On Tue, Sep 4, 2012 at 11:56 AM, kirby urner <email@example.com> wrote: > On Tue, Sep 4, 2012 at 3:33 AM, Paul Tanner <firstname.lastname@example.org> wrote: > >>> WIth the average kid, I can say (1/2)(1/3) is like starting with half >>> a third or 1/6, whereas (5/2)(1/3) is like 5(1/2)(1/3) or 5(1/6) i.e. >>> it's like 5 bucket fulls of marbles that are each 1/6 full. Since we >>> did (1/2)(1/3) earlier, it wasn't hard to turn (a/b)(c/d) into >>> (1/b)(c/d) first, and then multiply by a. (1/b) is taking an "nth" of >>> something (a fraction) whereas a(1/b)(c/d) is to repeatedly add >>> (c/bd). That's not stretching the meaning of repeated addition, just >>> breaking it down into steps. >> >> This most certainly stretching the original meaning. The original >> meaning is simply to take one of the given factors and add it to >> itself a number of times equal to 1 less than the other factor. We >> simply cannot do this with non-integer rationals. For them, we first >> have to derive two new factors to do what I just said. >> > > And what I'm saying is the "repeated addition" meme easily covers Q > and R. You disagree. I don't find your arguments persuasive. > >>> I don't think that's necessary to focus on in the case of >>> multiplication as repeated addition. >> >> It's necessary and inescapable because the irrationals and especially >> the transcendental irrationals are there, mucking up the repeated >> addition works. >> > > I don't think so. 3 * pi = pi + pi + pi. (1/3) * pi = adding a 3rd > of pi to 0 one time (not "repeated" addition, but sets the stage for > (2/3)*pi = pi * (1/3 + 1/3).
I do think so, since I'm talking about completely going into the irrationals. You're still using a rational as one of the factors. Let's see you do this when both factors are irrationals, especially non-algebraic ones.
> > Once you accept that repeated addition works in Q (which you don't, > you restrict it to Z integers whereas advocates of repeated addition > don't necessarily follow that restriction -- something you apparently > can't or won't understand) it's easy to generalize to R, regardless of > the fact that Q is but a subset of R.
You can't generalize it into that set that makes the reals different and at least an entire cardinality greater than the rationals, the irrationals, specifically the non-algebraic ones.
> >> Again, this does not appreciate just how different the irrationals and >> especially the transcendental irrationals are from the rationals. The >> irrationals are not merely repeating decimals like some rationals are >> in their decimal expansion. > > I don't think it's important or necessary to dwell on these > differences when talking about multiplication as repeated addition, > but our divergence comes earlier in you don't think Q is covered at > all.
It's covered if and only if repeated addition does not mean that you have to use as an addend one of the original factors, which is what the original meaning is in the naturals.
> > You say only Z is covered
With respect to the original meaning, that's right.
> and we need to switch to "scaling" for Q.
We need to show scaling right away even in the naturals if we want students to see the scaling that is inherent in the original meaning of repeated addition, where the original factors are used, and where the unit for the scaling is one of the original factors.
And we most certainly do need to show scaling for Q when we restrict ourselves to the original meaning of repeated addition, where one of the original factors is the repeated addend - we cannot apply the original meaning to Q.
> """Because the result of scaling by whole numbers can be thought of as > consisting of some number of copies of the original, whole-number > products greater than 1 can be computed by repeated addition; for > example, 3 multiplied by 4 (often said as "3 times 4") can be > calculated by adding 4 copies of 3 together.... Multiplication of > rational numbers (fractions) and real numbers is defined by systematic > generalization of this basic idea.""" > > http://en.wikipedia.org/wiki/Multiplication > > That's Wikipedia. I assume it represents a broad consensus. Notice > "Multiplication of Q and R is defined by systematic generalization of > this basic idea." (paraphrase). That's what I've been focusing on: > the generalization. You haven't been able to follow the > generalization, as you are stuck in Z, don't even make it to R.
You don't know what they had in mind when they were talking about this "generalization" - and besides, the article has a disclaimer by Wikipedia itself:
"his article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (April 2012)"
And so I don't accept the article as any kind of authoritative source.
Even the articles with no disclaimer should be taken (as I've said many times) only as good introductions to the subject.
>> irrationals are not even algebraic - meaning almost all of them are >> transcendental, where the conceptual difference between algebraic and >> non-algebraic numbers is absolutely fundamental. >> > > Again: even if there's a fundamental and important difference between > A and B, it's not a problem to dwell on the similarities of A and B > when it comes to making generalization X.
It is a problem when there exists no similarity that makes possible the generalization in the first place, since one is then dwelling on something that does not exist.
> > I agree with this author: > > """ > Now, the only problem with defining multiplication as repeated > addition comes when the definition of addition is incomplete. Anyone > saying that multiplication as a scaling factor and multiplication as > repeated addition are not the same thing seems to be making an > arbitrary point. Why limit the definition of addition? It can be > easily connected to scale because it is connected to scale. And > whoever decided that fractions and negative numbers are > counterexamples to the repeated addition explanation unfortunately > does not seem to understand either. > """ > > http://math4teaching.com/2011/12/14/can-we-define-multiplication-as-repeated-addition/
I don't for lots of reasons, and I notice that she leaves multiplication of two irrationals - especially two transcendental irrationals - out of the loop.
> >>> I think you're changing the subject, getting off onto something else. >>> >> >> No I'm not. See the below. >> > >> The differences are all-important if you are tying to salvage the >> (false) claim that multiplication is repeated addition in even the >> reals. > > Will you argue that scaling as an analogy or mental model works with > R, but repeated addition does not? > > You can scale a length e by pi (make it roughly 3 times longer).
That you have to introduce a "roughly" part demonstrates the truth of what I'm saying when both factors are irrationals.
When both factors are irrationals, now we have an even further stretch beyond the original meaning: Not only can we not use one of the original factors as the repeated addend, we cannot even talk about things precisely any more.
> > I'm arguing, like many writers, that scaling and repeated addition may > be seen as analogous / isomorphic. > > You can use volume: (4/5) * pi
Stop right there. We're back to having one of the original factors be a rational.
> I'm saying the "repeated addition" meme can be extended to Q and from > Q to R.
The extension cannot be made at all to R when we take into full account what R actually is: Almost all of R is made up of non-algebraic irrationals. This "generalization" to multiplication in R cannot be said to be made at all if the only way to make this "generalization" is to restrict the domain of one of the two factor variables to Q, which has a cardinality of at least an entire order lower than R.
And my points about cardinality are not irrelevant, my trying to change the subject as you say. The fact that the cardinal numbers for Q and R are different exposes how fundamentally different rationals and irrationals - especially a non-algebraic irrationals - are. The very nature of the proofs that Q is countable is such that we cannot apply the same reasoning to the irrationals we cannot do this because of the nature of each irrational - especially each transcendental irrational - is just not the same animal at all when we compare it to any rational. And it's the irrationals that make R.
The crux of all this is that those who think that "multiplication is repeated addition" is just fine for R seem to treat the irrationals as not all that important, just a side curiosity, just "those numbers over there" that "don't really count" for what is really important (which seems to be finite computation for those who think this way about multiplication), and all that. But again, it's the irrationals and especially the non-algebraic irrationals that make up almost all of R, and is the basis of calculus and much of everything else higher up in analysis and algebra and topology. (This "movement" of a tiny handful of people to ban irrationals is going nowhere, since it is not an answer to anything - it bans so much of analysis and algebra and topology. And so we see the "solution" for the repeated addition folks to the fact that repeated addition just cannot apply to two irrational factors is to, well, kill the messenger - ban those pesky irrationals causing the problem.)