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#
# rational.rb -
# $Release Version: 0.5 $
# $Revision: 1.7 $
# $Date: 1999/08/24 12:49:28 $
# 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 # -> -1
# (-1.75).to_i # -> -1
#
# In other words:
# Rational(-7,4) == -1.75 # -> true
# Rational(-7,4).to_i == (-1.75).to_i # -> true
#
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
#
# Converts the rational to a Float.
#
def to_f
@numerator.fdiv(@denominator)
end
#
# 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
#
# Returns +self+.
#
def to_r
self
end
#
# Returns a reconstructable string representation:
#
# Rational(5,8).inspect # -> "Rational(5, 8)"
#
def inspect
sprintf("Rational(%s, %s)", @numerator.inspect, @denominator.inspect)
end
#
# Returns a hash code for the object.
#
def hash
@numerator.hash ^ @denominator.hash
end
attr :numerator
attr :denominator
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
while min > 0
tmp = min
min = max % min
max = tmp
end
max
end
#
# Returns the <em>lowest common multiple</em> (LCM) of the two arguments
# (+self+ and +other+).
#
# Examples:
# 6.lcm 7 # -> 42
# 6.lcm 9 # -> 18
#
def lcm(other)
if self.zero? or other.zero?
0
else
(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?
[gcd, 0]
else
[gcd, (self.div(gcd) * other).abs]
end
end
end
class Fixnum
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
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