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Re: [xmca] a minus times a plus

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?


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
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