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neg * neg = pos; why?
Posted:
Jul 22, 1997 3:29 PM
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Prof Klespis remarks that secondary math methods courses have many
students who have had calculus but cannot give you a cogent argument or
example of why the product of two negative numbers is positive. I am
curious about what he would hope his students would say in defense of
this; something that would be convincing to the person-in-the-street I
hope. I think it seems quite arbitrary to many ordinary people that a
product of two negative numbers should be positive -- they don't see the
sense in it. Indeed, I have seen volumes written by people we would
likely call cranks who argue that we mathematicians got it wrong when we
defined multiplication that way, and that it has caused untold trouble.
The usual purely algebraic argument for "neg*neg is pos" depends on
assuming that the distributive law holds for ALL triples of numbers. But
of course that just begs the question of why we decided to define
multiplication for negative numbers so that the distributive law would
continue to work (having earlier checked that it works for positive
numbers). What would you like your students/teachers to argue about
this? Should their argument start with "Yes it is arbitrary, but not
unthinking. Here is why the conventional choice is a good one. ..."?
On the other hand, suppose we take a geometric approach to numbers as
recently advocated by Hy Bass at the Wash. Conf. sponsored by MSEB and
NCTM on the Nature and Role of Algebra in the K-14 Curriculum. We take
any Euclidean line, mark point 0 on it, choose a unit length and mark
point 1, and then the non-negative "numbers" are associated (identified)
with all the points on the ray from 0 through 1in the usual way. The
negative numbers are then identified with the remaining points on the
opposite ray in the usual way so that we have now a Cartesian coordinate
number line L. Now choose a second line M through 0 and mark a 1 on it
at the same unit distance as we used on L, and complete this so M is also
a Cartesian coordinate line. Now for any number b (point) on L, and
any nonzero number c on M, the product point b*c can be found: let J be
the line from 1 on L to c on M, and K the line parallel to J through b on
L, then b*c is the intersection of K and M. Now just check what this
gives for all cases of pos and neg numbers b, c. [Of course if M is
parallel to L and their positive directions are the same, and you do the
same construction but using 0 on L instead of 1, then the intersection
point is b+c.] Do secondary math methods books point these things out
and discuss them as alternative ways to think about the questions? Is
this any more convincing than an argument in favor of preserving the
distributive law?
Ladnor Geissinger
Math Prof at UNC Chapel Hill & Math Chair at IAT
email: ladnor.iat@mhs.unc.edu
or ladnor_geissinger@unc.edu
phone: 919-405-1925
address: Institute for Academic Technology
2525 Meridian Parkway, Suite 400
Durham NC 27713 USA
IAT phone: 919-560-5031
IAT fax: 919-560-5047
IAT web home page: www.iat.unc.edu
LEARN NC home page: www.learnnc.org
Mathwright Library: ike.engr.washington.edu/mathwright/
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