[Numeracy 528] Re: Another perspective on numbers, operations, and negatives
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Wed Sep 1 01:21:31 EDT 2010
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The notes below are directed to Michael Gyori and anyone else who
views numbers as essentially representing quantities.
I recommend the book Negative Mathematics: How Mathematical Rules Can
Be Positively Bent,
An easy introduction to the study of developing algebraic rules to
describe relations among things, by Alberto Martinez, Princeton Univ.
The first 5 chapters are mainly devoted to the history of numerical
algebra. He points out that negative numbers have been widely used at
least since the mid-1500s and there was lots of fuss about them even
among mathematicians and scientists until about the mid-1800s by which
time everything had been agreed upon and their utility was no longer in
MacLaurin justified the introduction of signed numbers into algebra on
the basis of physical utility. Math deals with more than magnitudes, it
must also handle other concepts of physical significance.
In DeMorgan's book of 1831 he uses signs to represent directions, and
says "rules of operation are the results of experience, not of abstract
In chapter 5 on page 103 Martinez says "It was because of symbolical
algebra that mathematics on the whole ceased to be defined as the
_science of quantity_."
Note that among other things, DeMorgan was recognizing that when
negative numbers were introduced and people were deciding how to extend
multiplication to all numbers, they were not forced by pure reason to
define (-1)*(-1) = 1. Along the way some pretty good mathematicians had
decided that they couldn't stomach that and had preferred to define
multiplication so that (-1)*(-1) = -1. But that meant that the
distributive law of multiplication over addition no longer held
universally, which meant that one had to be very careful in doing long
calculations, it led to considering many special cases, and calculating
with variables was difficult. This was so complicated and so
inefficient that eventually everyone decided that always having the
distributive law work was so much easier and gave us a much more useful
new tool for analysis.
On 8/30/2010 5:00 AM, Michael Gyori wrote:
> I really appreciate Ladnor's response below. I believe I
> understand the points being made and have no issue with them whatsoever.
> My question, again, ties in with how I teach math (not a
> primary undertaking in the context of my work). I've stated often that
> I teach numbers as representing quantities, back to when this list got
> underway and I contended that negative integers do not exist - a
> contention that itself triggered some discussion.
> Yes, natural numbers do not even include zero, I agree, but at least
> we can demonstrate that nothing is left.
> I have too many students who have learned to "hate" math. Most of
> them become more favorably inclined towards the subject after I work
> with them. One reason is that I demonstrate the relationship between
> math and issues that can arise in daily life that require math to
> solve problems (along the lines of EFF). I continue to struggle with
> establishing such a relationship with negative integers, which is the
> reason I address this audience in pursuit of some insight that, in
> turn, might benefit me and, in turn, my students.
> Thanks again,
> Michael A. Gyori
> Maui International Language School
> www.mauilanguage.com <http://www.mauilanguage.com/>
Ladnor Geissinger, Emer. Prof. Mathematics
Univ. of North Carolina, Chapel Hill NC 27599 USA
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