both complex and rational are now builtin classes.

git-svn-id: svn+ssh://ci.ruby-lang.org/ruby/trunk@15783 b2dd03c8-39d4-4d8f-98ff-823fe69b080e

3 files changed, 117 insertions, 1040 deletions

diff --git a/lib/complex.rb b/lib/complex.rb
index 9a621c033f..505b0120e3 100644
--- a/lib/complex.rb
+++ b/lib/complex.rb
@@ -1,473 +1,90 @@
-#
-#   complex.rb - 
-#   	$Release Version: 0.5 $
-#   	$Revision: 1.3 $
-#   	by Keiju ISHITSUKA(SHL Japan Inc.)
-#
-# ----
-#
-# complex.rb implements the Complex class for complex numbers.  Additionally,
-# some methods in other Numeric classes are redefined or added to allow greater
-# interoperability with Complex numbers.
-#
-# Complex numbers can be created in the following manner:
-# - <tt>Complex(a, b)</tt>
-# - <tt>Complex.polar(radius, theta)</tt>
-#   
-# Additionally, note the following:
-# - <tt>Complex::I</tt> (the mathematical constant <i>i</i>)
-# - <tt>Numeric#im</tt> (e.g. <tt>5.im -> 0+5i</tt>)
-#
-# The following +Math+ module methods are redefined to handle Complex arguments.
-# They will work as normal with non-Complex arguments.
-#    sqrt exp cos sin tan log log10
-#    cosh sinh tanh acos asin atan atan2 acosh asinh atanh
-#
-
-
-#
-# Numeric is a built-in class on which Fixnum, Bignum, etc., are based.  Here
-# some methods are added so that all number types can be treated to some extent
-# as Complex numbers.
-#
-class Numeric
-  #
-  # Returns a Complex number <tt>(0,<i>self</i>)</tt>.
-  #
-  def im
-    Complex(0, self)
-  end
-  
-  #
-  # The real part of a complex number, i.e. <i>self</i>.
-  #
-  def real
-    self
-  end
-  
-  #
-  # The imaginary part of a complex number, i.e. 0.
-  #
-  def image
-    0
-  end
-  alias imag image
-  
-  #
-  # See Complex#arg.
-  #
-  def arg
-    if self >= 0
-      return 0
-    else
-      return Math::PI
-    end
-  end
-  alias angle arg
-  
-  #
-  # See Complex#polar.
-  #
-  def polar
-    return abs, arg
-  end
-  
-  #
-  # See Complex#conjugate (short answer: returns <i>self</i>).
-  #
-  def conjugate
-    self
-  end
-  alias conj conjugate
-end
-
-
-#
-# Creates a Complex number.  +a+ and +b+ should be Numeric.  The result will be
-# <tt>a+bi</tt>.
-#
-def Complex(a, b = 0)
-  if b == 0 and (a.kind_of?(Complex) or defined? Complex::Unify)
-    a
-  elsif a.scalar? and b.scalar?
-    # Don't delete for -0.0
-    Complex.new(a, b)
-  else
-    Complex.new( a.real-b.imag, a.imag+b.real )
-  end
-end
-
-#
-# The complex number class.  See complex.rb for an overview.
-#
-class Complex < Numeric
-  @RCS_ID='-$Id: complex.rb,v 1.3 1998/07/08 10:05:28 keiju Exp keiju $-'
-
-  undef step
-  undef <, <=, <=>, >, >=
-  undef between?
-  undef div, divmod, modulo
-  undef floor, truncate, ceil, round
-
-  def scalar?
-    false
-  end
-
-  def Complex.generic?(other) # :nodoc:
-    other.kind_of?(Integer) or
-    other.kind_of?(Float) or
-    (defined?(Rational) and other.kind_of?(Rational))
-  end
-
-  #
-  # Creates a +Complex+ number in terms of +r+ (radius) and +theta+ (angle).
-  #
-  def Complex.polar(r, theta)
-    Complex(r*Math.cos(theta), r*Math.sin(theta))
-  end
-
-  #
-  # Creates a +Complex+ number <tt>a</tt>+<tt>b</tt><i>i</i>.
-  #
-  def Complex.new!(a, b=0)
-    new(a,b)
-  end
-
-  def initialize(a, b)
-    raise TypeError, "non numeric 1st arg `#{a.inspect}'" if !a.kind_of? Numeric
-    raise TypeError, "`#{a.inspect}' for 1st arg" if a.kind_of? Complex
-    raise TypeError, "non numeric 2nd arg `#{b.inspect}'" if !b.kind_of? Numeric
-    raise TypeError, "`#{b.inspect}' for 2nd arg" if b.kind_of? Complex
-    @real = a
-    @image = b
-  end
-
-  #
-  # Addition with real or complex number.
-  #
-  def + (other)
-    if other.kind_of?(Complex)
-      re = @real + other.real
-      im = @image + other.image
-      Complex(re, im)
-    elsif Complex.generic?(other)
-      Complex(@real + other, @image)
-    else
-      x , y = other.coerce(self)
-      x + y
-    end
-  end
-  
-  #
-  # Subtraction with real or complex number.
-  #
-  def - (other)
-    if other.kind_of?(Complex)
-      re = @real - other.real
-      im = @image - other.image
-      Complex(re, im)
-    elsif Complex.generic?(other)
-      Complex(@real - other, @image)
-    else
-      x , y = other.coerce(self)
-      x - y
-    end
-  end
-  
-  #
-  # Multiplication with real or complex number.
-  #
-  def * (other)
-    if other.kind_of?(Complex)
-      re = @real*other.real - @image*other.image
-      im = @real*other.image + @image*other.real
-      Complex(re, im)
-    elsif Complex.generic?(other)
-      Complex(@real * other, @image * other)
-    else
-      x , y = other.coerce(self)
-      x * y
-    end
-  end
-  
-  #
-  # Division by real or complex number.
-  #
-  def / (other)
-    if other.kind_of?(Complex)
-      self*other.conjugate/other.abs2
-    elsif Complex.generic?(other)
-      Complex(@real/other, @image/other)
-    else
-      x, y = other.coerce(self)
-      x/y
-    end
-  end
+class Integer
 
-  def quo(other)
-    Complex(@real.quo(1), @image.quo(1)) / other
-  end
-  
-  #
-  # Raise this complex number to the given (real or complex) power.
-  #
-  def ** (other)
-    if other == 0
-      return Complex(1)
-    end
-    if other.kind_of?(Complex)
-      r, theta = polar
-      ore = other.real
-      oim = other.image
-      nr = Math.exp!(ore*Math.log!(r) - oim * theta)
-      ntheta = theta*ore + oim*Math.log!(r)
-      Complex.polar(nr, ntheta)
-    elsif other.kind_of?(Integer)
-      if other > 0
-	x = self
-	z = x
-	n = other - 1
-	while n != 0
-	  while (div, mod = n.divmod(2)
-		 mod == 0)
-	    x = Complex(x.real*x.real - x.image*x.image, 2*x.real*x.image)
-	    n = div
-	  end
-	  z *= x
-	  n -= 1
-	end
-	z
-      else
-	if defined? Rational
-	  (Rational(1) / self) ** -other
-	else
-	  self ** Float(other)
-	end
-      end
-    elsif Complex.generic?(other)
-      r, theta = polar
-      Complex.polar(r**other, theta*other)
-    else
-      x, y = other.coerce(self)
-      x**y
-    end
-  end
-  
-  #
-  # Remainder after division by a real or complex number.
-  #
-
-=begin
-  def % (other)
-    if other.kind_of?(Complex)
-      Complex(@real % other.real, @image % other.image)
-    elsif Complex.generic?(other)
-      Complex(@real % other, @image % other)
-    else
-      x , y = other.coerce(self)
-      x % y
-    end
-  end
-=end
-
-#--
-#    def divmod(other)
-#      if other.kind_of?(Complex)
-#        rdiv, rmod = @real.divmod(other.real)
-#        idiv, imod = @image.divmod(other.image)
-#        return Complex(rdiv, idiv), Complex(rmod, rmod)
-#      elsif Complex.generic?(other)
-#        Complex(@real.divmod(other), @image.divmod(other))
-#      else
-#        x , y = other.coerce(self)
-#        x.divmod(y)
-#      end
-#    end
-#++
-  
-  #
-  # Absolute value (aka modulus): distance from the zero point on the complex
-  # plane.
-  #
-  def abs
-    Math.hypot(@real, @image)
-  end
-  
-  #
-  # Square of the absolute value.
-  #
-  def abs2
-    @real*@real + @image*@image
-  end
-  
-  #
-  # Argument (angle from (1,0) on the complex plane).
-  #
-  def arg
-    Math.atan2!(@image, @real)
-  end
-  alias angle arg
-  
-  #
-  # Returns the absolute value _and_ the argument.
-  #
-  def polar
-    return abs, arg
-  end
-  
-  #
-  # Complex conjugate (<tt>z + z.conjugate = 2 * z.real</tt>).
-  #
-  def conjugate
-    Complex(@real, -@image)
-  end
-  alias conj conjugate
-
-  #
-  # Test for numerical equality (<tt>a == a + 0<i>i</i></tt>).
-  #
-  def == (other)
-    if other.kind_of?(Complex)
-      @real == other.real and @image == other.image
-    elsif Complex.generic?(other)
-      @real == other and @image == 0
-    else
-      other == self
+  def gcd(other)
+    min = self.abs
+    max = other.abs
+    while min > 0
+      tmp = min
+      min = max % min
+      max = tmp
     end
+    max
   end
 
-  #
-  # Attempts to coerce +other+ to a Complex number.
-  #
-  def coerce(other)
-    if Complex.generic?(other)
-      return Complex.new!(other), self
+  def lcm(other)
+    if self.zero? or other.zero?
+      0
     else
-      super
+      (self.div(self.gcd(other)) * other).abs
     end
   end
 
-  #
-  # FIXME
-  #
-  def denominator
-    @real.denominator.lcm(@image.denominator)
-  end
-  
-  #
-  # FIXME
-  #
-  def numerator
-    cd = denominator
-    Complex(@real.numerator*(cd/@real.denominator),
-	    @image.numerator*(cd/@image.denominator))
-  end
-  
-  #
-  # Standard string representation of the complex number.
-  #
-  def to_s
-    if @real != 0
-      if defined?(Rational) and @image.kind_of?(Rational) and @image.denominator != 1
-	if @image >= 0
-	  @real.to_s+"+("+@image.to_s+")i"
-	else
-	  @real.to_s+"-("+(-@image).to_s+")i"
-	end
-      else
-	if @image >= 0
-	  @real.to_s+"+"+@image.to_s+"i"
-	else
-	  @real.to_s+"-"+(-@image).to_s+"i"
-	end
-      end
+  def gcdlcm(other)
+    gcd = self.gcd(other)
+    if self.zero? or other.zero?
+      [gcd, 0]
     else
-      if defined?(Rational) and @image.kind_of?(Rational) and @image.denominator != 1
-	"("+@image.to_s+")i"
-      else
-	@image.to_s+"i"
-      end
+      [gcd, (self.div(gcd) * other).abs]
     end
   end
-  
-  #
-  # Returns a hash code for the complex number.
-  #
-  def hash
-    @real.hash ^ @image.hash
-  end
-  
-  #
-  # Returns "<tt>Complex(<i>real</i>, <i>image</i>)</tt>".
-  #
-  def inspect
-    sprintf("Complex(%s, %s)", @real.inspect, @image.inspect)
-  end
-
-  
-  #
-  # +I+ is the imaginary number.  It exists at point (0,1) on the complex plane.
-  #
-  I = Complex(0,1)
-  
-  # The real part of a complex number.
-  attr_reader :real
-
-  # The imaginary part of a complex number.
-  attr_reader :image
-  alias imag image
-  
-end
-
-class Integer
-
-  unless defined?(1.numerator)
-    def numerator() self end
-    def denominator() 1 end
-
-    def gcd(other)
-      min = self.abs
-      max = other.abs
-      while min > 0
-        tmp = min
-        min = max % min
-        max = tmp
-      end
-      max
-    end
-
-    def lcm(other)
-      if self.zero? or other.zero?
-        0
-      else
-        (self.div(self.gcd(other)) * other).abs
-      end
-    end
-
-  end
 
 end
 
 module Math
-  alias sqrt! sqrt
+
   alias exp! exp
   alias log! log
   alias log10! log10
-  alias cos! cos
+  alias sqrt! sqrt
+
   alias sin! sin
+  alias cos! cos
   alias tan! tan
-  alias cosh! cosh
+
   alias sinh! sinh
+  alias cosh! cosh
   alias tanh! tanh
-  alias acos! acos
+
   alias asin! asin
+  alias acos! acos
   alias atan! atan
   alias atan2! atan2
-  alias acosh! acosh
+
   alias asinh! asinh
-  alias atanh! atanh  
+  alias acosh! acosh
+  alias atanh! atanh
+
+  def exp(z)
+    if Complex.generic?(z)
+      exp!(z)
+    else
+      Complex(exp!(z.real) * cos!(z.image),
+	      exp!(z.real) * sin!(z.image))
+    end
+  end
+
+  def log(*args)
+    z, b = args
+    if Complex.generic?(z) and z >= 0 and (b.nil? or b >= 0)
+      log!(*args)
+    else
+      r, theta = z.polar
+      a = Complex(log!(r.abs), theta)
+      if b
+	a /= log(b)
+      end
+      a
+    end
+  end
+
+  def log10(z)
+    if Complex.generic?(z)
+      log10!(z)
+    else
+      log(z) / log!(10)
+    end
+  end
 
-  # Redefined to handle a Complex argument.
   def sqrt(z)
     if Complex.generic?(z)
       if z >= 0
@@ -481,41 +98,29 @@ module Math
       else
 	r = z.abs
 	x = z.real
-	Complex( sqrt!((r+x)/2), sqrt!((r-x)/2) )
+	Complex(sqrt!((r + x) / 2), sqrt!((r - x) / 2))
       end
     end
   end
-  
-  # Redefined to handle a Complex argument.
-  def exp(z)
+
+  def sin(z)
     if Complex.generic?(z)
-      exp!(z)
+      sin!(z)
     else
-      Complex(exp!(z.real) * cos!(z.image), exp!(z.real) * sin!(z.image))
+      Complex(sin!(z.real) * cosh!(z.image),
+	      cos!(z.real) * sinh!(z.image))
     end
   end
-  
-  # Redefined to handle a Complex argument.
+
   def cos(z)
     if Complex.generic?(z)
       cos!(z)
     else
-      Complex(cos!(z.real)*cosh!(z.image),
-	      -sin!(z.real)*sinh!(z.image))
+      Complex(cos!(z.real) * cosh!(z.image),
+	      -sin!(z.real) * sinh!(z.image))
     end
   end
-    
-  # Redefined to handle a Complex argument.
-  def sin(z)
-    if Complex.generic?(z)
-      sin!(z)
-    else
-      Complex(sin!(z.real)*cosh!(z.image),
-	      cos!(z.real)*sinh!(z.image))
-    end
-  end
-  
-  # Redefined to handle a Complex argument.
+
   def tan(z)
     if Complex.generic?(z)
       tan!(z)
@@ -528,7 +133,8 @@ module Math
     if Complex.generic?(z)
       sinh!(z)
     else
-      Complex( sinh!(z.real)*cos!(z.image), cosh!(z.real)*sin!(z.image) )
+      Complex(sinh!(z.real) * cos!(z.image),
+	      cosh!(z.real) * sin!(z.image))
     end
   end
 
@@ -536,7 +142,8 @@ module Math
     if Complex.generic?(z)
       cosh!(z)
     else
-      Complex( cosh!(z.real)*cos!(z.image), sinh!(z.real)*sin!(z.image) )
+      Complex(cosh!(z.real) * cos!(z.image),
+	      sinh!(z.real) * sin!(z.image))
     end
   end
 
@@ -544,42 +151,23 @@ module Math
     if Complex.generic?(z)
       tanh!(z)
     else
-      sinh(z)/cosh(z)
-    end
-  end
-  
-  # Redefined to handle a Complex argument.
-  def log(z)
-    if Complex.generic?(z) and z >= 0
-      log!(z)
-    else
-      r, theta = z.polar
-      Complex(log!(r.abs), theta)
-    end
-  end
-  
-  # Redefined to handle a Complex argument.
-  def log10(z)
-    if Complex.generic?(z)
-      log10!(z)
-    else
-      log(z)/log!(10)
+      sinh(z) / cosh(z)
     end
   end
 
-  def acos(z)
+  def asin(z)
     if Complex.generic?(z) and z >= -1 and z <= 1
-      acos!(z)
+      asin!(z)
     else
-      -1.0.im * log( z + 1.0.im * sqrt(1.0-z*z) )
+      -1.0.im * log(1.0.im * z + sqrt(1.0 - z * z))
     end
   end
 
-  def asin(z)
+  def acos(z)
     if Complex.generic?(z) and z >= -1 and z <= 1
-      asin!(z)
+      acos!(z)
     else
-      -1.0.im * log( 1.0.im * z + sqrt(1.0-z*z) )
+      -1.0.im * log(z + 1.0.im * sqrt(1.0 - z * z))
     end
   end
 
@@ -587,7 +175,7 @@ module Math
     if Complex.generic?(z)
       atan!(z)
     else
-      1.0.im * log( (1.0.im+z) / (1.0.im-z) ) / 2.0
+      1.0.im * log((1.0.im + z) / (1.0.im - z)) / 2.0
     end
   end
 
@@ -595,7 +183,7 @@ module Math
     if Complex.generic?(y) and Complex.generic?(x)
       atan2!(y,x)
     else
-      -1.0.im * log( (x+1.0.im*y) / sqrt(x*x+y*y) )
+      -1.0.im * log((x + 1.0.im * y) / sqrt(x * x + y * y))
     end
   end
 
@@ -603,7 +191,7 @@ module Math
     if Complex.generic?(z) and z >= 1
       acosh!(z)
     else
-      log( z + sqrt(z*z-1.0) )
+      log(z + sqrt(z * z - 1.0))
     end
   end
 
@@ -611,7 +199,7 @@ module Math
     if Complex.generic?(z)
       asinh!(z)
     else
-      log( z + sqrt(1.0+z*z) )
+      log(z + sqrt(1.0 + z * z))
     end
   end
 
@@ -619,49 +207,47 @@ module Math
     if Complex.generic?(z) and z >= -1 and z <= 1
       atanh!(z)
     else
-      log( (1.0+z) / (1.0-z) ) / 2.0
+      log((1.0 + z) / (1.0 - z)) / 2.0
     end
   end
 
-  module_function :sqrt!
-  module_function :sqrt
   module_function :exp!
   module_function :exp
   module_function :log!
   module_function :log
   module_function :log10!
   module_function :log10
-  module_function :cosh!
-  module_function :cosh
-  module_function :cos!
-  module_function :cos
-  module_function :sinh!
-  module_function :sinh
+  module_function :sqrt!
+  module_function :sqrt
+
   module_function :sin!
   module_function :sin
+  module_function :cos!
+  module_function :cos
   module_function :tan!
   module_function :tan
+
+  module_function :sinh!
+  module_function :sinh
+  module_function :cosh!
+  module_function :cosh
   module_function :tanh!
   module_function :tanh
-  module_function :acos!
-  module_function :acos
+
   module_function :asin!
   module_function :asin
+  module_function :acos!
+  module_function :acos
   module_function :atan!
   module_function :atan
   module_function :atan2!
   module_function :atan2
-  module_function :acosh!
-  module_function :acosh
+
   module_function :asinh!
   module_function :asinh
+  module_function :acosh!
+  module_function :acosh
   module_function :atanh!
   module_function :atanh
-  
-end
 
-# Documentation comments:
-#  - source: original (researched from pickaxe)
-#  - a couple of fixme's
-#  - RDoc output for Bignum etc. is a bit short, with nothing but an
-#    (undocumented) alias.  No big deal.
+end
diff --git a/lib/mathn.rb b/lib/mathn.rb
index 724d37ea6f..f3be55eb6d 100644
--- a/lib/mathn.rb
+++ b/lib/mathn.rb
@@ -127,7 +127,7 @@ class Rational
     if other.kind_of?(Rational)
       other2 = other
       if self < 0
-	return Complex.new!(self, 0) ** other
+	return Complex.__send__(:new!, self, 0) ** other
       elsif other == 0
 	return Rational(1,1)
       elsif self == 0
@@ -175,7 +175,7 @@ class Rational
 	num = 1
 	den = 1
       end
-      Rational.new!(num, den)
+      Rational(num, den)
     elsif other.kind_of?(Float)
       Float(self) ** other
     else
@@ -187,7 +187,7 @@ class Rational
   def power2(other)
     if other.kind_of?(Rational)
       if self < 0
-	return Complex(self, 0) ** other
+	return Complex.__send__(:new!, self, 0) ** other
       elsif other == 0
 	return Rational(1,1)
       elsif self == 0
@@ -219,7 +219,7 @@ class Rational
 	num = 1
 	den = 1
       end
-      Rational.new!(num, den)
+      Rational(num, den)
     elsif other.kind_of?(Float)
       Float(self) ** other
     else
@@ -306,4 +306,3 @@ end
 class Complex
   Unify = true
 end
-
diff --git a/lib/rational.rb b/lib/rational.rb
index 59588528ab..b12bf7ef38 100644
--- a/lib/rational.rb
+++ b/lib/rational.rb
@@ -1,469 +1,23 @@
-#
-#   rational.rb -
-#       $Release Version: 0.5 $
-#       $Revision: 1.7 $
-#       by Keiju ISHITSUKA(SHL Japan Inc.)
-#
-# Documentation by Kevin Jackson and Gavin Sinclair.
-# 
-# When you <tt>require 'rational'</tt>, all interactions between numbers
-# potentially return a rational result.  For example:
-#
-#   1.quo(2)              # -> 0.5
-#   require 'rational'
-#   1.quo(2)              # -> Rational(1,2)
-# 
-# See Rational for full documentation.
-#
-
-#
-# Creates a Rational number (i.e. a fraction).  +a+ and +b+ should be Integers:
-# 
-#   Rational(1,3)           # -> 1/3
-#
-# Note: trying to construct a Rational with floating point or real values
-# produces errors:
-#
-#   Rational(1.1, 2.3)      # -> NoMethodError
-#
-def Rational(a, b = 1)
-  if a.kind_of?(Rational) && b == 1
-    a
-  else
-    Rational.reduce(a, b)
-  end
-end
-
-#
-# Rational implements a rational class for numbers.
-#
-# <em>A rational number is a number that can be expressed as a fraction p/q
-# where p and q are integers and q != 0.  A rational number p/q is said to have
-# numerator p and denominator q.  Numbers that are not rational are called
-# irrational numbers.</em> (http://mathworld.wolfram.com/RationalNumber.html)
-#
-# To create a Rational Number:
-#   Rational(a,b)             # -> a/b
-#   Rational.new!(a,b)        # -> a/b
-#
-# Examples:
-#   Rational(5,6)             # -> 5/6
-#   Rational(5)               # -> 5/1
-# 
-# Rational numbers are reduced to their lowest terms:
-#   Rational(6,10)            # -> 3/5
-#
-# But not if you use the unusual method "new!":
-#   Rational.new!(6,10)       # -> 6/10
-#
-# Division by zero is obviously not allowed:
-#   Rational(3,0)             # -> ZeroDivisionError
-#
-class Rational < Numeric
-  @RCS_ID='-$Id: rational.rb,v 1.7 1999/08/24 12:49:28 keiju Exp keiju $-'
-
-  #
-  # Reduces the given numerator and denominator to their lowest terms.  Use
-  # Rational() instead.
-  #
-  def Rational.reduce(num, den = 1)
-    raise ZeroDivisionError, "denominator is zero" if den == 0
-
-    if den < 0
-      num = -num
-      den = -den
-    end
-    gcd = num.gcd(den)
-    num = num.div(gcd)
-    den = den.div(gcd)
-    if den == 1 && defined?(Unify)
-      num
-    else
-      new!(num, den)
-    end
-  end
-
-  #
-  # Implements the constructor.  This method does not reduce to lowest terms or
-  # check for division by zero.  Therefore #Rational() should be preferred in
-  # normal use.
-  #
-  def Rational.new!(num, den = 1)
-    new(num, den)
-  end
-
-  private_class_method :new
-
-  #
-  # This method is actually private.
-  #
-  def initialize(num, den)
-    if den < 0
-      num = -num
-      den = -den
-    end
-    if num.kind_of?(Integer) and den.kind_of?(Integer)
-      @numerator = num
-      @denominator = den
-    else
-      @numerator = num.to_i
-      @denominator = den.to_i
-    end
-  end
-
-  #
-  # Returns the addition of this value and +a+.
-  #
-  # Examples:
-  #   r = Rational(3,4)      # -> Rational(3,4)
-  #   r + 1                  # -> Rational(7,4)
-  #   r + 0.5                # -> 1.25
-  #
-  def + (a)
-    if a.kind_of?(Rational)
-      num = @numerator * a.denominator
-      num_a = a.numerator * @denominator
-      Rational(num + num_a, @denominator * a.denominator)
-    elsif a.kind_of?(Integer)
-      self + Rational.new!(a, 1)
-    elsif a.kind_of?(Float)
-      Float(self) + a
-    else
-      x, y = a.coerce(self)
-      x + y
-    end
-  end
-
-  #
-  # Returns the difference of this value and +a+.
-  # subtracted.
-  #
-  # Examples:
-  #   r = Rational(3,4)    # -> Rational(3,4)
-  #   r - 1                # -> Rational(-1,4)
-  #   r - 0.5              # -> 0.25
-  #
-  def - (a)
-    if a.kind_of?(Rational)
-      num = @numerator * a.denominator
-      num_a = a.numerator * @denominator
-      Rational(num - num_a, @denominator*a.denominator)
-    elsif a.kind_of?(Integer)
-      self - Rational.new!(a, 1)
-    elsif a.kind_of?(Float)
-      Float(self) - a
-    else
-      x, y = a.coerce(self)
-      x - y
-    end
-  end
-
-  #
-  # Returns the product of this value and +a+.
-  #
-  # Examples:
-  #   r = Rational(3,4)    # -> Rational(3,4)
-  #   r * 2                # -> Rational(3,2)
-  #   r * 4                # -> Rational(3,1)
-  #   r * 0.5              # -> 0.375
-  #   r * Rational(1,2)    # -> Rational(3,8)
-  #
-  def * (a)
-    if a.kind_of?(Rational)
-      num = @numerator * a.numerator
-      den = @denominator * a.denominator
-      Rational(num, den)
-    elsif a.kind_of?(Integer)
-      self * Rational.new!(a, 1)
-    elsif a.kind_of?(Float)
-      Float(self) * a
-    else
-      x, y = a.coerce(self)
-      x * y
-    end
-  end
-
-  #
-  # Returns the quotient of this value and +a+.
-  #   r = Rational(3,4)    # -> Rational(3,4)
-  #   r / 2                # -> Rational(3,8)
-  #   r / 2.0              # -> 0.375
-  #   r / Rational(1,2)    # -> Rational(3,2)
-  #
-  def / (a)
-    if a.kind_of?(Rational)
-      num = @numerator * a.denominator
-      den = @denominator * a.numerator
-      Rational(num, den)
-    elsif a.kind_of?(Integer)
-      raise ZeroDivisionError, "division by zero" if a == 0
-      self / Rational.new!(a, 1)
-    elsif a.kind_of?(Float)
-      Float(self) / a
-    else
-      x, y = a.coerce(self)
-      x / y
-    end
-  end
-
-  #
-  # Returns this value raised to the given power.
-  #
-  # Examples:
-  #   r = Rational(3,4)    # -> Rational(3,4)
-  #   r ** 2               # -> Rational(9,16)
-  #   r ** 2.0             # -> 0.5625
-  #   r ** Rational(1,2)   # -> 0.866025403784439
-  #
-  def ** (other)
-    if other.kind_of?(Rational)
-      Float(self) ** other
-    elsif other.kind_of?(Integer)
-      if other > 0
-	num = @numerator ** other
-	den = @denominator ** other
-      elsif other < 0
-	num = @denominator ** -other
-	den = @numerator ** -other
-      elsif other == 0
-	num = 1
-	den = 1
-      end
-      Rational.new!(num, den)
-    elsif other.kind_of?(Float)
-      Float(self) ** other
-    else
-      x, y = other.coerce(self)
-      x ** y
-    end
-  end
-
-  def div(other)
-    (self / other).floor
-  end
-
-  #
-  # Returns the remainder when this value is divided by +other+.
-  #
-  # Examples:
-  #   r = Rational(7,4)    # -> Rational(7,4)
-  #   r % Rational(1,2)    # -> Rational(1,4)
-  #   r % 1                # -> Rational(3,4)
-  #   r % Rational(1,7)    # -> Rational(1,28)
-  #   r % 0.26             # -> 0.19
-  #
-  def % (other)
-    value = (self / other).floor
-    return self - other * value
-  end
-
-  #
-  # Returns the quotient _and_ remainder.
-  #
-  # Examples:
-  #   r = Rational(7,4)        # -> Rational(7,4)
-  #   r.divmod Rational(1,2)   # -> [3, Rational(1,4)]
-  #
-  def divmod(other)
-    value = (self / other).floor
-    return value, self - other * value
-  end
-
-  #
-  # Returns the absolute value.
-  #
-  def abs
-    if @numerator > 0
-      self
-    else
-      Rational.new!(-@numerator, @denominator)
-    end
-  end
-
-  #
-  # Returns +true+ iff this value is numerically equal to +other+.
-  #
-  # But beware:
-  #   Rational(1,2) == Rational(4,8)          # -> true
-  #   Rational(1,2) == Rational.new!(4,8)     # -> false
-  #
-  # Don't use Rational.new!
-  #
-  def == (other)
-    if other.kind_of?(Rational)
-      @numerator == other.numerator and @denominator == other.denominator
-    elsif other.kind_of?(Integer)
-      self == Rational.new!(other, 1)
-    elsif other.kind_of?(Float)
-      Float(self) == other
-    else
-      other == self
-    end
-  end
-
-  #
-  # Standard comparison operator.
-  #
-  def <=> (other)
-    if other.kind_of?(Rational)
-      num = @numerator * other.denominator
-      num_a = other.numerator * @denominator
-      v = num - num_a
-      if v > 0
-	return 1
-      elsif v < 0
-	return  -1
-      else
-	return 0
-      end
-    elsif other.kind_of?(Integer)
-      return self <=> Rational.new!(other, 1)
-    elsif other.kind_of?(Float)
-      return Float(self) <=> other
-    elsif defined? other.coerce
-      x, y = other.coerce(self)
-      return x <=> y
-    else
-      return nil
-    end
-  end
-
-  def coerce(other)
-    if other.kind_of?(Float)
-      return other, self.to_f
-    elsif other.kind_of?(Integer)
-      return Rational.new!(other, 1), self
-    else
-      super
-    end
-  end
-
-  #
-  # Converts the rational to an Integer.  Not the _nearest_ integer, the
-  # truncated integer.  Study the following example carefully:
-  #   Rational(+7,4).to_i             # -> 1
-  #   Rational(-7,4).to_i             # -> -2
-  #   (-1.75).to_i                    # -> -1
-  #
-  # In other words:
-  #   Rational(-7,4) == -1.75                 # -> true
-  #   Rational(-7,4).to_i == (-1.75).to_i     # false
-  #
-
-  def floor()
-    @numerator.div(@denominator)
-  end
-
-  def ceil()
-    -((-@numerator).div(@denominator))
-  end
-
-  def truncate()
-    if @numerator < 0
-      return -((-@numerator).div(@denominator))
-    end
-    @numerator.div(@denominator)
-  end
-
-  alias_method :to_i, :truncate
-
-  def round()
-    if @numerator < 0
-      num = -@numerator
-      num = num * 2 + @denominator
-      den = @denominator * 2
-      -(num.div(den))
-    else
-      num = @numerator * 2 + @denominator
-      den = @denominator * 2
-      num.div(den)
-    end
-  end
+class Fixnum
 
-  #
-  # Converts the rational to a Float.
-  #
-  def to_f
-    @numerator.quof(@denominator)
-  end
+  alias quof fdiv
 
-  #
-  # Returns a string representation of the rational number.
-  #
-  # Example:
-  #   Rational(3,4).to_s          #  "3/4"
-  #   Rational(8).to_s            #  "8"
-  #
-  def to_s
-    if @denominator == 1
-      @numerator.to_s
-    else
-      @numerator.to_s+"/"+@denominator.to_s
-    end
-  end
+  alias power! **
+  alias rpower **
 
-  #
-  # Returns +self+.
-  #
-  def to_r
-    self
-  end
+end
 
-  #
-  # Returns a reconstructable string representation:
-  #
-  #   Rational(5,8).inspect     # -> "Rational(5, 8)"
-  #
-  def inspect
-    sprintf("Rational(%s, %s)", @numerator.inspect, @denominator.inspect)
-  end
+class Bignum
 
-  #
-  # Returns a hash code for the object.
-  #
-  def hash
-    @numerator.hash ^ @denominator.hash
-  end
+  alias quof fdiv
 
-  attr :numerator
-  attr :denominator
+  alias power! **
+  alias rpower **
 
-  private :initialize
 end
 
 class Integer
-  #
-  # In an integer, the value _is_ the numerator of its rational equivalent.
-  # Therefore, this method returns +self+.
-  #
-  def numerator
-    self
-  end
-
-  #
-  # In an integer, the denominator is 1.  Therefore, this method returns 1.
-  #
-  def denominator
-    1
-  end
 
-  #
-  # Returns a Rational representation of this integer.
-  #
-  def to_r
-    Rational(self, 1)
-  end
-
-  #
-  # Returns the <em>greatest common denominator</em> of the two numbers (+self+
-  # and +n+).
-  #
-  # Examples:
-  #   72.gcd 168           # -> 24
-  #   19.gcd 36            # -> 1
-  #
-  # The result is positive, no matter the sign of the arguments.
-  #
   def gcd(other)
     min = self.abs
     max = other.abs
@@ -475,10 +29,6 @@ class Integer
     max
   end
 
-  # Examples:
-  #   6.lcm 7        # -> 42
-  #   6.lcm 9        # -> 18
-  #
   def lcm(other)
     if self.zero? or other.zero?
       0
@@ -486,15 +36,7 @@ class Integer
       (self.div(self.gcd(other)) * other).abs
     end
   end
-  
-  #
-  # Returns the GCD _and_ the LCM (see #gcd and #lcm) of the two arguments
-  # (+self+ and +other+).  This is more efficient than calculating them
-  # separately.
-  #
-  # Example:
-  #   6.gcdlcm 9     # -> [3, 18]
-  #
+
   def gcdlcm(other)
     gcd = self.gcd(other)
     if self.zero? or other.zero?
@@ -503,55 +45,5 @@ class Integer
       [gcd, (self.div(gcd) * other).abs]
     end
   end
-end
-
-class Fixnum
-  alias quof quo
-  remove_method :quo
-
-  # If Rational is defined, returns a Rational number instead of a Float.
-  def quo(other)
-    Rational.new!(self, 1) / other
-  end
-  alias rdiv quo
 
-  # Returns a Rational number if the result is in fact rational (i.e. +other+ < 0).
-  def rpower (other)
-    if other >= 0
-      self.power!(other)
-    else
-      Rational.new!(self, 1)**other
-    end
-  end
-end
-
-class Bignum
-  alias quof quo
-  remove_method :quo
-
-  # If Rational is defined, returns a Rational number instead of a Float.
-  def quo(other)
-    Rational.new!(self, 1) / other
-  end
-  alias rdiv quo
-
-  # Returns a Rational number if the result is in fact rational (i.e. +other+ < 0).
-  def rpower (other)
-    if other >= 0
-      self.power!(other)
-    else
-      Rational.new!(self, 1)**other
-    end
-  end
-end
-
-unless defined? 1.power!
-  class Fixnum
-    alias power! **
-    alias ** rpower
-  end
-  class Bignum
-    alias power! **
-    alias ** rpower
-  end
 end