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*Subject*: Re: [xmca] a minus times a plus*From*: Mike Cole <lchcmike@gmail.com>*Date*: Mon, 27 Apr 2009 17:06:28 -0700*Cc*: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Delivered-to*: xmca@weber.ucsd.edu*Dkim-signature*: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=gamma; h=domainkey-signature:mime-version:received:reply-to:in-reply-to :references:date:message-id:subject:from:cc:content-type; bh=eDja8Gz0e/zx35p+XikosjyWgydXOGnvuHdNuHEt98I=; b=eG/iB7LSmphX7WngV0id7zQkWb52pk2HKm2DjCOHYXtE2Az6I2Rp8ze3ftjuChHxaS Zev4K52wuodqrPMbhCuXwxhSQt3XPZg/CBGN9uZrCpIHUNs78vJ+G1BZ1QldV0QtZRPp xXRT9DBGd6xxnzY+zw/zi9GX4EDP9Xuvfx0m0=*Domainkey-signature*: a=rsa-sha1; c=nofws; d=gmail.com; s=gamma; h=mime-version:reply-to:in-reply-to:references:date:message-id :subject:from:cc:content-type; b=pbm1fa+zCXz+z04OG7IRTTulRPr+Zy+olhLgVn7fZTltBhnrwA9SYjJkhQhv7Grkau QudCUH/qd0tS7PmYe6Ty9U4vT2qAu5xgO2K3MErkFsBGhspooby7AaWzVcCECx6Pex/9 UpcHqxLQRde+RyQ6tJEN1lICyHVr7V1geXVoY=*In-reply-to*: <2C46D7A7-AD94-441C-AABD-269045835E3D@umich.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>*Reply-to*: mcole@weber.ucsd.edu, "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Sender*: xmca-bounces@weber.ucsd.edu

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 a number line that ranges along both x and y axes from (say) -10 to +10 its pretty easy of visualize the relations involved. And there are games that kids can play that provide them with a lot of practice in getting a strong sense of how positive and negative positions along these lines work. What might there be of a similar nature that would help kids and old college professors 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, the > negative integers (and, if you wish, zero) were added to the natural numbers > in a way that preserves (in a sense) their (the natural numbers) usual > arithmetical regularities. It would be unfortunate if something that was > true in the natural numbers was no longer true in the integers, which is a > extension that includes them. Perhaps the easiest way to the negative x > positive business is as follows (and, of course, this can be made opaquely > precise - smile): > > 3 x 1 = 3 > 2 x 1 = 2 > 1 x 1 = 1 > 0 x 1 = 0 > > so what, given regularity in the naturals + zero) do you think happens > next? 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 number equations of > the form > > y = b ('b' a natural number) > > > Multiplying these two equations in the usual fashion within the natural > numbers 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 (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 explanation >> that is representable in a manner understandable to old college professors >> and 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

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

**RE: [xmca] a minus times a plus***From:*"Hasty, James E." <hastyj@clarke.k12.ga.us>

**Re: [xmca] a minus times a plus***From:*Andy Blunden <ablunden@mira.net>

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

**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>

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