Rational, complex and mathn

rational e$B$Oe(B floore$B!"e(Btruncatee$B!"e(Bceile$B!"e(Bround
e$B$rDj5A$7$F$$$^$;$s!#e(BNumeric
e$B$N$^$^Ds6!$7$F$$$^$9!#$7$+$7!"e(BNumeric
e$B$N$b$N$O!"IbF0>[email protected]?t$KJQ49$7$Fe(B
e$B$7$^$&$?$a$^$H$b$J7k2L$,[email protected]$i$l$k$H$O8B$j$^$;$s!#e(B[ruby-dev:32201]

e$B8=>u!"e(Bfloor e$B$N$+$o$j$Ke(B to_i
e$B$r$D$+$C$FLdBj$,$J$$$N$O!"e(Bto_i e$B$NDj5A$,e(B
floor e$B$K$J$C$F$$$k$+$i$G$9!#e(BFloat#to_i e$B$NDj5A$+$i!“e(BNumeric
e$B$Ne(B to_i e$B$Oe(B
truncate e$B$G$”[email protected]$H;W$$$^$9!#e(B

Complex(1,2).numerator e$B$,%(%i!<$K$J$j$^$9!#e(Brational
e$B$rFI$s$G$$$k$H%(%i!<e(B
e$B$K$J$j$^$;$s!#e(B

$ ruby19 -r complex -e ‘Complex(1,2).numerator’
/usr/local/ruby19/lib/ruby/1.9.0/complex.rb:345:in denominator': undefined methoddenominator’ for 1:Fixnum (NoMethodError)
from /usr/local/ruby19/lib/ruby/1.9.0/complex.rb:352:in
numerator' from -e:1:in

Complex#quo e$B$,5!G=$7$^$;$s!#[email protected]?t3d$j$7$F$7$^$$$^$9!#e(B

$ ruby19 -r complex -e ‘p Complex(1,2).quo(2)’
Complex(0, 1)

$ ruby19 -r complex -r rational -e ‘p Complex(1,2).quo(2)’
Complex(0, 1)

complex
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between?e$B!"e(Bfloore$B!"e(Bceile$B!"e(Brounde$B!"e(Btruncate
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e$B$”$H!“e(B% e$B$O0UL#$,$”$k$N$G$7$g$&$+!#e(B

mathn e$B$G!“e(BRational#inspect
e$B$r=q$-49$($F$$$k$,!”$5$9$,$KM>7W$J$*@$OC$Ne(B
e$B$h$&$J5$$b$7$^$9!#$I$&$7$F$b$7$?$1$l$P!"$=$&$$$&$3$H$O!“e(Birb
e$B$J$I%”%W%je(B
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e$B0J2<[email protected]!#e(Bfloore$B!"e(Bceile$B!"e(Btruncatee$B!"e(Bround
e$B$O86$5$s$Ne(B rational e$B$+$i%Q%/e(B
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Index: lib/rational.rb

— lib/rational.rb (revision 15409)
+++ lib/rational.rb (working copy)
@@ -238,6 +238,10 @@
end
end

  • def div(other)
  • (self / other).floor
  • end
  • Returns the remainder when this value is divided by +other+.

@@ -249,7 +253,7 @@

r % 0.26 # -> 0.19

def % (other)

  • value = (self / other).to_i
  • value = (self / other).floor
    return self - other * value
    end

@@ -261,7 +265,7 @@

r.divmod Rational(1,2) # -> [3, Rational(1,4)]

def divmod(other)

  • value = (self / other).to_i
  • value = (self / other).floor
    return value, self - other * value
    end

@@ -270,7 +274,7 @@

def abs
if @numerator > 0

  •  Rational.new!(@numerator, @denominator)
    
  •  self
    
    else
    Rational.new!(-@numerator, @denominator)
    end
    @@ -345,10 +349,37 @@

    Rational(-7,4) == -1.75 # -> true

    Rational(-7,4).to_i == (-1.75).to_i # false

  • def to_i
  • Integer(@numerator.div(@denominator))
  • def floor()

  • @numerator.div(@denominator)
    end

  • def ceil()

  • -((-@numerator).div(@denominator))

  • end

  • def truncate()

  • if @numerator < 0

  •  return -(([email protected]).div(@denominator))
    
  • end

  • @numerator.div(@denominator)

  • end

  • alias_method :to_i, :truncate

  • def round()

  • if @numerator < 0

  •  num = [email protected]
    
  •  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.

@@ -476,10 +507,11 @@

class Fixnum
alias quof quo

  • undef quo
  • If Rational is defined, returns a Rational number instead of a

Fixnum.

  • remove_method :quo
  • If Rational is defined, returns a Rational number instead of a

Float.
def quo(other)

  • Rational.new!(self,1) / other
  • Rational.new!(self, 1) / other
    end
    alias rdiv quo

@@ -488,26 +520,18 @@
if other >= 0
self.power!(other)
else

  •  Rational.new!(self,1)**other
    
  •  Rational.new!(self, 1)**other
    
    end
    end
  • unless defined? 1.power!
  • alias power! **
  • alias ** rpower
  • end
    end

class Bignum

  • unless defined? Complex
  • alias power! **
  • end
  • alias quof quo
  • remove_method :quo
  • alias quof quo
  • undef quo
  • If Rational is defined, returns a Rational number instead of a

Bignum.

  • If Rational is defined, returns a Rational number instead of a

Float.
def quo(other)

  • Rational.new!(self,1) / other
  • Rational.new!(self, 1) / other
    end
    alias rdiv quo

@@ -519,8 +543,15 @@
Rational.new!(self, 1)**other
end
end
+end

  • unless defined? Complex
    +unless defined? 1.power!
  • class Fixnum
  • alias power! **
    alias ** rpower
    end
  • class Bignum
  • alias power! **
  • alias ** rpower
  • end
    end
    Index: lib/mathn.rb
    ===================================================================
    — lib/mathn.rb (revision 15409)
    +++ lib/mathn.rb (working copy)
    @@ -121,11 +121,6 @@
    class Rational
    Unify = true
  • remove_method :inspect

  • def inspect

  • format “%s/%s”, numerator.inspect, denominator.inspect

  • end

  • alias power! **

    def ** (other)
    Index: lib/complex.rb
    ===================================================================
    — lib/complex.rb (revision 15409)
    +++ lib/complex.rb (working copy)
    @@ -104,6 +104,10 @@
    @RCS_ID=’-$Id: complex.rb,v 1.3 1998/07/08 10:05:28 keiju Exp keiju
    $-’

    undef step

  • undef <, <=, <=>, >, >=

  • undef between?

  • undef divmod, modulo

  • undef floor, truncate, ceil, round

    def scalar?
    false
    @@ -199,6 +203,10 @@
    x/y
    end
    end

  • def quo(other)

  • Complex(@real.quo(1), @image.quo(1)) / other

  • end

    Raise this complex number to the given (real or complex) power.

@@ -248,6 +256,8 @@

Remainder after division by a real or complex number.

+=begin
def % (other)
if other.kind_of?(Complex)
Complex(@real % other.real, @image % other.image)
@@ -258,7 +268,8 @@
x % y
end
end

+=end
+
#–

def divmod(other)

if other.kind_of?(Complex)

@@ -312,8 +323,6 @@
end
alias conj conjugate

  • undef <=>
  • Test for numerical equality (a == a + 0i).

@@ -410,9 +419,35 @@

end

+class Integer

  • unless defined?(1.numerator) # temporal

  • 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

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e$B$k$H$$$($k$N$+$b$7$l$^$;$s!#e(Bcomplex e$B$bF1MM$G$9$,!“e(Brational
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Complex(1,2) ** -2 #=> Complex(Rational(-3, 25), Rational(-4, 25))

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Complex(1,2) ** Rational(-2) #=> Complex(-0.12, -0.16)

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$ ruby19 -r rational -e ‘1 ** Rational(1)’
/usr/local/ruby19/lib/ruby/1.9.0/rational.rb:489:in power!': super: no superclass methodpower!’ for Rational(1, 1):Rational (NoMethodError)
from /usr/local/ruby19/lib/ruby/1.9.0/rational.rb:489:in
rpower' from -e:1:in

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require ‘rational’
require ‘complex’
require ‘bigdecimal’

class Numeric

def numerator() to_r.numerator end
def denominator() to_r.denominator end

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

end

class Float

def decode
f, n = Math.frexp(self)
f = Math.ldexp(f, Float::MANT_DIG).to_i
n -= Float::MANT_DIG
return f, n
end

private :decode

def to_r
f, n = decode
f * Float::RADIX ** n
end

end

class BigDecimal

def to_r
s, f, b, e = split
Rational(s * f.to_i * b ** (e - f.size))
end

end

class Complex

undef numerator
undef denominator
undef floor
undef ceil
undef truncate
undef round

end

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$ ruby19 -r complex -e ‘Complex(1).floor’
-e:1:in floor': can't convert Complex into Float (TypeError) from -e:1:in

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$ ruby19 -r complex -e ‘Complex(1).div(1)’
-e:1:in div': can't convert Complex into Float (TypeError) from -e:1:in

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From: Tadayoshi F. [email protected]
Subject: [ruby-dev:33668] Re: rational, complex and mathn
Date: Sat, 9 Feb 2008 15:41:54 +0900
Message-ID: [email protected]

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In message “Re: [ruby-dev:33664] Re: rational, complex and mathn”
on Sat, 9 Feb 2008 08:43:16 +0900, Tadayoshi F. [email protected]
writes:

|rational e$B$OCY$$$G$9!#86$5$s$Ne(B rational e$B$r$D$+$&$HB.$/$J$j$^$9!#e(B

e$B86$5$s$Ne(Brationale$B$OF3F~M=Dj$,$"$j$^$9$N$G!"$3$N5!2q$K$b$&0lEYe(B
e$BH`$r$D$D$$$F$_$F$/[email protected]$5$$!#e(B

e$B$^$D$b$He(B e$B$f$-$R$m$G$9e(B

In message “Re: [ruby-dev:33662] rational, complex and mathn”
on Sat, 9 Feb 2008 08:06:02 +0900, Tadayoshi F. [email protected]
writes:
|
|rational e$B$Oe(B floore$B!"e(Btruncatee$B!"e(Bceile$B!"e(Bround e$B$rDj5A$7$F$$$^$;$s!#e(BNumeric
|e$B$N$^$^Ds6!$7$F$$$^$9!#$7$+$7!"e(BNumeric e$B$N$b$N$O!"IbF0>[email protected]?t$KJQ49$7$Fe(B
|e$B$7$^$&$?$a$^$H$b$J7k2L$,[email protected]$i$l$k$H$O8B$j$^$;$s!#e(B[ruby-dev:32201]
|
|e$B8=>u!"e(Bfloor e$B$N$+$o$j$Ke(B to_i e$B$r$D$+$C$FLdBj$,$J$$$N$O!"e(Bto_i e$B$NDj5A$,e(B
|floor e$B$K$J$C$F$$$k$+$i$G$9!#e(BFloat#to_i e$B$NDj5A$+$i!“e(BNumeric e$B$Ne(B to_i e$B$Oe(B
|truncate e$B$G$”[email protected]$H;W$$$^$9!#e(B
|
|Complex(1,2).numerator e$B$,%(%i!<$K$J$j$^$9!#e(Brational e$B$rFI$s$G$$$k$H%(%i!<e(B
|e$B$K$J$j$^$;$s!#e(B
|
|$ ruby19 -r complex -e ‘Complex(1,2).numerator’
|/usr/local/ruby19/lib/ruby/1.9.0/complex.rb:345:in denominator': undefined methoddenominator’ for 1:Fixnum (NoMethodError)
| from /usr/local/ruby19/lib/ruby/1.9.0/complex.rb:352:in numerator' | from -e:1:in
|
|Complex#quo e$B$,5!G=$7$^$;$s!#[email protected]?t3d$j$7$F$7$^$$$^$9!#e(B
|
|$ ruby19 -r complex -e ‘p Complex(1,2).quo(2)’
|Complex(0, 1)
|
|$ ruby19 -r complex -r rational -e ‘p Complex(1,2).quo(2)’
|Complex(0, 1)
|
|complex e$B$G$O!“ITMW$K;W$($k%a%=%C%I$,J|CV$7$F$”$j$^$9!#e(B>e$B!"e(B>=e$B!"e(B<e$B!"e(B<=e$B!"e(B
|between?e$B!"e(Bfloore$B!"e(Bceile$B!"e(Brounde$B!"e(Btruncate e$B$J$I!#e(B1.8 e$B$G$O!“e(Bstepe$B!“e(B<=> e$B$b!#e(B
|e$B$”$H!“e(B% e$B$O0UL#$,$”$k$N$G$7$g$&$+!#e(B
|
|mathn e$B$G!“e(BRational#inspect e$B$r=q$-49$($F$$$k$,!”$5$9$,$KM>7W$J$*@$OC$Ne(B
|e$B$h$&$J5$$b$7$^$9!#$I$&$7$F$b$7$?$1$l$P!”$=$&$$$&$3$H$O!“e(Birb e$B$J$I%”%W%je(B
|e$B%1!<%7%g%sB&$G$d$l$P$$$$$h$&$J5$$,$7$^$9!#e(B
|
|e$B0J2<[email protected]!#e(Bfloore$B!"e(Bceile$B!"e(Btruncatee$B!"e(Bround e$B$O86$5$s$Ne(B rational e$B$+$i%Q%/e(B
|e$B$j$^$7$?!#e(B

e$B$I$&$b%a%s%F%J!<$NH?1~$,0-$$$N$G!“e(Btrunke$B$K%3%_%C%H$7$F$/[email protected]$5$$!#e(B
e$BITK~$,$”[email protected]$5$s<+?H$Ke(Breverte$B$7$F$b$i$$$^$7$g$&!#e(B

e$B86$5$s$Ne(Brationale$B$OF3F~M=Dj$,$"$j$^$9$N$G!"$3$N5!2q$K$b$&0lEYe(B
e$BH`$r$D$D$$$F$_$F$/[email protected]$5$$!#e(B

e$B$H$$$&;v$J$N$G!"6qBNE*$JOC$r$7$^$7$g$&e(B > e$B86$5$se(B

e$B$1$$$8$e!w$$$7$D$+$G$9e(B.

In [ruby-dev :33706 ] the message: "[ruby-dev:33706] Re: rational,
complex and mathn ", on Feb/12 11:04(JST) Yukihiro M. writes:

e$B$^$D$b$He(B e$B$f$-$R$m$G$9e(B

e$B$I$&$b%a%s%F%J!<$NH?1~$,0-$$$N$G!“e(Btrunke$B$K%3%_%C%H$7$F$/[email protected]$5$$!#e(B
e$BITK~$,$”[email protected]$5$s<+?H$Ke(Breverte$B$7$F$b$i$$$^$7$g$&!#e(B

e$B?=$7Lu$J$$e(B.

e$B%3%_%C%H$7$F$/[email protected]$5$C$F$/[email protected]$5$C$F7k9=$G$9e(B.

[email protected](B,

|mathn e$B$G!“e(BRational#inspect e$B$r=q$-49$($F$$$k$,!”$5$9$,$KM>7W$J$*@$OC$Ne(B
|e$B$h$&$J5$$b$7$^$9!#$I$&$7$F$b$7$?$1$l$P!"$=$&$$$&$3$H$O!“e(Birb e$B$J$I%”%W%je(B
|e$B%1!<%7%g%sB&$G$d$l$P$$$$$h$&$J5$$,$7$^$9!#e(B

e$B$N$H$3$m$O;D$7$F$*[email protected]$1$k$H$"$j$,$?$$$J$!e(B.

__
---------------------------------------------------->> [email protected](B
e$B7=<ye(B <<—
---------------------------------->> e-mail: [email protected] <<—

[email protected]$5$s$,=P$F$3$i$l$F$h$+$C$?!#e(B

|mathn e$B$G!“e(BRational#inspect e$B$r=q$-49$($F$$$k$,!”$5$9$,$KM>7W$J$*@$OC$Ne(B
|e$B$h$&$J5$$b$7$^$9!#$I$&$7$F$b$7$?$1$l$P!"$=$&$$$&$3$H$O!“e(Birb e$B$J$I%”%W%je(B
|e$B%1!<%7%g%sB&$G$d$l$P$$$$$h$&$J5$$,$7$^$9!#e(B

e$B$N$H$3$m$O;D$7$F$*[email protected]$1$k$H$"$j$,$?$$$J$!e(B.

e$B$=$3$r5$$K$5$l$k$H$OM=A[$7$F$$$^$;$s$G$7$?!#$G$-$l$PM}M3$rCN$j$?$$$N$Ge(B
e$B$9$,!#e(B

e$B$1$$$8$e!w$$$7$D$+$G$9e(B.

In [ruby-dev :33717 ] the message: "[ruby-dev:33717] Re: rational,
complex and mathn ", on Feb/12 21:22(JST) Tadayoshi F. writes:

[email protected]$5$s$,=P$F$3$i$l$F$h$+$C$?!#e(B

e$B?=$7Lu$J$$e(B. e$B:G6a%P%?%P%?$7$F$$$?$b$N$Ge(B…
e$B$^[email protected];[email protected]|$K$OJV;v$G$-$k$h$&EXNO$7$^$9e(B.

|mathn e$B$G!“e(BRational#inspect e$B$r=q$-49$($F$$$k$,!”$5$9$,$KM>7W$J$@$OC$Ne(B
|e$B$h$&$J5$$b$7$^$9!#$I$&$7$F$b$7$?$1$l$P!"$=$&$$$&$3$H$O!“e(Birb e$B$J$I%”%W%je(B
|e$B%1!<%7%g%sB&$G$d$l$P$$$$$h$&$J5$$,$7$^$9!#e(B
e$B$N$H$3$m$O;D$7$F$
[email protected]$1$k$H$"$j$,$?$$$J$!e(B.

e$B$=$3$r5$$K$5$l$k$H$OM=A[$7$F$$$^$;$s$G$7$?!#$G$-$l$PM}M3$rCN$j$?$$$N$Ge(B
e$B$9$,!#e(B

e$B$He(B. e$B;W$C$?$N$G$9$,e(B,
e$B$3$3$O;d$N%*%j%8%J%k$G$O$J$$$G$9$Me(B.
rivision 1363 e$B$G>>K$5$s$,DI2C$5$l$F$$$^$9$Me(B.
e$B>>K$5$s$NDI2C$J$N$Ge(B, [email protected](B
e$B$l$+$+$i$N%j%/%(%9%[email protected]$H;W$$$^$9$,e(B…

e$B;d$NJ}?K$O$^$5$K2<5-$NDL$j$Ge(B:

|mathn e$B$G!“e(BRational#inspect e$B$r=q$-49$($F$$$k$,!”$5$9$,$KM>7W$J$*@$OC$Ne(B
|e$B$h$&$J5$$b$7$^$9!#$I$&$7$F$b$7$?$1$l$P!"$=$&$$$&$3$H$O!“e(Birb e$B$J$I%”%W%je(B
|e$B%1!<%7%g%sB&$G$d$l$P$$$$$h$&$J5$$,$7$^$9!#e(B

irb -m e$B$GBP1~$9$k$h$&$K$7$F$$$^$9e(B.

e$B$H$$$&$3$H$Ge(B, e$B:o$C$F$+$^$$$^$;$se(B. e$B$H$$$&$+e(B,
e$B$9$G$K:o$C$F$"$j$^$9$Me(B.

__
---------------------------------------------------->> [email protected](B
e$B7=<ye(B <<—
---------------------------------->> e-mail: [email protected] <<—

e$B$=$3$r5$$K$5$l$k$H$OM=A[$7$F$$$^$;$s$G$7$?!#$G$-$l$PM}M3$rCN$j$?$$$N$Ge(B
e$B$9$,!#e(B

e$B$He(B. e$B;W$C$?$N$G$9$,e(B, e$B$3$3$O;d$N%*%j%8%J%k$G$O$J$$$G$9$Me(B.
(e$BCfN,e(B)
e$B$H$$$&$3$H$Ge(B, e$B:o$C$F$+$^$$$^$;$se(B. e$B$H$$$&$+e(B, e$B$9$G$K:o$C$F$"$j$^$9$Me(B.

e$B$J$k$[$I!#$"$j$,$H$&$4$6$$$^$9!#e(B

e$B86$G$9!#e(B

Tadayoshi F. e$B$5$s$O=q$-$^$7$?e(B:

e$B86$5$s$Ne(Brationale$B$OF3F~M=Dj$,$"$j$^$9$N$G!"$3$N5!2q$K$b$&0lEYe(B
e$BH`$r$D$D$$$F$_$F$/[email protected]$5$$!#e(B

e$B$H$$$&;v$J$N$G!"6qBNE*$JOC$r$7$^$7$g$&e(B > e$B86$5$se(B

e$B$7$^$7$g$&!#e(B

e$BA0$K$U$J$P$5$s$H8D?ME*$J%a!<%k$N$d$j$H$j$G!“7k6I$^$?;d$,e(Brationale$B$r$^$He(B
e$B$a$k$3$H$K$J$C$?$N$G$9$,!”$N$S$N$S$K$J$C$F$7$^$C$?$3$H$H!":#2s$U$J$P$5e(B
e$B$s$,e(Bnu*e$B%7%j!<%:$r$^$H$a$i$l$?$3$H$,$"$C$FBg$-$/>u67$OJQ$o$C$?$H;W$$$^$9!#e(B
e$B$G!";d$K$O8=;[email protected]$G<!$NFsDL$j$,9M$($i$l$^$9!#e(B

e$B$U$J$P$5$s$Ne(Bnurate$B$re(BRationale$B$H$7$F$$$:$le(Bruby-1.9e$B$KF~$l$F$b$i$&!#e(B
e$B%a%s%F%J$O$U$J$P$5$s$H$7$^$9!#$b$A$m$s8D!9$N4X?t$N%A%e!<%K%s%0e(B
e$B$J$I$N$*<jEA$$$O$5$;[email protected]$-$^$9!#e(B

e$B$3$N$^$^;d$Ne(Brational.ce$B$r2~NI$7$Fe(Bruby-1.9e$B$KF~$l$F$b$i$&!#e(B
e$B%a%s%F%J$O;d$H$7$^$9!#[email protected]$7$3$N>l9g!"$U$J$P$5$s$Ne(Bnurat.c
e$B$r85$K$7$F!"e(Brational.ce$B$r=q$-D>$9$D$b$j$G$9!#e(B

e$B$I$&$7$^$7$g$&$+!#$[$s$H$K$I$C$A$G$b$$$$$N$G$9$,!"e(B2.
[email protected]$H$^$?;~4V$,$+$+e(B
e$B$j$=$&$J5$$,!De(B

e$B86$G$9!#e(B

Tadayoshi F. wrote:

e$BK<AE*$JItJ,$K3d9~$`$h$&$J<jF0:GE,2=$O$7$J$$$D$b$j$G$$$^$9!#$?$V$s!"$3e(B
e$B$NJ}?K$G$d$C$F$$$k$H!"$b$&$"$^$jB.$/$O$J$i$J$$$H;W$$$^$9!#$=$l$G$b!"e(B
lib/rational.rb e$B$KHf$Y!"e(B1.9e$B!"e(B1.8 e$B$H$b$KB.$/$J$C$F$$$^$9!#e(B

e$B$=$l$J$j$KB.$/$O$J$k$7!"C1=c$J$H$3$m$G$=$l$J$j$KK~B-$7$F$$$^$9$,!“0lEY!“e(B
e$B86$5$s$K8=>u$N%3!<%I$r8+$F$b$i$C$F!”$=$l$GK\Ev$K$D$+$$$b$N$K$J$k$+H=CGe(B
e$B$7$F$b$i$$$?$$$N$G$9$,!”$I$&$G$7$g$&e(B > e$B86$5$se(B

e$B$U$J$P$5$s$N%3!<%I$K$O4{$KEDCf$5$s$Ne(B1.9e$BBP1~$b40N;$7$F$$$F!"$d$O$j>h$C$+e(B
e$B$j$?$$$G$9!#e(B

e$B=q$-D>$9$J$i!"$D$+$($k$H$3$m$O$D$+$C$F$b$i$C$F$$$$$G$9$7!"$^$?!"$$$E$le(B
e$B$K$7$F$b!"<jEA$($k$H$3$m$O<jEA$$$^$9!#e(B

e$B$"$j$,$H$&$4$6$$$^$9!#e(B

e$B$=$l$G$O!"$U$J$P$5$s$Ne(B nurat_core.c
e$B$r>/$7$:$D=q$-49$($k7A$G!“4{$K$”$ke(B
e$B;d$Ne(B rational.c
e$B$NFbMF$r>h$;$F$$$3$&$H;W$$$^$9!#[email protected]$7!"=q$-49$($?7k2L!“e(B
e$B$”$kDxEY$O$C$-$j$7$?2~A1$,$J$1$l$P!"$=$NItJ,$N=q$-49$($O85$KLa$7$^$9!#e(B

e$B=q$-49$($N7P2a$O$*CN$i$;$9$k$N$G!"$*IU$-9g$$$/[email protected]$5$$!#e(B

e$B%a%s%F%J$O;d$H$7$^$9!#[email protected]$7$3$N>l9g!"$U$J$P$5$s$Ne(Bnurat.c
e$B$r85$K$7$F!"e(Brational.ce$B$r=q$-D>$9$D$b$j$G$9!#e(B

e$B$I$&$7$^$7$g$&$+!#$[$s$H$K$I$C$A$G$b$$$$$N$G$9$,!"e(B2. [email protected]$H$^$?;~4V$,$+$+e(B
e$B$j$=$&$J5$$,!De(B

nurat e$B$N$[$&$G$9$,!"%F%9%H$r6/2=$7$D$D!">/$7<jF0:GE,2=e(B
(e$B$H$$$&Dx$N$b$Ne(B
e$B$G$b$J$$$,e(B)
e$B$7;O$a$^$7$?!#<j4V$N3d$K$O8z2L$,$"$k$h$&$J5$$,$7$^$9!#e(B

date
e$B$H0l=o$K$D$+$C$?$H$3$m$G$O!“86$5$s$N$H$”$^$jB=?’$J$$$h$&$J46$8$Je(B
e$B$N$G$9$,!"%Y%s%A%^!<%/E*$J;[email protected]$H!“86$5$s$[$&$,%O%C%-%j$HB.$$!”$H$$$&e(B
e$B$3$H$,H=$j$^$9!#e(B

e$BK<AE*$JItJ,$K3d9~$`$h$&$J<jF0:GE,2=$O$7$J$$$D$b$j$G$$$^$9!#$?$V$s!"$3e(B
e$B$NJ}?K$G$d$C$F$$$k$H!"$b$&$"$^$jB.$/$O$J$i$J$$$H;W$$$^$9!#$=$l$G$b!"e(B
lib/rational.rb e$B$KHf$Y!"e(B1.9e$B!"e(B1.8
e$B$H$b$KB.$/$J$C$F$$$^$9!#e(B

e$B$=$l$J$j$KB.$/$O$J$k$7!"C1=c$J$H$3$m$G$=$l$J$j$KK~B-$7$F$$$^$9$,!“0lEY!“e(B
e$B86$5$s$K8=>u$N%3!<%I$r8+$F$b$i$C$F!”$=$l$GK\Ev$K$D$+$$$b$N$K$J$k$+H=CGe(B
e$B$7$F$b$i$$$?$$$N$G$9$,!”$I$&$G$7$g$&e(B > e$B86$5$se(B

e$B=q$-D>$9$J$i!"$D$+$($k$H$3$m$O$D$+$C$F$b$i$C$F$$$$$G$9$7!"$^$?!"$$$E$le(B
e$B$K$7$F$b!"<jEA$($k$H$3$m$O<jEA$$$^$9!#e(B

1.8 e$B$K$be(B r15446 e$B$NJQ99$r$7$^$9!#e(B1.8
e$B$G$O$*$=$i$/[email protected]$N$?$a$+!“e(B
Complex#<=> e$B$J$I$O$=$N$^$^J|CV$5$l$F$$$^$9!#$=$l$K4XO”$9$kItJ,$He(B
mathn
e$B$Ne(B inspect e$B$r=|$$$FJQ99$7$^$9!#e(B

e$B$=$l$J$j$KB.$/$O$J$k$7!"C1=c$J$H$3$m$G$=$l$J$j$KK~B-$7$F$$$^$9$,!“0lEY!“e(B
e$B86$5$s$K8=>u$N%3!<%I$r8+$F$b$i$C$F!”$=$l$GK\Ev$K$D$+$$$b$N$K$J$k$+H=CGe(B
e$B$7$F$b$i$$$?$$$N$G$9$,!”$I$&$G$7$g$&e(B > e$B86$5$se(B

e$B$U$J$P$5$s$N%3!<%I$K$O4{$KEDCf$5$s$Ne(B1.9e$BBP1~$b40N;$7$F$$$F!"$d$O$j>h$C$+e(B
e$B$j$?$$$G$9!#e(B

e$B$=$&$G$9$+!#$^$"!"[email protected]$1Lr$KN)$D$+$o$+$j$^$;$s$,!#e(B

e$B$=$l$G$O!"$U$J$P$5$s$Ne(B nurat_core.c e$B$r>/$7$:$D=q$-49$($k7A$G!“4{$K$”$ke(B
e$B;d$Ne(B rational.c e$B$NFbMF$r>h$;$F$$$3$&$H;W$$$^$9!#[email protected]$7!"=q$-49$($?7k2L!“e(B
e$B$”$kDxEY$O$C$-$j$7$?2~A1$,$J$1$l$P!"$=$NItJ,$N=q$-49$($O85$KLa$7$^$9!#e(B

e$B=q$-49$($N7P2a$O$*CN$i$;$9$k$N$G!"$*IU$-9g$$$/[email protected]$5$$!#e(B

[email protected]|$"[email protected]$j$D$1$F!"$"$?$i$7$$$N$r=P$9$D$b$j$G$9!#e(B

e$BHf3SEy$N$?$a$K!“86$5$s$Ne(B rational
e$B$bIiC4$K$J$i$J$$DxEY$K%a%s%F$7$?$[$&e(B
e$B$,$$$$$H;W$$$^$9$N$G!”:#8=:_!"$"$-$i$+$K%P%0$C$]$$$H$3$m$r$CN$i$;$7$Fe(B
e$B$
$-$^$9!#e(B

Rational(-1,3) ** -1 #=> Rational(3, -1)
Rational(-1,3) ** -3 #=> Rational(27, -1)
Rational(1).integer? #=> true

e$BHyL/$J$b$Ne(B:

Rational.new!(1, -2) #=> Rational(1, -2) # e$B%*%j%8%J%k$Oe(B
Rational(-1,2) e$B$K$J$k!#e(B

e$B$?$V$s!"e(BRational(-1,3) e$B$,e(B Rational(3,-1)
e$B$K$J$k860x$+$J$H;W$&$N$G$9$,!#e(B

e$B$1$$$8$e!w$$$7$D$+$G$9e(B.

In [ruby-dev :33797 ] the message: "[ruby-dev:33797] Re: rational,
complex and mathn ", on Feb/15 22:46(JST) Tadayoshi F. writes:

e$BAGD>$KD>$=$&$H$9$k$H!"7k6Ie(B Float#to_r e$B$rDj5A$9$k$3$H$K$J$k$s$8$c$J$$$+e(B
e$B$H;W$$$^$9!#$7$+$7!"J#AG?t$Ne(B numeratore$B!"e(Bdenominator e$B$r<h$k$H$$$&$N$,$he(B
e$B$/$o$+$i$J$/$F!"e(BCommon Lisp e$B$r;O$a!“IaDL$OJ#AG?t$KBP$7$F$O5!G=$7$J$$$Ge(B
e$B$9$7!”@PDM$5$s$K!"0U?^$rJ9$$$?$[$&$,$$$$$G$9$M!#e(B

e$B$?$7$+$Ke(B, numeratore$B!"e(Bdenominator
e$BC1[email protected]$H0UL#$,$J$$$+$be(B…

e$BJ#AG?te(B: z = (a/b)+(c/d)i
e$B$re(B z.numerator/z.denominator

e$B$H$$$&J,[email protected]?t$G$=$m$($?7A<0$KJQ49$7$?$$$H$-$K;H$&[email protected]$H;W$$$^e(B
e$B$9e(B.
e$BCf3X$H$+9b9;$N;~$K$O$=$&$d$C$FJ,Jl$r$=$m$($F$$$^$7$?$h$Me(B(^^;
e$B7W;;e(B
e$BCf$K$O;H$&$3$H$O$J$$$H;W$$$^$9$,e(B,e$BI=<($9$k$H$-$K$O;H$&$+$be(B.

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e$B$-$^$7$?e(B. Complex#normal_rational_form e$B$H$+e(B.

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