[ruby-dev:42006] Re: [Bug #3676] CMath.cbrt(-8)の結果が複素数にならない

=1B$B$`$i$?$G$9!#=1B(B

On 2010/08/11, at 1:08, Yusuke ENDOH wrote:

> 2010=1B$BG/=1B(B8=1B$B7n=1B(B10=1B$BF|=1B(B23:22 Tadayoshi Funaba =
<[e-mail]>:
>>> =
=1B$B8=3D:_$N=1B(BCMath.cbrt(-8)=1B$B$N7k2L$O=1B(B-2=1B$B$G!"$3$l$O=1B(B3=1B=
$B>h$9$k$H=1B(B-8=1B$B$K$J$k$H$$=1B(B
>>> =
=1B$B$&0UL#$G@5$7$$CM$K;W$($k$N$G$9$,!#=1B(BCmath.cbrt(-8)=1B$B$,JV$9$Y$-=1B=
(B
>>> =1B$B!V@5$7$$!WCM$O$J$s$@$H$*9M$($G$9$+!)=1B(B
>>=20
>> =1B$B@dBPE*$K@5$7$$CM$O$J$$$H;W$$$^$9$,!"$*$*$h$=3D!"=1B(BComplex(-8) =
** (1.0/3) =1B$B$G=1B(B
>> =1B$B$"$k=1B(B (1.0+1.73205080756888i) =1B$B$G$O$J$$$G$7$g$&$+!#=1B(B
>=20
> =1B$B9b9;%l%Y%k$NJ#AG?t$NCN<1$7$+$J$$$N$G!"=1B(BComplex#**(Rational) =
=1B$B$,=1B(B
> n =1B$B>h:,$N$I$l$rJV$9$+$OITDj$@$H;W$C$F$$$^$7$?!#=1B(B
> =1B$B6D3Q$,@5$G:G>.$N$b$N$,!V$*$*$h$=3D@5$7$$!W$s$G$9$+$M!#=1B(B

=1B$B6D3Q$,@5$G:G>.$N$b$N$r<gCM=1B(B (principal value) =1B$B$H8@$$!"=1B(B
=1B$BB?2A4X?t$r0l2A4X?t$H$9$k$?$a$K;HMQ$5$l$^$9!#=1B(B
http://ja.wikipedia.org/wiki/%E4%B8%BB%E5%80%A4

n=1B$B>h:,$N<gCM$O=1B(B1=1B$B$N86;O=1B(Bn=1B$B>h:,$N<B?tG\$K$J$C$F$$$^$9!#=
=1B(B
http://ja.wikipedia.org/wiki/1%E3%81%AE%E5%86%AA%E6%A0%B9

=1B$B$4B8CN$NDL$j!"$3$N<gCM$5$($"$l$P$9$Y$F$N2r$r=3DgHV$K@8@.$G$-$^$9=1B(B=
:

  a =3D Complex(-8)**(1.0/3)  # =1B$B<gCM=1B(B
  w =3D a / a.abs  # 1=1B$B$N86;O=1B(B3=1B$B>h:,=1B(B
  a * w    #=3D> (-0.9999999999999996+1.7320508075688776i)
  a * w**2 #=3D> (-2.0+7.771561172376096e-16i)

=1B$B$7$+$7!"Ii?t$N4q?t>h:,$O$+$J$i$:<B2r$r;}$A$^$9$+$i!"=1B(B
=
=1B$B<gCM$G$O$J$/<B2r$r%@%$%l%/%H$K<j$KF~$l$kJ}K!$,$"$C$F$bNI$$$H;W$$$^$9!=
#=1B(B
=
=1B$B$J$<$J$i!">e$NJ}K!$G$OIbF0>.?tE@1i;;$K$h$k8m:9$r4^$s$G$7$^$&$3$H$H!"=1B=
(B
=1B$B%@%$%l%/%H$K5a$a$i$l$l$PI,MW$J$+$C$?7W;;$rI,MW$H$7$F$$$k$+$i$G$9!#=1B=
(B

(=1B$B0zMQ$N=3Dg=3Dx$rF~$lBX$($^$9=1B(B)

> 2010=1B$BG/=1B(B8=1B$B7n=1B(B11=1B$BF|=1B(B0:35 Tadayoshi Funaba =
<[e-mail]>:
>> =
=1B$B$^$D$b$H$5$s$,8@$C$F$$$k$N$O$=3D$&$$$&0UL#$8$c$J$$$H;W$$$^$9$,!"Dj5A0=
h$H=1B(B
>> =1B$BCM0h$,$"$C$F4X?t$,$"$k!"=1B(BMath =
=1B$B$O<B?t$KBP$7$FDj5A$,$"$k!#=1B(BCMath =1B$B$OJ#AG?t$K=1B(B
>> =
=1B$BBP$7$FDj5A$5$l$k!"$H$$$&$N$O>o<1E*$J;v$G!"5DO@$NM>CO$O$J$$$/$i$$$K;W$=
$=1B(B
>> =1B$B$^$9!#=1B(BCMath =
=1B$B$O@)8B$NE1GQ!"$"$k$$$ODj5A0hCM0h$N3HBg$@$H;W$$$^$9!#=1B(B
>=20
> =1B$B;d$O$^$D$b$H$5$s$HF1$8463P$G$7$?=1B(B (CMath =1B$B$O=1B(B Math =
=1B$B$N5sF0$r3HD%$9$k=1B(B
> =1B$B$@$1$GJQ$($O$7$J$$=1B(B) =1B$B!#=1B(B[ruby-core:31234] =
=1B$B$N?M$bF1$8463P$@$H;W$$$^$9!#=1B(B
> =1B$B$A$J$_$K8e$GEP>l$9$k$K0c$$$J$$$H;W$$$^$9$,!"$`$i$1$s$5$s$O$U$J$P=1B=
(B
> =1B$B$5$s$HF1$8463P$N$h$&$G$9!#=1B(B
> =1B$B$3$N5!2q$K%3%s%;%s%5%9=1B(B (=1B$B$d>-MhE*$JJ}?K=1B(B) =
=1B$B$r7h$a$H$/$H$h$5$=3D$&$G$9$M!#=1B(B


=1B$B;d$O!"$I$A$i$+$H$$$($P$U$J$P$5$s$HF1$8N)>l$G$9!#=1B(B
=
=1B$B$7$+$7>e$G=3Dq$$$?$h$&$K<B2r$r%@%$%l%/%H$KF@$kJ}K!$,$"$C$F$bNI$$$H;W$=
$$^$9!#=1B(B

=1B$B$?$H$($P=1B(B CMath.cbrt =1B$B$K%*%W%7%g%s0z?t$rDI2C$7$F=1B(B

  CMath.cbrt(-8) #=3D> =1B$B<gCM=1B(B
  CMath.cbrt(-8, :real_only =3D> true) #=3D> -2

=1B$B$N$h$&$K$9$k$H$+!#=1B(B

=1B$B$b$7$/$O5U$K=1B(B

  CMath.cbrt(-8)  #=3D> -2
  CMath.cbrt(-8, :principal_value =3D> true)  #=3D> =1B$B<gCM=1B(B

=1B$B$G$bNI$$$+$b$7$l$^$;$s$,!"<B2r$rF@$?$$$J$i8=3D:_$O=1B(B Math.cbrt =
=1B$B$r;H$($k$N$G=1B(B
CMath.cbrt =1B$B$N%G%U%)%k%H$NJV$jCM$O<gCM$N$[$&$,NI$$$H;W$$$^$9!#=1B(B

=1B$B$3$N$h$&$J!"%*%W%7%g%s0z?t$K$h$k5sF0$NJQ99$,5v$5$l$k$N$G$"$l$P!"=1B(B=

=1B$B$$$C$=3D=1B(B CMath =1B$B$H=1B(B Math =
=1B$B$rE}9g$7$F$7$^$&$N$b0l$D$NJ}K!$@$H;W$$$^$9!#=1B(B


> =1B$B>/$J$/$H$b=1B(B Complex#** =
=1B$B$N;EMM$H$7$F7h$^$C$F$$$k$J$i!"$G$-$l$P=1B(B
> rdoc =1B$B$K=3Dq$$$F$*$$$FM_$7$$$G$9!#=1B(B

=1B$B;d$O=1B(B ** =1B$B$N7k2L$O<gCM$K$J$C$F$$$k$[$&$,NI$$$H9M$($^$9!#=1B(B=

=1B$BM}M3$O$$$/$D$+$"$C$F!"=1B(B

(1) 1=1B$B$N86;O=1B(B3=1B$B>h:,$N<B?tG\$H$$$&?t3XE*$KFCJL$G$"$k=1B(B
(2) =1B$B%l%7!<%P$,=1B(B Complex =
=1B$B$J$s$@$+$i!"0lHL$K<B2r$O4|BT$G$-$J$$=1B(B
(3) Mathematica =1B$B$,$=3D$&$J$C$F$$$k=1B(B

=1B$B$J$I$G$9!#$J$*=1B(B Mathematica =1B$B$N5sF0$O=1B(B Wolfram|Alpha =
=1B$B$G8!>Z$G$-$^$9!#=1B(B
http://www.wolframalpha.com/input/?i=3D(-27)^(1/3)


--
Kenta Murata
OpenPGP FP =3D 1D69 ADDE 081C 9CC2 2E54  98C1 CEFE 8AFB 6081 B062

=1B$BK\$r=3Dq$-$^$7$?=1B(B!!
=1B$B!X=1B(BRuby =1B$B5U0z$-%l%7%T!Y=1B(B =
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