Subject: Re: [HM] A question around zero
From: Don Cook (tdctdc@surfsouth.com)
Date: Tue Jan 25 2000 - 18:03:56 EST
Ralph Rami wrote:
I recently reviewed the Standards for school mathematics published
by all the States in the U.S.A. (The results are published by the
Fordham Foundation, if anyone is interested, on their website.)
Many of these Standards documents include a Glossary, for no good
purpose that I could see, particularly as the definitions were often
mathematically illiterate. Others were more amusing than confusing.
One state *defined* "odd number" as "a whole number whose decimal
expression ends in 1, 3, 5, 7, or 9." You can't be any clearer than
that, I think, unless you want to get into newmath (1955-1975)
language, and write "...ends in '1', '3', '5', '7', or '9'.
Dear Ralph,
When I taught newmath, 1, 3, 5, 7, and 9 were numerals and '1', '3', '5', '7',
and '9' were numbers - or was it the other way around? For those of you who
never had to go through this nonsense with 5th graders - "a numeral is the
name of a number". There were some wonderful aspects to newmath, one study
(I lost the reference) showed that the collection of students who came from
newmath produced our best generation of graduate students. In my experience,
newmath failed because the elementary school teachers were not prepared to
teach it.
Two questions:
1. Should we be having this debate in the year 2000 on whether zero is
even or not. In the set of integers, it is! Non mathematicians have trouble
with the concept that not all numbers are cardinal numbers. Sometimes they
will accept zero as a temperature between -1 and 1. A George Carlin joke
that I use to explain the difference between 'zero' and 'none' is, "Suppose
when you woke up the weather person said, 'It's no degrees outside'. What
would you wear?"
2. I need help with Euclid and zero. I thought, that for Euclid, all
numbers were lengths of line segments. Since there are no line segments with
either zero or negative length, these numbers do not exist. Will someone who
has more knowlege than me expand on this?
In An Introduction to Algebra, being the first part of a COURSE OF
MATHEMATICS adapted to the method of instruction in American Colleges,
Fourth Edition, 1827, Jeremiah Day, D.D. LL.D, President of Yale College
writes, "To one who has just entered on the study of algebra, there is
generally nothing more perplexing, than the use of what are called negative
quantities. He supposes he is about to be introduced into a class of
quantities which are entirely new; a sort of mathematical nothings, of which
he can form no distinct conception. ...this is owing to a misapprehension of
the term negative, as used in mathematics. A negative quantity is one which
is required to be subtracted. ... The terms positive and negative as used
in mathematics, are merely relative. They imply that there is, ... some such
opposition as requires that one should be subtracted from the other. But
this opposition is not that of existance and non-existance, nor of one thing
greater than nothing, and another less than nothing. For, in many cases,
many of the signs may be, indifferently and at pleasure, applied to the very
same quantity; ... in determining the progress of a ship, for instance, her
easting may be marked +, and her westing -; or the westing may be +..."
Peace,
Don Cook
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