[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*: Tue, 28 Apr 2009 11:05:39 +1000*Delivered-to*: xmca@weber.ucsd.edu*Domainkey-signature*: a=rsa-sha1; s=2007001; d=ucsd.edu; c=simple; q=dns; b=b28YoTTMS8GhM/PbQZYsZE7ddG9DgAPjefjS8vFdeSznfnEv0TKgLwG6eAcbPkIrX XzwGReOgzVnSHQPZsloKQ==*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*: 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)

From the worst ex-maths teacher in the world ...

Andy 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 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 someoneplease 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

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

- 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):