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MichaelWhat isn't clear to me in all this is where you are, in a manner of speaking, coming from and going to. I certainly see matrix algebra as, in a sense, an extension of the integers. What I don't see yet - although it seems possible - is how you are going to go from some primitive, yet to be defined, model (I'm using this in the technical sense) of matrix algebra to the integers. You, without some care, are going to lose commutativity.
On the other hand, as you seem to be thinking of your transformation as a kind of mapping, I happen to agree that functions provide a somewhat 'deeper' way to look at all this (and I happen to use such when I teach grad courses that waver in this direction). Unfortunately, being 'deeper' tends to cloak some, I think, interesting aspects of integers, or, perhaps one might say, the number line. So there is a trade off. I seem to remember an article by Halliday on unpacking where, in a sense, he touches on this tradeoff.
That you could do something like this and 'personally' come to a degree of verification is what is both powerful and perhaps seductive about mathematics.
Ed On Apr 28, 2009, at 8:51 AM, Wolff-Michael Roth wrote:
Hi Ed,I think it is very helpful to think and look at similarities with other matrix operations, for example, to look at the determinant of the 1-dimensional matrix, which is -1, which means, the sense is inversed. Thus, when you take 1 and 5, 1 is the smaller, 5 the larger, then multiplying each with -1 you get the results inversed, -1 is larger than -5. Thus, even if it were not mathematical, we could learn a lot of looking at multiplication as a TRANSFORMATION, whereby some set of numbers comes to be mapped back onto itself. :-)I think that the mathematical idea of transformation (mapping, function...) is one of the most powerful in our culture.Michael On 27-Apr-09, at 5:59 PM, Ed Wall wrote: MichaelThe reason why a physicist's thinking works this way is because they are immersed in our number system and hence facts can be used to prove, in a sense, themselves. In writing A = -1 you are, in a sense, making such a move. The unfortunate thing is that when you do this you, in a sense, gloss over the very structure you are trying to uncover. There are also the equally unhelpful - and, please, note that these are my opinions - of the sort (they can be made somewhat nicer): you earn a negative five dollars for three days, what do you have at the end of 3 days. The negative times the negative stories are really arcane and I must admit to be unsure just what is going. I, by the have no problem with this physicist's take as illustrative of the consequence of the the structure of the naturals and their extension to the integers (and the next extension is, one might say, the rationals). However, it ignores, in a sense, the structure of the naturals and I happen to think that structure is crucial to children's understanding.Ed On Apr 27, 2009, at 8:34 PM, Wolff-Michael Roth wrote:Can't you think like this---perhaps it is too much of a physicist's thinking. We can think of the following general function (operator in physics) that produces an image y of x operated upon by A.y = Axif x is from the domain of positive integers, then A = -1 would produce an image that is opposite to the one when A = +1, the identity operation.Conceptually you would then not think in terms of a positive times a negative number, but in terms of a positive number that is projected opposite of the origin on a number line, and, if the number is unequal to 1, like -2, then it is also stretched.The - would then not be interpreted in the same way as the + Cheers, Michael On 27-Apr-09, at 4:16 PM, Ed Wall wrote: MikeIt 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 = 0so 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 formy = b ('b' a natural number)Multiplying these two equations in the usual fashion within the natural numbers givesxy + 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 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_______________________________________________ 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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