Re: Re: Is -1^(2/5) really undefined in R?
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- Subject: [mg69147] Re: [mg69094] Re: [mg69075] Is -1^(2/5) really undefined in R?
- From: János <janos.lobb at yale.edu>
- Date: Thu, 31 Aug 2006 04:39:00 -0400 (EDT)
- References: <200608290847.EAA00784@smc.vnet.net> <200608301032.GAA27866@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
I do not want to split electrons here, but for me the question in this case is this: What has higher precedence: Sign/Times ? or Power ? I vote for Power :) According to my vote -1^x is not the same as (-1)^x , so Ben's second equation: >> (-1^2)^(1/5) = 1^(1/5) is not correct. According to Mathematica: In[13]:= -1^2 Out[13]= -1 János On Aug 30, 2006, at 6:32 AM, Murray Eisenberg wrote: > The difficulty is, of course, exactly what the meaning of z^(m/n) is > when m and n are integers and z is negative. > > By the same kind of reasoning as Ben gave, > > (-1)^(2/5) = ( (-1)^(1/5) )^2 = (-1)^2 = 1. > > [Because (-1)^5 = -1, it seems correct to say that (-1)^(1/5) = -1.] > > So that's consistent with what Ben calculated. (I inserted missing > parentheses in Ben's calculation.) > > But is it correct (as was asked)? > > Any real value of (-1)^(2/5) must also be a complex value (with zero > imaginary part). So let's see what the complex values are. The > definition of the multi-valued power z^a is, in Mathematica syntax, > the > set of all complex numbers of the form > > Exp[a ( Log[Abs[z]] + I (Arg[z] + 2n Pi) )] > > where n can be any integer. > > Let's evaluate this in Mathematica, noting from further calculations > that additional values of n outside the range {0,1,2,3} just give the > same values again: > > a = 2/5; z = -1; > vals = Table[ > Exp[a ( Log[Abs[z] ] + I (Arg[z] + 2 n Pi) )], > {n, 0, 3} > ] // ComplexExpand > {-1/4 + Sqrt[5]/4 + (I/2)*Sqrt[(5 + Sqrt[5])/2], > -1/4 - Sqrt[5]/4 - (I/2)*Sqrt[(5 - Sqrt[5])/2], 1, > -1/4 - Sqrt[5]/4 + (I/2)*Sqrt[(5 - Sqrt[5])/2]} > > (I actually converted the output to InputForm to make the result > display > "linearly".) > > And if you either take N of these values or plot them -- most easily > using David Park's Cardano3`ComplexGraphics` routines > > ComplexGraphics[{PointSize[0.025], ComplexPoint /@ vals}]; > > -- you see that you have 4 distinct values. > > Moreover, exactly one of these 4 values is real, namely, 1. So it IS > correct to say that (-1)^(2/5) = 1 -- provided you're looking only for > real roots. > > And Mathematica seems to agree: > > <<Miscellaneous`RealOnly` > > (-1)^(2/5) > 1 > > > Ben wrote: >> Is -1^(2/5) really undefined in R? >> >> Mathematica seems to think so, I guess since it looks like a >> negative square root, but >> >> (-1)^(2/5) = ((-1)^2)^(1/5) = 1^(1/5) = 1 >> >> Is this correct mathematically? >> >> cheers, >> >> BC >> >> > > -- > Murray Eisenberg murray at math.umass.edu > Mathematics & Statistics Dept. > Lederle Graduate Research Tower phone 413 549-1020 (H) > University of Massachusetts 413 545-2859 (W) > 710 North Pleasant Street fax 413 545-1801 > Amherst, MA 01003-9305 ---------------------------------------------- Trying to argue with a politician is like lifting up the head of a corpse. (S. Lem: His Master Voice)
- References:
- Is -1^(2/5) really undefined in R?
- From: Ben <ben.carbery@spam.me>
- Re: Is -1^(2/5) really undefined in R?
- From: Murray Eisenberg <murray@math.umass.edu>
- Is -1^(2/5) really undefined in R?