> These examples show our cleverness at inventing unlikely scenarios for
> extending the application of integer arithmetic to the multiplication of
> negative numbers. The original question remains: is there any "real world"
> (of whichever kind) problem for which that operation is really useful?
What seems to be sought here is a heuristic argument for the rules set up to
allow operations with negative numbers. Some contributors here feel that none
have yet been adduced, others feel that none should be needed. I am, I believe,
with the latter camp.
The heuristics are in why it is desirable to define negative numbers in the
first place. The world got along without formal negative numbers for a long
time, and did not believe in their reality even after accepting the need to
work with them. Even the great DeCartes would talk about true (positive) and
false (negative) roots of polynomial equations.
With the benefit of (several hundred years of) hindsight, we can now see the
logical process involved in introducing the negative numbers. Once you have a
consistent set of positive numbers (integers, rationals, even irrationals) to
work with -- and noticing that this set is not closed under the operation of
subtraction -- the bold step is taken of adjoining a new unit, which is labeled
for distinction -1 (could be colored red, instead, if this were practical).
Rules for extending the field operations to include this new object are set up,
which imply the creation as well of a whole raft of negative numbers -- one for
each positive number. And it turns out that it is logically necessary that the
product of two negative nubmers be a postive number. If one were formally
constructing the number system, one might first create the negative integers as
equivalence classes, much as the rationals are constructed as equivalence
classes of "fractions". But that is modern formalism (though it is mightily
satisfying).
At any rate the negative numbers turn out to be no less (and no more) grounded
in reality than the positive numbers, even though they might be further removed
from models that can be pointed to in satisfaction, as though they corroborated
something (they don't, really).
The next step, of course, was to realize that the set of signed "real" numbers
was still not closed under the extraction of roots, and to adjoin yet a further
unit 1i, and extend the field rules to also include this new object. A formal
model for this extension is to introduce pairs of numbers (x,y) with a
particular, peculiar rule for multiplication of two pairs. The geometric model
of the complex plane provides a more satisfying way to understand this rule.
And where is the "real-world" example to explain why the tricky rules for
working with complex numbers work? Why isn't one being demanded as well?
Probably because complex numbers have not yet been _accepted_ by the general
public in the way that signed numbers have. They are very, very useful in
certain contexts, though, and no one is going to suggest that we get rid of
them (though I am waiting for a politician to make this argument any day now).
And they are no less real than the numbers on the number line (I often ask my
students when was the last time they saw a "2" walking down the street).
It is a lot easier to explore complex arithmetic with the aid of a modern
calculator (e.g a Texas Instruments TI-83). Will we see the topic of complex
numbers move "downward" in the curriculum the way that many other topics that
used to be considered "advanced" have? Will the general idea of working with
complex numbers be in possession of the average educated person at some point
in the future? I, myself, sincerely hope so, but I have given up on the idea
that it will happen in my lifetime. Why, what we are really still arguing about
is whether negative numbers exist, isn't it?
RWW Taylor
National Technical Institute for the Deaf
Rochester Institute of Technology
Rochester NY 14623
>>>> The plural of mongoose begins with p. <<<<
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