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Re: NEGxNEG=POS:Once more with feeling
Posted:
Apr 3, 1996 7:07 PM
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Even in 2*3, the 3 refers to the number of objects, while the 2 refers to
the number of groups of 3. So the interpretations of the two numbers are
necessarily different even with whole numbers.
When considering (-2)*(-3), the -3 refers to a directed amount, and the
-2 refers to a "directed number" of groups of -3. (Refer to my exciting
post from yesterday about the mathematical basis for integers, and how
integer multiplication is not the same as whole number multiplication.)
I continue to argue that the conceptual basis for this is non-trivial,
and many of us (myself included) have rather tentative understandings of
integer multiplication. Which is why we have a tough time teaching it.
Gary
At 3:10 PM 4/3/96, Lou Talman wrote:
>Tim hendrix wrote:
>
>> An important part of the Postman approach or any other story
>> approach to unearth the concept of off-setting properties of multiplying
>> two negatives is that each integer in the product must have a different
>> interpretation:
>
>This observation is itself important; it ties in nicely with my observation
>about the way we overload the "-" sign in our discussions of arithmetic with
>signed numbers. The fact that we must give different interpretations to the
>"-" signs when we give plausibility arguments to justify (-)*(-) = (+) is an
>indication of the fundamental artificiality of those plausibility arguments.
>Notice that in my earlier post I *did not* overload the "-" sign. I used "-"
>as the unary negation operator *only*. That's one of the reasons I say that
>this is the "real" reason why (-)*(-) = (+).
>
>--Lou Talman
W. Gary Martin
Curriculum Research and Development Group
University of Hawaii
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