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
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
_______________________________________________
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