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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 for things like this by starting before numbers (math isn't really "about" numbers anyway, it's about patterns), like with simple transformations. Think of the simple system of transformations to a thing consisting of "Leave It Alone" and "Reflect it Through a Mirror". Then ask, how do the transformations in this simple system combine? Well, if ILeave It Alone, Leave It Alone, the net effect is that I've Left It Alone - I * I = I Leave It Alone, then Reflect it Through a Mirror, the net effect is that I've Reflected it Through a Mirror - I * M = M Reflect it Through a Mirror, then Leave It Alone, the net effect is that 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 the net effect is that I've Left It Alone - M * M = IOf course, this can be illustrated for anyone who doesn't "see" it or believe it, but the pattern is strictly pre-numerical. And it shows up in lots and lots of places in mathematics.So when we encounter multiplication by positive and negative numbers ... 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 have posted on this topic have already demonstrated... whereas multiplying a number by a positive number "leaves it alone" with respect to which side 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 a number linethat ranges along both x and y axes from (say) -10 to +10 its prettyeasy 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 senseof how positive and negative positions along these lines work.What might there be of a similar nature that would help kids and old collegeprofessors 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 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 opaquelyprecise - 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 happens next? This thinking works for, of course, for negative times negative. Theopaque proof is more or less as follows.Negative numbers are solutions to natural number equations of the form (I'msimplifying all this a little) x + a = 0 ('a' a natural number)>>and likewise positive numbers are solutions to natural number equations ofthe form y = b ('b' a natural number)Multiplying these two equations in the usual fashion within the naturalnumbers 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, could someoneplease post an explanation of whymultiplying a negative number by a positive numbers yields a negativenumber? What I would really love is an explanationthat is representable in a manner understandable to old college professorsand 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
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