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Topic: Bringing the Discussion to Order
Replies: 67   Last Post: May 1, 2010 1:03 AM

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 Paul A. Tanner III Posts: 5,920 Registered: 12/6/04
Re: Bringing the Discussion to Order
Posted: Apr 18, 2010 4:09 PM

Pam wrote:

> You are defining it according
> to how it works for whole numbers, but I am giving it a more
> universal definition:  To use RA to represent
> multiplication, we add to/subtract from 0 (with
> multiplication being movement away from 0 and division being
> movement toward 0) a series of numbers that are based on one
> of the factors.  In other words, if I multiply 1/2 by
> 2  1/2, I can add 0 + 1/2 + 1/2 + 1/4, but the 1/4 has
> a very important relationship to the 1/2.  Just as I
> expand my understanding of number to include parts of
> numbers, so, too, I expand my understanding of RA and
> multiplication.
>
> Alternately, I can do the multiplication/RA using an
> improper fraction:  1/2 * 5/2 = (1/2)/2 (what can I
> subtract two times from 1/2 to equal 0?), then add my result
> 5 times to 0.

You're applying part of the little field theorem I gave. (In a field like Q, dividing by a number is multiplying by the multiplicative inverse of that number.)

And you are doing as I said, with is that with "MIRA" or "MARA" (multiplication as repeated addition) in Q, you are using multiplication as part of the mix to represent multiplication. (Again, in a field like Q, dividing by a number is multiplying by the multiplicative inverse of that number.)

But with MIRA or MARA in N, there is none of this using multiplication to represent multiplication.

And so the fundamental jump from MIRA or MARA in N to "MIRA" or "MARA" in Q is that in N, multiplication is not used to represent multiplication, but in Q, it is. That is the fundamental change in the "redefinition" or changing of the meaning of MIRA or MARA.

I'm just saying that going from not having to use multiplication to represent multiplication to having to use multiplication to represent multiplication is just too much of a fundamental change in the meaning of MIRA or MARA, to much in the fundamental meaning of what it means to represent multiplication.

In other words, isn't it too much of a stretch to say that you're using the thing being represent
w numbers behave doesn't come from
> properties, properties come from how numbers behave, don't
> they?

Sometimes how numbers behave does come from properties. That's exactly what Devlin was saying when he talked about behaviors deriving from properties in the definitions of abstract structures like fields and ringoids - behaviors like repeated addition showing itself as a property of or a property associated with multiplication.

This ties back to what I said at the top, which is that you're using that little field theorem I gave without knowing that you're using it. Here's how what you do in an application of that theorem:

You are doing

(a/b)(c/d) = [(a/b)/d]c = [(a/b)c]/d

or, using RA notation,

(a/b)(c/d) = [(a/b)/d]_1 + ... + [(a/b)/d]_c = [(a/b)_1 + ... + (a/b)_c]/d

which, when we use the fact that division is multiplying by a multiplicative inverse:

(a/b)(c/d) = [(a/b)(1/d)]c = [(a/b)c](1/d)

or, using RA notation,

(a/b)(p/q) = [(a/b)(1/]_1 + ... + [(a/b)(1/d)]_c = [(a/b)_1 + ... + (a/b)_c](1/d)

Let (1/d) = t and c = n, and let c/d = y. Note that

y = c/d = c(1/d) = (1/d)_1 + ... + (1/d)_c = t_1 + ... + t_n

And, where a/b = x, the part of the little field theorem you are using is:

xy = x(t_1 + ... + t_n) = (xt)_1 + ... + (xt)_n = (x_1 + ... + x_n)t

(See further below this part of the theorem in its more fully stated context.)

So I repeat my point: Isn't it too much to call this multiplication xy being represented as repeated addition?

Even if it's OK to call it multiplication xy being represented as repeated addition, it still proves Devlin right when he says that how numbers behave in terms of the any type of repeated addition stuff derives from those properties in those definitions of those general algebraic structures called fields and ringoids, and that it is not because multiplication is somehow intrinsically addition, repeated or otherwise, because multiplication is not.

To try to make more sure that you see how all this repeated addition stuff derives from those properties in thos
nitions of those general algebraic structures, and how any pattern of repeated addition in the more general context of ringoids forms the basis for similar patterns in the less general context of fields, here again is that unfolding (apologies for all the extra or repeated information if it's too much for some):

(I know the following is "wordy" in terms of writing out lots of trivial algebra and so on, but I write it out anyway to try to make the unfolding more rather than less explicitly clear. My apologies to all put off by this "wordiness".)

A VERY LITTLE RINGOID THEOREM:

Let V be a ringoid with a multiplicative identity denoted u. Let all variables denote elements in V. Let W be the subset of V that is the set of all u + u.

For multiplication in V as V x V -> V, f : a,b -> f(a,b) = ab, we can consider subsets of this multiplication, replacing V with W:

(1) For V x W -> V, we have b = u + u and

ab = a(u + u) = au + au = a + a

(2) For W x W -> V, we have a = u + u and b = u + u and

ab = (u + u)(u + u) = (u + u)u + (u + u)u = (uu + uu) + (uu + uu) = (u + u) + (u + u)

(In (1) and (2) we already have a limited form of multiplication as "repeated addition" such that a product is equal to one of the factors summed. We also have in (2) an anticipation of the familiar identity in the natural numbers N of 2 x 2 = 2 + 2.)

A LITTLE RINGOID THEOREM:

Let V be a ringoid with a multiplicative identity denoted u and with addition being associative. Let all variables denote elements in V except k,m,n, which are natural numbers. For any element r, let r_1 = ... = r_k = r. Since we have included the additive associative property to have summation: Let W be the subset of V that is the set of all summations of u, the set of all u_1 + ... + u_k.

For multiplication in V as V x V -> V, f : a,b -> f(a,b) = ab, we can consider subsets of this multiplication, replacing V with W:

(1) For V x W -> V, we have b = u_1 + ... + u_n and

ab = a(u_1 + ... + u_n) = (au)_1 + ... + (au)_n = a_1 + ... + a_n

(2) For W x W -> V, we have a = u_1 + ... + u_m and b = u_1 + ... + u_n and

ab = (u_1 + ... + u_m)(u_1 + ... + u_n) = u_1(u_1 + ... + u_n) + ... + u_m(u_1 + ... + u_n) = u_1u_1 + ... + u_mu_n = u_1 + ... + u_{m*n}

(3) By (u_1 + ... + u_m)(u_1 + ... + u_n) = u_1 + ... + u_{m*n}, we have that W x W is closed, W x W -> W. And so W x W -> W is a subset of V x V -> V.

(We have, as a specific instance of (3), natural number multiplicationN x N -> N being a subset of real number multiplication R x R -> R. Some say that natural number multiplication should be specified axiomatically and taught in no way that reflects the fact that natural number multiplication is a subset of real number multiplication. I say that they should reconsider this in full view of this closure result at this most general level of ringoids; natural number multiplication being a subset of real number multiplication and all that that implies is far, far too important to be ignored, and should instead be embraced to the nth degree. One final note here: Natural number multiplication as in xk is not used to write these summations as in x_1 + ... + x_k because this theorem covers all ringoids with a multiplicative identity and associative under addition - natural numbers are not elements of all such ringoids.)

A LITTLE FIELD THEOREM:

Let F be a field with the multiplicative identity denoted u. Let all variables denote elements in F except k,m,n, which are natural numbers. For any element r, let r_1 = ... = r_k = r.

Since the set of all nonzero elements in a field is a multiplicative group, p = xq has a unique solution in x for all nonzero p,q. And so for all nonzero p and for all k such that such that u_1 + ... + u_k is nonzero there is some x such that

p = x(u_1 + ... + u_k) = (xu)_1 + ... + (xu)_k = x_1 + ... + x_k

Let G be the subset of F that is the set of all such p, the set of all nonzero elements in F that are summations of an element in F with the upper bound of summation k being such that u_1 + ... + u_k is nonzero.

For multiplication in F as F x F -> F, f : a,b -> f(a,b) = ab,
ith G:

(1) For F x G -> F, we have b = t_1 + ... + t_n and

ab = a(t_1 + ... + t_n) = (at)_1 + ... + (at)_n = (a_1 + ... + a_n)t

(2) For G x G -> F, we have a = s_1 + ... + s_m and b = t_1 + ... + t_n and

ab = (s_1 + ... + s_m)(t_1 + ... + t_n) = s_1(t_1 + ... + t_n) + ... + s_m(t_1 + ... + t_n) = s_1t_1 + ... + s_mt_n = (s*t)_1 + ... + (s*t)_{m*n}

(3) By (s_1 + ... + s_m)(t_1 + ... + t_n) = (s*t)_1 + ... + (s*t)_{m*n}, we have that G x G is closed, G x G -> G. And so G x G -> G is a subset of F x F -> F.

(The requirement that the upper bound of summation k being such that u_1 + ... + u_k is nonzero is to take into account that some fields have characteristic 0 and some fields have characteristic nonzero. The point is to stay for this theorem within the multiplicative group of a field.)

Message was edited by: Paul A. Tanner III

Date Subject Author
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