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

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

>>> =
=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)

>> =
=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 =
http://www.amazon.co.jp/dp/4798119881/mrkn-22

E-mail: mrkn / mrkn.jp