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*To*: mcole@weber.ucsd.edu, "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Subject*: Re: [xmca] a minus times a plus*From*: Ed Wall <ewall@umich.edu>*Date*: Mon, 27 Apr 2009 20:44:12 -0400*Cc*:*Delivered-to*: xmca@weber.ucsd.edu*Domainkey-signature*: a=rsa-sha1; s=2007001; d=ucsd.edu; c=simple; q=dns; b=rSvxZgM0JGSp+TWzZtKv2DjeumsfJ7icjyISf1MiengbQS+kwAysLOdca+sGA3IAD MSx5GBmX5nLrV0F+TR6tw==*In-reply-to*: <30364f990904271706l114497cax5f814ffa09a51893@mail.gmail.com>*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>*Reply-to*: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Sender*: xmca-bounces@weber.ucsd.edu

Mike

0 x a = 0 1 x a = a a x b = b x a a x (b x c) = (a x b) x c a x (b + c ) = a x b + a x c

\ and then

3 x a = 2 x a + 1 x a 4 x a = 2 x (2 x a)

8 x a = 2 x ( 2 x (2 x a))) 7 x 6 x 9 x

Ed

Ed On Apr 27, 2009, at 8:06 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

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**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>

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