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

*To*: Jerry Balzano <gjbalzano@ucsd.edu>, "mcole@weber.ucsd.edu" <mcole@weber.ucsd.edu>, "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Subject*: Re: [xmca] a minus times a plus*From*: Jim Levin <jalevin@ucsd.edu>*Date*: Mon, 27 Apr 2009 20:44:36 -0700*Cc*:*Delivered-to*: xmca@weber.ucsd.edu*Domainkey-signature*: a=rsa-sha1; s=2007001; d=ucsd.edu; c=simple; q=dns; b=YY2paErrzAyQvqqc+cI2V72QBgK1kTCFiTg5Pkfh7ZQ54zop1WHSOl3Cx8MEtI/Rt mNBAndh/8GBG2og5RNW6w==*In-reply-to*: <6B313151-6CFA-4CDD-B489-5DCBB63FAF4C@ucsd.edu>*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> <30364f990904271706l114497cax5f814ffa09a51893@mail.gmail.com> <6B313151-6CFA-4CDD-B489-5DCBB63FAF4C@ucsd.edu>*Reply-to*: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Sender*: xmca-bounces@weber.ucsd.edu

I "see" that Jerry and I "mirror" the way we think about negative number ;-) At 8:37 PM -0700 4/27/09, Jerry Balzano wrote:

I can't resist this one - I would say that one develops intuitions forthings like this by starting before numbers (math isn't really "about"numbers anyway, it's about patterns), like with simpletransformations. Think of the simple system of transformations to athing consisting of "Leave It Alone" and "Reflect it Through aMirror". Then ask, how do the transformations in this simple systemcombine? Well, if ILeave It Alone, Leave It Alone, the net effect is that I've Left ItAlone - I * I = ILeave It Alone, then Reflect it Through a Mirror, the net effect isthat I've Reflected it Through a Mirror - I * M = MReflect it Through a Mirror, then Leave It Alone, the net effect isthat I've Reflected it Through a Mirror - M * I = Mand finally, the hard/interesting one - if IReflect it Through a Mirror, then Reflect it Through a Mirror thenet effect is that I've Left It Alone - M * M = IOf course, this can be illustrated for anyone who doesn't "see" it orbelieve it, but the pattern is strictly pre-numerical. And it showsup in lots and lots of places in mathematics.So when we encounter multiplication by positive and negativenumbers ... whoops, there it is again, that pattern. And no wonder -what does multiplying by a negative number do, but "flip" (reflect)the number through the origin of the number line, as others who haveposted on this topic have already demonstrated... whereas multiplyinga number by a positive number "leaves it alone" with respect to whichside of the origin the number is on.-Jerry Balzano On Apr 27, 2009, at 5:06 PM, Mike Cole wrote:Great!! Thanks Ed and Eric and please, anyone else with other ways of explaining the underlying concepts.Now, we appear to have x and y coordinates here. If I am using anumber linethat ranges along both x and y axes from (say) -10 to +10 its prettyeasy of visualize the relations involved. And there are games thatkids canplay that provide them with a lot of practice in getting a strongsenseof how positive and negative positions along these lines work.What might there be of a similar nature that would help kids and oldcollegeprofessors understand why -8*8=64 while -8*-8=64?Might the problem of my grand daughter, doing geometry, saying,"Well, duh,grandpa, its just a fact!) arise from the fact (is it a fact?) that they learn multiplication "facts" before they learn about algebra and grokable explanations that involve even simple equations such as y+a=0 are unintelligible have become so fossilized that the required reorganization of understanding is blocked? mike On Mon, Apr 27, 2009 at 4:16 PM, Ed Wall <ewall@umich.edu> wrote:Mike It is simply (of course, it isn't simple by the way) because, thenegative integers (and, if you wish, zero) were added to thenatural numbersin a way that preserves (in a sense) their (the natural numbers)usualarithmetical regularities. It would be unfortunate if somethingthat wastrue in the natural numbers was no longer true in the integers,which is aextension that includes them. Perhaps the easiest way to thenegative xpositive business is as follows (and, of course, this can be madeopaquelyprecise - 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 thinkhappensnext? This thinking works for, of course, for negative timesnegative. Theopaque proof is more or less as follows.Negative numbers are solutions to natural number equations of theform (I'msimplifying all this a little) x + a = 0 ('a' a natural number)>>and likewise positive numbers are solutions to natural numberequations ofthe form y = b ('b' a natural number)Multiplying these two equations in the usual fashion within thenaturalnumbers gives xy + 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(whichI'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, couldsomeoneplease post an explanation of whymultiplying a negative number by a positive numbers yields anegativenumber? 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

**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:*Mike Cole <lchcmike@gmail.com>

**Re: [xmca] a minus times a plus***From:*Jerry Balzano <gjbalzano@ucsd.edu>

- 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):