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Re: Non-Euclidean Arithmetic
Posted:
Sep 8, 2012 10:26 PM
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Too much gibberish to make mathematical sense of "multiplication."
As with all human languages, there is no "correct" meaning of any term or
phrase ... only the human challenge of the transmitter and the receiver
trying to use the same the meanings of words/phrases/expositions about
whatever is being attended.
Crabtree's (elusive) meanings of "multiplication" not withstanding, one
might ask of what are the usual meanings of "multiplication" within each
curricular theory of arithmetic ... and of what comprehensive meaning might
globally encompass all of those contextual particulars.
In global perspective, "multiplication" seems to normally be equated with
the (unary) "of" composition of functions .. including "multiplications" by
complex numbers -- so called because those invoke "multiplications" within
lesser (4-quadrant) systems of numbers. Likewise for "multiplications" of
fractions, integers, rationals, and reals. That "of" meaning appears to be
pervasive.
Within the arithmetics of real numbers (and subordinate systems), the global
(mx) meaning of "multiplications" is all about using using multipliers, m,
as per-1 rates/slopes of the mx *proportions* through the origin ... "run"
x, at the rate of m-per-+1 ... and the resulting value is mx.
That describes grade-2 "multiplication" of Arabic digits ... and the
multiplications of complex and modular numbers ... and everything in
between. So, it would seem that curricular education in "multiplications"
might best focus on the mx functions (the proportions) and their
per-pos-unit rates. To teach "multiplications" within more myopic contexts
... of particular kinds of uses ... induces students' later needs for later
*redefining* the meaning of the term. Such transitions work only when the
instruction gives special attention to how/why the meaning of
"multiplications" is being repeatedly re-defined.
Cordially,
Clyde.
- --------------------------------------------------
From: "Jonathan Crabtree" <sendtojonathan@yahoo.com.au>
Sent: Saturday, September 08, 2012 4:31 PM
To: <math-teach@mathforum.org>
Subject: Re: Non-Euclidean Arithmetic
> A note to all.
>
> Please keep replies relevant to the topic.
>
> My post is about arithmetic and defining operations.
>
> Philosophy is down the hall and politics is in building B.
>
> Thank you.
>
> ########
>
> I have defined multiplication for the rationals as mk = the combination of
> m either added to or subtracted from zero k times ie multiplication is the
> combination of a multiplicand either added to or subtracted from zero as
> many times as there are unities in the multiplier.
>
> Scaling can give you the same numeric answer as multiplication, yet that
> does not make the operations the same.
>
> The symbol is : and it results in a change of magnitude not change of
> magnitude not multitude. Toy planes, trains and automobiles are scaled as
> you know. Therefore the concept is already familiar to children. already
> familiar to children
>
> My definition of mathematical scaling is 'a change of magnitude created by
> making the multiplier the unit of measure.
>
> Multiplication involves many nstances of the same object in units or parts
> and relates to multi-tude.
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