4)
+        self.assertTrue(2 <= R(4, 2))
+        self.assertFalse(2 <= R(3, 2))
+        self.assertTrue(R(4, 2) <= 2)
+        self.assertFalse(R(5, 2) <= 2)
+
+    def testBigFloatComparisons(self):
+        # Because 10**23 can't be represented exactly as a float:
+        self.assertFalse(R(10**23) == float(10**23))
+        # The first test demonstrates why these are important.
+        self.assertFalse(1e23 < float(R(trunc(1e23) + 1)))
+        self.assertTrue(1e23 < R(trunc(1e23) + 1))
+        self.assertFalse(1e23 <= R(trunc(1e23) - 1))
+        self.assertTrue(1e23 > R(trunc(1e23) - 1))
+        self.assertFalse(1e23 >= R(trunc(1e23) + 1))
+
+    def testBigComplexComparisons(self):
+        self.assertFalse(R(10**23) == complex(10**23))
+        self.assertTrue(R(10**23) > complex(10**23))
+        self.assertFalse(R(10**23) <= complex(10**23))
+
+    def testMixedEqual(self):
+        self.assertTrue(0.5 == R(1, 2))
+        self.assertFalse(0.6 == R(1, 2))
+        self.assertTrue(R(1, 2) == 0.5)
+        self.assertFalse(R(1, 2) == 0.4)
+        self.assertTrue(2 == R(4, 2))
+        self.assertFalse(2 == R(3, 2))
+        self.assertTrue(R(4, 2) == 2)
+        self.assertFalse(R(5, 2) == 2)
+
+    def testStringification(self):
+        self.assertEquals("rational.Rational(7,3)", repr(R(7, 3)))
+        self.assertEquals("(7/3)", str(R(7, 3)))
+        self.assertEquals("7", str(R(7, 1)))
+
+    def testHash(self):
+        self.assertEquals(hash(2.5), hash(R(5, 2)))
+        self.assertEquals(hash(10**50), hash(R(10**50)))
+        self.assertNotEquals(hash(float(10**23)), hash(R(10**23)))
+
+    def testApproximatePi(self):
+        # Algorithm borrowed from
+        # http://docs.python.org/lib/decimal-recipes.html
+        three = R(3)
+        lasts, t, s, n, na, d, da = 0, three, 3, 1, 0, 0, 24
+        while abs(s - lasts) > R(1, 10**9):
+            lasts = s
+            n, na = n+na, na+8
+            d, da = d+da, da+32
+            t = (t * n) / d
+            s += t
+        self.assertAlmostEquals(math.pi, s)
+
+    def testApproximateCos1(self):
+        # Algorithm borrowed from
+        # http://docs.python.org/lib/decimal-recipes.html
+        x = R(1)
+        i, lasts, s, fact, num, sign = 0, 0, R(1), 1, 1, 1
+        while abs(s - lasts) > R(1, 10**9):
+            lasts = s
+            i += 2
+            fact *= i * (i-1)
+            num *= x * x
+            sign *= -1
+            s += num / fact * sign
+        self.assertAlmostEquals(math.cos(1), s)
+
+def test_main():
+    run_unittest(RationalTest)
+
+if __name__ == '__main__':
+    test_main()