[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]

*To*: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Subject*: Re: [xmca] a minus times a plus*From*: Andy Blunden <ablunden@mira.net>*Date*: Wed, 29 Apr 2009 14:59:07 +1000*Delivered-to*: xmca@weber.ucsd.edu*Domainkey-signature*: a=rsa-sha1; s=2007001; d=ucsd.edu; c=simple; q=dns; b=p2Gus5yKxHsi5OZMgoH1Y6eEl/PteIbQjjXMn2DIWsEkMjcPw7R+0LUM+RCUtlcEL /214SSFnVrvSIaU/s3hxg==*In-reply-to*: <2B2B3D4D-462E-4654-8457-A1C4F21B2874@uvic.ca>*List-archive*: <http://dss.ucsd.edu/mailman/private/xmca>*List-help*: <mailto:xmca-request@weber.ucsd.edu?subject=help>*List-id*: "eXtended Mind, Culture, Activity" <xmca.weber.ucsd.edu>*List-post*: <mailto:xmca@weber.ucsd.edu>*List-subscribe*: <http://dss.ucsd.edu/mailman/listinfo/xmca>, <mailto:xmca-request@weber.ucsd.edu?subject=subscribe>*List-unsubscribe*: <http://dss.ucsd.edu/mailman/listinfo/xmca>, <mailto:xmca-request@weber.ucsd.edu?subject=unsubscribe>*References*: <30364f990904271547o5b4df21eifca69bf8318483f2@mail.gmail.com> <2C46D7A7-AD94-441C-AABD-269045835E3D@umich.edu> <0193A85F-03A7-4811-B912-217722A5770B@uvic.ca> <BDD722EC-3E52-48C1-829B-AD5053D51B78@umich.edu> <2B2B3D4D-462E-4654-8457-A1C4F21B2874@uvic.ca>*Reply-to*: ablunden@mira.net, "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Sender*: xmca-bounces@weber.ucsd.edu*User-agent*: Thunderbird 2.0.0.14 (Windows/20080421)

Andy Wolff-Michael Roth wrote:

Hi Ed,I think it is very helpful to think and look at similarities with othermatrix operations, for example, to look at the determinant of the1-dimensional matrix, which is -1, which means, the sense is inversed.Thus, when you take 1 and 5, 1 is the smaller, 5 the larger, thenmultiplying each with -1 you get the results inversed, -1 is larger than-5.Thus, even if it were not mathematical, we could learn a lot of lookingat multiplication as a TRANSFORMATION, whereby some set of numbers comesto be mapped back onto itself. :-)I think that the mathematical idea of transformation (mapping,function...) is one of the most powerful in our culture.Michael On 27-Apr-09, at 5:59 PM, Ed Wall wrote: MichaelThe reason why a physicist's thinking works this way is becausethey are immersed in our number system and hence facts can be used toprove, in a sense, themselves. In writing A = -1 you are, in a sense,making such a move. The unfortunate thing is that when you do this you,in a sense, gloss over the very structure you are trying to uncover.There are also the equally unhelpful - and, please, note that these aremy opinions - of the sort (they can be made somewhat nicer): you earn anegative five dollars for three days, what do you have at the end of 3days. The negative times the negative stories are really arcane and Imust admit to be unsure just what is going.I, by the have no problem with this physicist's take asillustrative of the consequence of the the structure of the naturalsand their extension to the integers (and the next extension is, onemight say, the rationals). However, it ignores, in a sense, thestructure of the naturals and I happen to think that structure iscrucial to children's understanding.Ed On Apr 27, 2009, at 8:34 PM, Wolff-Michael Roth wrote:Can't you think like this---perhaps it is too much of a physicist'sthinking. We can think of the following general function (operator inphysics) that produces an image y of x operated upon by A.y = Axif x is from the domain of positive integers, then A = -1 wouldproduce an image that is opposite to the one when A = +1, the identityoperation.Conceptually you would then not think in terms of a positive times anegative number, but in terms of a positive number that is projectedopposite of the origin on a number line, and, if the number is unequalto 1, like -2, then it is also stretched.The - would then not be interpreted in the same way as the + Cheers, Michael On 27-Apr-09, at 4:16 PM, Ed Wall wrote: MikeIt is simply (of course, it isn't simple by the way) because, thenegative integers (and, if you wish, zero) were added to the naturalnumbers in a way that preserves (in a sense) their (the naturalnumbers) usual arithmetical regularities. It would be unfortunate ifsomething that was true in the natural numbers was no longer true inthe integers, which is a extension that includes them. Perhaps theeasiest way to the negative x positive business is as follows (and, ofcourse, this can be made opaquely precise - smile):3 x 1 = 3 2 x 1 = 2 1 x 1 = 1 0 x 1 = 0so what, given regularity in the naturals + zero) do you think happensnext? This thinking works for, of course, for negative times negative.The opaque proof is more or less as follows.Negative numbers are solutions to natural number equations of the form(I'm simplifying all this a little)x + a = 0 ('a' a natural number)and likewise positive numbers are solutions to natural numberequations of the formy = b ('b' a natural number)Multiplying these two equations in the usual fashion within thenatural numbers givesxy + ay = 0 or substituting for y xy + ab = 0 so, by definition, xy is a negative number.Notice how all this hinges on the structure of the natural numbers(which I've somewhat assumed in all this).Ed On Apr 27, 2009, at 6:47 PM, Mike Cole wrote:Since we have some mathematically literate folks on xmca, could someone please post an explanation of why multiplying a negative number by a positive numbers yields a negative number? What I would really love is an explanationthat is representable in a manner understandable to old collegeprofessorsand young high school students alike. mike _______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca_______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca _______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca_______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca _______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca

-- ------------------------------------------------------------------------ Andy Blunden http://home.mira.net/~andy/ Hegel's Logic with a Foreword by Andy Blunden: From Erythrós Press and Media <http://www.erythrospress.com/>. _______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca

**Follow-Ups**:**Re: [xmca] a minus times a plus***From:*Ng Foo Keong <lefouque@gmail.com>

**References**:**[xmca] a minus times a plus***From:*Mike Cole <lchcmike@gmail.com>

**Re: [xmca] a minus times a plus***From:*Ed Wall <ewall@umich.edu>

**Re: [xmca] a minus times a plus***From:*Wolff-Michael Roth <mroth@uvic.ca>

**Re: [xmca] a minus times a plus***From:*Ed Wall <ewall@umich.edu>

**Re: [xmca] a minus times a plus***From:*Wolff-Michael Roth <mroth@uvic.ca>

- Prev by Date:
**Re: [xmca] a minus times a plus** - Next by Date:
**Re: [xmca] a minus times a plus** - Previous by thread:
**Re: [xmca] a minus times a plus** - Next by thread:
**Re: [xmca] a minus times a plus** - Index(es):