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Re: Non-Euclidean Arithmetic
Posted:
Sep 2, 2012 7:50 AM
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> Any of these repeated-addition definitions for product ab does not
> work in a field, especially for the set of all real numbers but even
> for the set of all rational numbers, unless we restrict the
> multiplication such that a or b is an integer.
>
Are we going to dive into this yet again? Maybe so, why not.
I'm happy to concede the "repeated addition" meme makes plenty of
sense in many a field, thanks to the distributive property, including
Q and R.
(a/b)(c/d) = a (bc/d) so you can always isolate an integer and then
say you're adding (bc/d) to itself a times. (bc/d) is just any member
of Q i.e. a rational. We're happy adding 2.99487 repeatedly. Jump to
Reals.
> It seems to me that this fact is what motivated Devlin to imply that
> at the very least we ought to come up with a general enough definition
> of multiplication that if not replaces then at least supplements these
> repeated-something "definitions" of binary multiplication.
>
I prefer an approach which isolates multiplication in a game that has
no concept of addition. You can still make sense of such a game.
And there's still "repeated multiplication" (of a thing by itself) and
that leads to powering, interesting in itself.
I also think Algebra is where we need a lot more multiplication and
addition modulo N. Computers make that easy and even fun.
We currently move from the GCD to LCM to four operations in Q too quickly.
We should pause at GCD and do more with "relatively prime" (stranger)
and "totatives" in Z, circling Fermat's Little Theorem.
Distilling totatives of N and involving those in modulo N operations
in Z is spiraling towards RSA, a point on the horizon for our digital
/ computational / integrated mathematics track ala STEM.
> That is, with such a general enough definition students would from the
> beginning have a mental model of binary multiplication that holds up
> on any set on which absolute value is defined. And this includes the
> natural numbers all the way up through the real numbers.
>
I think you'll find "repeated addition" can usually be stretched to
the reals by an avid teacher of that meme. The transition from Q to R
is quasi seamless.
Curriculum segments to look at more, and found oft times with the
physics track, have to do with wiring up English prepositions "in"
"per" "for" "from" and so on.
We say a particle m has momentum mv for a distance d, i.e. a car going
20 mph for one foot is using "for" in a multiplicative sense. Given v
(velocity), all this happens "in" or "per" a time interval (1/t).
So in physics we have momentum (mv) for a distance (d) giving
"action". "For d" implies a corresponding t i.e. a time frame for the
action to happen "in". mvd/t then provide our units of "energy".
Per time e.g. per second (1/t), then becomes a unit like hertz i.e.
frequency. Dividing by time is multiplying by frequency.
Seconds go by at rate f, like buckets or film frames, each containing
some action (mvd) -- call that "a scenario" (action packed).
Action at a frequency describes an energy level. Speed up the
frequency and more energy goes by. The notion of energy per time
interval, or "power" enters the story.
This is all basic STEM of course, Newtonian units, primitive mechanics.
E = h f where h is unit of action (mvd) and f = 1/t i.e. frequency.
E/t = power.
Since we're going to anchor a lot of calculus in this arena, it's
important to do lots of figuring around scenarios and their energy
transformations.
How sun energy gets impounded by means of photosynthesis and expended
through animal metabolism is at the backbone of STEM.
We link energy to currency ($) such that $ starts to come across as an
energy unit -- helpful when designing circuits that channel funds
(credits).
> His idea was to think of positive multiplication as scaling.
>
> That is, to visualize this idea, we could imagine that at point 0 on
> the number line we have a typical measuring tape machine such that we
> could pull out measuring tape attached to a spring such that if we let
> go, all the measuring tape gets pulled back into the machine.
>
There's no hard line between scaling and repeated addition, as
repeated addition "of what" may always be answered "of some distance".
One of the big redundancies of the Algebra curriculum is (x,y,z) is
painstakingly erected, usually with a lot of work in (x,y) before a z
axis is hoisted, yet with the concept of Vector not formally
introduced.
Then we go back over all the same material, this time with a thin
wrapper called the "vector" concept, and its added dot and cross
products.
More integration of "vector" with "length", and with the idea of an
"energy vector" (v has direction) would also tie in "edge" as in
"vertex, edge, face" in topology talk (V + F == E + 2 per Descartes
and Euler).
A directed energy flow along some edge: this is our model for
electrical circuits, even if the energy flow is alternating (back and
forth).
> For product xb, with can think of pulling the tape out to length b,
> and then pulling it out further from there by a scaling factor of x to
> a point represented by the product xb.
>
> Of course we have to explain what we mean by scaling, that we think of
> length b as a unit, but this is good thing: It creates a mental
> foundation for proportional reasoning very early on. (Using the number
> line as model regardless has the same effect, in my view. It helps the
> mind see the scaling inherent in the moving away from 0 when we are
> multiplying positive elements.)
>
Again, it's not important to pick fights between proponents of
"repeated addition" and "scaling vectors" as if these have to be
treated very differently.
You can "repeatedly add" a unit vector to itself 0.8899 times to get a
shorter "scaled" vector: Is that scaling or repeatedly adding? I'd
say there's no basic difference.
> Likewise we could think of positive division as the inverse of this:
>
> For division a/b for a = xb, with can think of starting at point a
> that is a product obtained from multiplying xb = a, and then letting
> the tape machine pull back the tape by an inverse scaling factor of b
> to arrive back at point x.
>
The complex numbers are a challenge to the repeated addition school as
we have a mental picture of adding c to itself 3 times i.e. the
scaling model is intact, but what is c + d?
How does vector addition relate to vector multiplication on the complex plane?
Complex number multiplication opens back on trigonometry as we
discover in Algebra 2. We get "clock hands" and rotation.
After fractals and the development of "chaos mathematics" did K-12 do
anything to adapt its complex number curriculum segments to this new
knowledge?
Were computers enlisted and fractals generated, per z = z * z + c,
thereby reinforcing ideas about * and + in the field C?
Amazingly enough, big wood pulp publishing was not hugely enthusiastic
about fractals. These happen on screens and screens are the big
competitor to the printed page.
Research idea: compare STEM curricula that are strong on Mandelbrot
Set versus those that are weak and see if there's a relationship to
how much spent on wood pulp textbooks.
> Side note: Yes, I think that it's a good idea to teach that a/b can
> always be viewed as (xb)/b for some x since dividend a always equals
> xb for some x, and I think that it's a good idea to teach that this x
> is where we end up on the number line (or more generally in the set of
> elements that is the domain of our variables) when we divide a or xb
> by b.
>
> Back to the point: What about negative numbers or 0? Even with them,
> the model I just gave above on the positive numbers still works - just
> apply the rules of signs to get there such that these rules would come
> into play after we taught positive multiplication and positive
> division.
>
Repeated addition has no real problem with doing the same things in the mirror.
> But Devlin I think had another beef with these repeated-something
> "definitions" of binary multiplication: The term "binary" should give
> way the beef. That is, for multiplication on a set on which absolute
> value is defined, for product xb, the term "binary" suggests that we
> have two elements x and b such that on the number line we should be
> able to use one of the elements x to get from the other element point
> b to the product point xb in just one step. That is,
> repeated-something "definitions" give us no mental model at all for
> how to get from b to xb in just one step.
>
I don't know that I'd start down this road, as v1 * v2 is typically
related to Area. Since v1 can be sliced up, v2 (slice + slice + slice
+ slice) becomes a picture of repeated addition.
We can start with v1 and v2 spread apart at 60 degrees instead of 90.
v2 starts at full length 10, while v1 is just 1 unit. The resulting
area, of triangle v1 * v2, is 1/10th.
Each time v1 gets 1 unit longer, another triangle is drawn. They all
have the same base and altitude, even though angles are varying, and
10 of them fill in an equilateral triangle of sides v1 == v2. Area =
1.
Doing these area and volume operations with two and three vectors,
getting containers of three and six edges respectively, is of course
spiraling towards tetravolumes, a concept which only a minority of
STEM teachers are tracking in 2012 (the more alert ones and/or better
positioned ones, on average).
> But even so, if we want a repeated-something model to include in our
> mix of models, I still prefer a somewhat broader alternative I've laid
> out here a few times here at math forum - and yes, it address the k-1
> problem. Here it is in action, explaining the right side in the
> identity p/q = p(1/q) for integer p and non-zero integer q - note that
> it allows using the generalized associative law for addition on a sum
> of n elements, x_1 + ... + x_n:
>
> Multiplication p(1/q) means to add together p instances of (1/q).
> (Yes, we can use the rules of signs to make this fit.)
>
Back to repeated addition once again.
> I like this language of adding together a number of instances of
> something - it does not require a particular ordering or grouping of
> the elements. (Yes, I think that what is meant with the term
> "instances" can be easily explained and understood.)
>
You come full circle to repeated addition.
"Instances" is a good word to use, as STEM brings in "objects and
events" pretty early, both in a generic sense and in the context of
theater / computer (actors / agents / workers / threads / processes/
scenarios).
An instance of a class or type is a special case thereof, whereas the
type itself is more the blueprint, the template.
dog = Dog( ) # a dog is born
is an expression of this "incarnation" relationship, assigning the
name 'dog' to the newly birthed "dog instance" (it came from Dog's
"birth method" which optionally accepts arguments).
Of course the syntax varies with notation, but most of them use "dot
notation" to then ascribe attributes to instances e.g. dog.name =
"Rover". dog.eat("bacon").
The use of "dot notation" as integral with other math notation is a
hallmark / signature of STEM in that software engineering is bleeding
over into math notation with this new use of "dot".
Internet domain names and the concept of containership, with dots
pointing "into", go together. When we say "dot com" we mean "within
the com top-level domain".
Kirby
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