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Re: Some important demonstrations on negative numbers
Posted:
Nov 30, 2012 8:23 PM
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Re: laymen's under-enlightenment about quantities:
The word, "numbers" has various meanings ... to various persons. When all
of the various kinds of things that are commonly called "number systems" are
compared ... what surfaces is that the in-common meaning of "numbers" is
objects in any system that satisfies a specific list of (some field)
properties ... and those collectively constitute a postulational definition
for number-ness. [Unfortunately, the core curriculum commonly induces some
truly myopic meanings for "numbers" ... sometimes as meaning only the
Arabic numerals.]
Any system which meets those criteria so qualifies as "a number system." It
has nothing whatsoever to do with the intrinsic nature of such objects ...
only with how they relate to each other through the operations among them.
Anyone who chooses to limit their discourse to using some alternative or
more specific meaning for number-ness ... cardinal, ordinal, Arabic,
rational, etc. ... has the obligation of clarifying their own meaning(s) of
the term.
In particular, any system of *quantities* becomes a system of numbers, as
soon as one imposes any qualifying "multiplications" among those quantities.
[Scalar additions and multiplications are intrinsic to quantity-ness.] Most
of the familiar number-systems [integers, fractions, decimal, rationals, and
reals (as sequences of whole numbers)] are derived as quantities, or as
(vector) combinations of quantities whose numerators [J.C.'s "adjectives"]
are whole numbers.
Quantities are no more "physical" than one's *thought* about an apple is
edible. But quantities do often provide us with *mathematical* ways of
talking about the physical. Indeed, it can be argued that all of numeric
mathematics arose from mankind's natural dealings with "empirical"
quantities ... even via roundabout paths through initially nonsensical
formalities.
Bottom line: student's common-sensible mathematical encounters with
mathematical quantities are their only (conceptual understanding) means of
personally deriving arithmetic and functional "numeracy." Without
quantitative derivations, they are forced to play the (risky) game of tying
to play scholastic "conventions."
Cordially,
Clyde
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From: "Joe Niederberger" <niederberger@comcast.net>
Sent: Friday, November 30, 2012 5:51 PM
To: <math-teach@mathforum.org>
Subject: Re: Some important demonstrations on negative numbers
>>Joe, numbers are not quantities. Quantities are physical, they include a
>>number, a unit, a direction, if needed, and for goodness sake, a freaking
>>context.
>
> My goodness. I was speaking of the good old world, as Dickens might say,
> not your personal universe.
>
> Here is Isaac Newton:
> "Algebraic quantities are of two sorts, affirmative and negative; an
> affirmative quantity is greater than nothing, and is known by this sign +;
> a negative quantity is less than nothing, and is known by this sign -."
>
> He follows up with a money example.
>
> Here is Euler:
> "The calculation of imaginary quantities is of the greatest importance."
>
> Lest you think taking number as "quantity" is merely archaic usage, check
> these:
>
> * http://oxforddictionaries.com/definition/english/mathematics
> * http://dictionary.reference.com/browse/mathematics?s=t&ld=1122
> * http://en.wikipedia.org/wiki/Mathematics
> * http://en.wikipedia.org/wiki/Quantity#Quantity_in_mathematics
>
> I understand the distinction you are pointing to, though; nice as it is,
> it doesn't seem particularly germane in this context, that of
> understanding negative numbers and their rules.
>
> Saying that a negative number "is a mathematical concept" (well, by golly,
> its abstract!) does nothing to explain what it is. How are they different
> from the whole numbers a child already knows about? What good are they?
> Why are the rules (esp. the infamous one) such as they are?
>
> What's your lesson look like? What are the key points for a child?
>
> R.H. says:
>>Mathematics deals only with the number part of all that.
>
> And which part is that? Does your concept of number include separable
> components as well? What makes a real number real, but an imaginary number
> a figment of the imagination?
>
> And now, just for fun, some people who want to get real about math:
> http://web.maths.unsw.edu.au/~jim/structmath.html
> http://web.maths.unsw.edu.au/~jim/manifesto.html
>
> Cheers,
> Joe N
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