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Re: Non-Euclidean Arithmetic
Posted:
Sep 4, 2012 3:33 AM
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On Tue, Sep 4, 2012 at 12:00 AM, kirby urner <kirby.urner@gmail.com> wrote:
>>>>>
>>> It may seem like radical surgery to you but I assure you the avid
>>> teachers of the "repeated addition" meme took this step long ago.
>>
>> I'm talking about how it must seem to the average kid being hit with
>> this. If you think that having to break up one of the factors before
>> being able to apply the repeated addition model on two factors neither
>> of which is one of the two original factors is not for the average kid
>> a radical departure from being able to apply it directly to the two
>> original factors, then be my guest.
>>
>
> 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.
>>> You have an ungodly huge Avogadro Number of atoms and you divvy them
>>> so that 3.14159... of them (some trillions) get to be a "unit" in some
>>> way, with respect to some multiplier. Numbers like pi and e are just
>>> fractions that "never end" in this vague hand-wavy extension of Q to
>>> R.
>>
>> This does not appreciate at all what an irrational - especially a
>> transcendental irrational - is.
>>
>
> 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.
>
> We can say we have pi% (pi percent) of the balls, meaning 3.14159...% of them.
>
> That's pi% = pi * (1/100). it doesn't matter, in this context, that
> we approach a limit without being able to terminate the decimal
> expression (for pi). That's moving to a new topic, whereas when it
> comes to multiplication as repeated addition, we find that Reals (R)
> and Rationals {Q} don't present significantly different conceptual
> challenges.
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. No irrational has an infinitely recurring
finite block of digits in its infinite decimal expansion. Every
rational has an infinitely recurring finite block of digits starting
at some point after the decimal point in its infinite decimal
expansion. (For this latter I mean when we include such as 0 repeated
infinitely, as in for instance 0.1000...) That's an absolutely
fundamental difference conceptually, and this is just the tip of the
iceberg in terms of fundamental differences. Almost all of the
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.
>> I'd say that if there is any isolation going on here, it would be in
>> people being isolated from understanding just how fundamentally
>> different an animal an irrational number is from a rational number,
>> and this especially applies to transcendental irrationals, which are
>> so fundamentally different they are fundamentally different even in
>> comparison to algebraic irrationals.
>>
>
> I think you're changing the subject, getting off onto something else.
>
No I'm not. See the below.
>
> True, there are important differences between p/q rationals, members
> of Q, and numbers with no p/q expression (p, q both in Z).
>
> But when it comes to what is addition, and what is multiplication, I
> don't think the differences between Q and R are really that important.
>
The differences are all-important if you are tying to salvage the
(false) claim that multiplication is repeated addition in even the
reals. In Q, you can at least break down one of the factors to obtain
two new factors to apply the repeated addition model on. In R - namely
the irrationals, especially the transcendental irrationals, which make
up almost all the reals as I point out below - you can't do that at
all. And then there are the absolutely fundamental differences I
talked about above between the rationals and irrationals. Devlin said
and argued what he said and argued in part because of these
differences.
> Reals in the sense of
> infinitely precise numbers have no application in computing.
>
And that's where things go wrong when trying to funnel everything in
mathematics through a computer science bottleneck. So much of the
class of all objects studied by professional mathematicians just
cannot fit.
> We use
> floating point or extended precision decimal instead.
>
> Some mathematicians want to cast all mathematics in terms of rationals
> or at least tilt in that direction.
>
> Example: http://en.wikipedia.org/wiki/Rational_trigonometry
>
"Some" meaning "one and only one person who rejects most of
mathematics who's going nowhere with this."
(I checked out what's being said about him. It's not pretty.)
> Anyway, to wander off into the differences between Q and R is somewhat
> of a rhetorical fallacy if you're saying the similarities between Q
> and R, when it comes to multiplication as repeated addition, somehow
> should not be raised.
>
> Differences to not make similarities go away. Multiplying to
> rationals and multiplying two reals is not an appreciably different
> process,
It's not appreciably different in computer science since you are not
really dealing with irrationals at all in the first place - you are
dealing only with rational approximations.
>
>> Not only that, to see more of just how fundamentally different a
>> transcendental irrational actually is, the number of elements in the
>> set of all reals being an uncountable infinity and the set of all
>> reals being a continuum is due to the number of elements in the set of
>> all transcendental irrationals being an uncountable infinity. The
>> number of elements in the set of all algebraic numbers, which is the
>> union of the set of all rational numbers (all of which are algebraic)
>> and the set of all algebraic irrationals, is merely a countable
>> infinity, which means that this union cannot be a continuum. (A set is
>> a continuum only if it contains an uncountable infinity of elements.)
>>
>
> I see you wandering off into Cantorism but failing to make a point, as
> the differences between Q and R are not at issue here.
>
The point is that Devlin is right when he says that it is not the case
that multiplication in the reals is repeated addition, irrespective of
the fact that from using certain algebraic properties like the
distributive property we can derive models of multiplication as
repeated addition. I went over that before in my post
http://mathforum.org/kb/message.jspa?messageID=7883395
in this thread. That is, he's right that merely being able to derive
multiplication as repeated addition in the reals - as I did in that
post - does not mean that repeated addition is what multiplication is
in the reals. That is, the derivation is purely an artifact of the
algebraic properties, and has nothing to do with what multiplication
is. In the reals, it is as fundamental and elemental as addition.
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