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Topic: Multitudes, magnitudes, and multiplication
Replies: 67   Last Post: Mar 21, 2010 7:54 PM

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kirby urner

Posts: 2,762
Registered: 11/29/05
Re: Multiplication in R arises from ADDITION
Posted: Mar 16, 2010 2:48 PM
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On Mon, Mar 15, 2010 at 8:40 AM, Joe Niederberger
<niederberger@comcast.net> wrote:

<< SNIP >>

> So, the evidence as it stands, lends (strongly) towards the view that
> addition is basic, multiplication derived.  It says you can always
> correctly view it that way if you choose to.
>
> Joe N.
>
> p.s. Pam, this is a bit related to the little intuition you sent me in your
> last email.
>
>


Somehow I thought we were done with this question.

We don't only care about Real Numbers.

Sometimes, we don't care about Real Numbers at all.

Matrices multiply. Adding them may be pretty worthless.

Vectors add, but don't multiply (with each other), although quaternions do.

We preview these meanings of "multiply" with younger students,
not be drilling them in all kinds of arcane algorithms, by by giving
them a heads up as to the multiplicity of types in the "math objects
zoo".

Your nomenclature may vary, but the basic concepts are as
traditional and conservative as you need them to be.

At more advanced levels, it pays to show the family resemblances
between the different kinds of addition and multiplication.

At these levels a more abstract algebra approach enters in, and
we share about closure, inverse, neutral element (identity element),
associativity (matrices are associative, not commutative, w/r to
that binary operation we call "multiplication" w/r to them).

Using clever little logic games, encouraging free play and
discovery with finite groups is not out of reach. I'm not going to
make extravagant claims for their pedagogical value, but nor
am I going to sneer and jeer like some know-it-all pundit.

Back to my iconoclastic / radical agenda, here are pictures of
a robust classroom-ready 4-frequency tetrahedron made of plastic
icosahedra held together by strong springs:

http://www.flickr.com/photos/17157315@N00/4436951027/in/photostream/
http://www.flickr.com/photos/17157315@N00/4436951019/in/photostream/

(flextegrity by Sam Lanahan, sculpture by Glenn Stockton)

Of course no classroom has such a thing, as we're looking at a
prototype fresh from the factory, woven to suit. With 4-frequency,
you first get a nucleus.

The edge and face centers are easy to apprehend. 4-frequency
means five members along an edge, plastic icosahedra in this
case. So there's a middle one that's easy for students to grasp.

Having something heavy and robust adds measurably to the quality
of the kinesthetics upon passing this around -- makes a lasting
impression.

This is not salesmanship as I can't offer any for sale. More like
advertising my Radical Math agenda, a product I believe in
(obviously) and consider open source in large degree i.e. I'm
not the gatekeeper (nor bottleneck). The truth is out there.

What might be a real world application for getting into polyhedral
numbers ala sphere packing? The tetrahedral numbers? The
triangular numbers?

Well, here's a write-up of a recent nanotechnology lecture we got
in Portland, Oregon recently, where I think the connections are
pretty clear (note Coxeter paper for further reading, if you want
to go deeper into this stuff):

http://worldgame.blogspot.com/2010/03/towards-nanoscience.html

Kirby


Date Subject Author
3/7/10
Read Multitudes, magnitudes, and multiplication
Bill Marsh
3/8/10
Read Re: Multitudes, magnitudes, and multiplication
Joshua Fisher
3/8/10
Read Re: Multitudes, magnitudes, and multiplication
Joe Niederberger
3/9/10
Read Re: Multitudes, magnitudes, and multiplication
Bill Marsh
3/10/10
Read Re: Multitudes, magnitudes, and multiplication
kirby urner
3/9/10
Read Re: Multitudes, magnitudes, and multiplication
Bill Marsh
3/9/10
Read Re: Multitudes, magnitudes, and multiplication
Paul A. Tanner III
3/10/10
Read Re: Multitudes, magnitudes, and multiplication
Joe Niederberger
3/10/10
Read Re: Multitudes, magnitudes, and multiplication
Joe Niederberger
3/10/10
Read Re: Multitudes, magnitudes, and multiplication
kirby urner
3/11/10
Read Re: Multitudes, magnitudes, and multiplication
Joe Niederberger
3/11/10
Read Re: Multitudes, magnitudes, and multiplication
Joe Niederberger
3/11/10
Read Re: Multitudes, magnitudes, and multiplication
kirby urner
3/11/10
Read Re: Multitudes, magnitudes, and multiplication
Paul A. Tanner III
3/11/10
Read Re: Multitudes, magnitudes, and multiplication
Joe Niederberger
3/11/10
Read RE: Multitudes, magnitudes, and multiplication
Joshua Fisher
3/11/10
Read Re: Multitudes, magnitudes, and multiplication
Michael Paul Goldenberg
3/11/10
Read Re: Multitudes, magnitudes, and multiplication
Joe Niederberger
3/11/10
Read Re: Multitudes, magnitudes, and multiplication
Paul A. Tanner III
3/12/10
Read Re: Multitudes, magnitudes, and multiplication
Joe Niederberger
3/12/10
Read Re: Multitudes, magnitudes, and multiplication
Paul A. Tanner III
3/12/10
Read Re: Multitudes, magnitudes, and multiplication
Joe Niederberger
3/12/10
Read Re: Multitudes, magnitudes, and multiplication
Joshua Fisher
3/12/10
Read Re: [mirabusters] Re: Multitudes, magnitudes, and multiplication
Michael Paul Goldenberg
3/12/10
Read Re: Multitudes, magnitudes, and multiplication
Joe Niederberger
3/12/10
Read Re: Multitudes, magnitudes, and multiplication
Michael Paul Goldenberg
3/12/10
Read Re: Multitudes, magnitudes, and multiplication
Pam
3/12/10
Read Re: Multitudes, magnitudes, and multiplication
Michael Paul Goldenberg
3/13/10
Read Re: Multitudes, magnitudes, and multiplication
Joe Niederberger
3/12/10
Read Re: Multitudes, magnitudes, and multiplication
Joshua Fisher
3/12/10
Read Re: Multitudes, magnitudes, and multiplication
Paul A. Tanner III
3/12/10
Read Re: Multitudes, magnitudes, and multiplication
Joe Niederberger
3/13/10
Read Re: Multitudes, magnitudes, and multiplication
Joe Niederberger
3/13/10
Read Re: Multitudes, magnitudes, and multiplication
Michael Paul Goldenberg
3/12/10
Read Re: Multitudes, magnitudes, and multiplication
Paul A. Tanner III
3/13/10
Read Re: Multitudes, magnitudes, and multiplication
Paul A. Tanner III
3/14/10
Read Re: Multitudes, magnitudes, and multiplication
Joe Niederberger
3/14/10
Read Re: Multitudes, magnitudes, and multiplication
Joe Niederberger
3/15/10
Read Re: Multitudes, magnitudes, and multiplication
Paul A. Tanner III
3/15/10
Read Re: Multitudes, magnitudes, and multiplication
Joe Niederberger
3/15/10
Read Re: Multitudes, magnitudes, and multiplication
Michael Paul Goldenberg
3/15/10
Read Re: Multitudes, magnitudes, and multiplication
Joe Niederberger
3/15/10
Read Re: Multitudes, magnitudes, and multiplication
Joe Niederberger
3/15/10
Read Re: Multitudes, magnitudes, and multiplication
Joshua Fisher
3/15/10
Read Multiplication in R arises from ADDITION
Joe Niederberger
3/15/10
Read Re: Multiplication in R arises from ADDITION
Michael Paul Goldenberg
3/15/10
Read Re: Multiplication in R arises from ADDITION
Joshua Fisher
3/16/10
Read Re: Multiplication in R arises from ADDITION
Joe Niederberger
3/16/10
Read Re: Multiplication in R arises from ADDITION
Michael Paul Goldenberg
3/16/10
Read Re: Multiplication in R arises from ADDITION
Jonathan Groves
3/16/10
Read Re: Multiplication in R arises from ADDITION
Joe Niederberger
3/16/10
Read Re: Multiplication in R arises from ADDITION
Joe Niederberger
3/16/10
Read Re: Multiplication in R arises from ADDITION
Michael Paul Goldenberg
3/16/10
Read Math Terminology
Joe Niederberger
3/16/10
Read Re: Math Terminology
Michael Paul Goldenberg
3/16/10
Read Re: Math Terminology
kirby urner
3/16/10
Read Re: Multiplication in R arises from ADDITION
Paul A. Tanner III
3/16/10
Read Re: Multiplication in R arises from ADDITION
Joe Niederberger
3/17/10
Read Re: Multiplication in R arises from ADDITION
Joe Niederberger
3/18/10
Read Re: Multiplication in R arises from ADDITION
Michael Paul Goldenberg
3/19/10
Read Re: Multiplication in R arises from ADDITION
Paul A. Tanner III
3/16/10
Read Re: Multiplication in R arises from ADDITION
Paul A. Tanner III
3/16/10
Read Re: Multiplication in R arises from ADDITION
kirby urner
3/16/10
Read Re: Multiplication in R arises from ADDITION
Joshua Fisher
3/16/10
Read Re: Multiplication in R arises from ADDITION
Paul A. Tanner III
3/16/10
Read Re: Multiplication in R arises from ADDITION
Joshua Fisher
3/17/10
Read Re: Multiplication in R arises from ADDITION
Joshua Fisher
3/21/10
Read Re: Multiplication in R arises from ADDITION
Bill Marsh

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