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January 5th, 2010, 03:07 PM   #1
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Expanding the set of Integers to include more than pos/negs

As with my last threads, and the existence of mathematical branches to explain them, I have another question that will probably have an answer. I hope someone can point me towards the name of a field that applies to it.

In the set of integers a number has a certain property (I use this term in a purely referential sense), and that property is negativity or positivity. Since the construction of these properties is based off of addition and subtraction, why aren't these properties expanded to include multiplication, division and exponential, root, etc...

I may better explain using typical mathematical examples:
*a * (/a) = 0 or traditionally seen in integers: a + (-a) = 0
*a ^ (/b) = /c or: a * (-b) = -c, or a negative and a positive multiplied make a negative, and therefore a multiplicative put to the exponent of a divisive makes a divisive.
Essentially all the rules of integers would be in effect (though perhaps exceptions that I have not noted will need to be made), but to a more complicated extent.

This is a little flimsy so far as I have explained it, and my model already assumes a lot of things. I originally developed this idea with the M notation (an independent variation of the hyper operator) I was using earlier--and noted in the other thread, and it inferred some very strange things. Commutativity in exponents for example is traditionally not seen, but with divisive and multiplicative properties we do see it. The notation creates a heirarchy of numerals (what we traditionally list as numbers) and propreital numerals (or numbers of properties) and includes them together--by this I mean to infer [...'/'=-2, '-'=-1, '+'=1, '*'=2...], and that mathematics occurs on two levels. Such an idea can be viewed in a case as such: /a + (-b) = ? or What happens when you add a divisive and a negative number, or perform an operation on two numbers with properties not of the same symmetry? (where symmetry is when properties have the same absolute value (again: [...'/'=-2, '-'=-1, '+'=1, '*'=2...], and therefore +:- are symmetrical)).
Many variables get added in producing the end property of a sum in any equation: including the operator's level (?n) and its difference between the levels of each property (such as: ^6 ?7 -5, exponential six acting with the seventh level hyper operator on negative five).
There is a very intimate connection with hyper operators in this 'set.' It is easy to picture the set when you envision it as 'hyper integers' or 'hyper properties.' This sounds really complicated so I'll stop and continue in later replies if there are any questions.

I'd also like to point out I am aware of the need for a 'log' property, it would be very difficult to handle this, and I do not want to address it in this post.

It is very difficult to imagine this idea because it defies the traditional economic layout of mathematics we created. We usually visualize numbers as being linear, and expressable geometrically on a cartesian plane, but this disallows that.
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January 5th, 2010, 05:26 PM   #2
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Re: Expanding the set of Integers to include more than pos/negs

Quote:
Originally Posted by jamesuminator
It is very difficult to imagine this idea because it defies the traditional economic layout of mathematics we created.
This time, instead of referencing abstract algebra (still slightly relevant), I'll reference foundations of mathematics/formal logic/set theory.


[Here I originally had a point-by-point description of Peano arithmetic; I decided to delete it in favor of a less formal discussion.]

Let's start with the theory of addition on the natural numbers N = {0, 1, 2, ...}. Then define a structure (a, b) where a in in N and b is in {+, -}. Let (a, b) = (c, d) if a = b and c = d, but also let (0, b) = (0, d) regardless of b and d. (Otherwise they are unequal.) Then define (a, b) + (c, d) as (a + c, b) if b = d and (a, b) + (a, d) = (0, a) if b ? d.

This should be enough to give addition on the integers. The ordinary relations <, >, etc. can be defined in the obvious way. And of course we can then treat the whole structure (a, b) as a single number. Call the set of such numbers Z, as usual.


So now you're asking about extending this to a structure (n, a) with n in Z and a in {*, /}. I imagine you want equality to be (m, a) = (n, b) iff [(m = n and a = b) or m = n = 1], and multiplication to be (m, a) * (n, b) = (mn, a) if a = b and (m, a) * (m, b) = (1, a) if a ? b. Is that right?
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January 6th, 2010, 02:43 PM   #3
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Re: Expanding the set of Integers to include more than pos/negs

Yes, this is exactly what I mean; you express it far better than I do.

Edit:
Actually, I'm willing to argue another set if this one proves to have faults, such that:
m is in the set of N, and c is {/,*}, m = n and c = d; or as you described it.
(m, c) = (n, d); only if m = n, and c = d.
(m, c) * (n, d) = (m+n, c) such that c = d.
And the main conjecture:
(m, c) * (m, d) = (0,--) such that d =/= d.

This is clearly less complex, but I will let you continue with the first manner of description and consider the latter afterwards.
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January 7th, 2010, 05:32 AM   #4
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Re: Expanding the set of Integers to include more than pos/negs

Quote:
Originally Posted by jamesuminator
Actually, I'm willing to argue another set if this one proves to have faults, such that:
OK, let's go!

Quote:
Originally Posted by jamesuminator
(m, c) = (n, d); only if m = n, and c = d.
In particular, you have (1, /) ? (1, *).

Quote:
Originally Posted by jamesuminator
(m, c) * (n, d) = (m+n, c) such that c = d.
This is quite odd; is it what you intended? I would have expected
(m, c) * (n, d) = (m*n, c) if c = d
which is different in two places.

Quote:
Originally Posted by jamesuminator
And the main conjecture:
(m, c) * (m, d) = (0,--) such that d =/= d.
??

(0, --) is not a number in your system (call it J), since -- is not in {/, *}.
d ? d is always true.
You haven't defined what (m, c) * (n, d) is for c ? d, so nothing can be determined about that case.
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January 7th, 2010, 12:44 PM   #5
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Re: Expanding the set of Integers to include more than pos/negs

Quote:
In particular, you have (1, /) ? (1, *).
Yes, in the same way -1 =/= 1.

Quote:
This is quite odd; is it what you intended? I would have expected
(m, c) * (n, d) = (m*n, c) if c = d
which is different in two places.
This is an alternative method of developing the set; and may prove to be the better way given occam's razor.
It's easy to invision it as: when a number is divisive or multiplicative {*, /}, multiplication is to it, as addition is to integers {+, -}.

Quote:
d ? d is always true.
Sorry, it was c =/= d; typo.

Quote:
(0, --) is not a number in your system (call it J), since -- is not in {/, *}.
By that I just meant the number 0; and since 0 can have either/or properties I used that blank thing. It should be assumed to be in the system and its property is simply arbitrary: * or / works just as fine.

Quote:
You haven't defined what (m, c) * (n, d) is for c ? d, so nothing can be determined about that case.
(m, c) * (n, d) = (m-n, d) where c =/= d.
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January 7th, 2010, 02:31 PM   #6
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Re: Expanding the set of Integers to include more than pos/negs

OK, let me see if we're communicating. (I'm not entirely sure I understand you, or that you've communicated all you need.)

J is the set of ordered pairs (m, c) with m in N = {0, 1, 2, ...} and c = {/, *}.
For (m, c) and (n, d) in J, (m, c) = (n, d) iff m = n and c = d.
(m, c) * (n, c) = (m+n, c)
(m, c) * (n, d) = (m-n, d) where c ? d.

Is this right? (Please check carefully, because I'll make conclusions based on exactly what is written.)
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January 7th, 2010, 02:45 PM   #7
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Re: Expanding the set of Integers to include more than pos/negs

Yes exactly.
It is also worth noting,
(m, c) / (n, d) = (m-n, c) if c = d
(m, c) / (n, d) = (m+n, d) if c =/= d
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January 7th, 2010, 03:03 PM   #8
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Re: Expanding the set of Integers to include more than pos/negs

My first observation is that "=" is not an equivalence relation on J, since

(1, *) * (1, /) = (0, /) by Axiom 3
(1, /) * (1. *) = (0, *) by Axiom 3
(0, /) ? (0, *) by Axiom 1
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January 9th, 2010, 11:16 AM   #9
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Re: Expanding the set of Integers to include more than pos/negs

Hmm, I see what is wrong now.

I seem to have misappropriated something. Let me reanalyze:
(m, c) * (n, d) = (m-n, c) iff (c =/= d, and m > n)
(m, c) * (n, d) = (m-n, d) iff (c =/= d, and m < n)

such that:
(5, *) * (3, /) = (5-3, *) = (2, *)
(5, *) * (7, /) = (5-7, *) = (-2, *) = (2, /)

So therefore it would instill that c & d are dependent on the values of m & n.

You really have to imagine it as a whole set, so let me explain this using a general grammar.
M(0) is successorship, M(1) is addition, and M(-1) is subtraction, etc etc...

Let m & n belong to the set of N; and let c & d belong to the same set of indexes as M(n), such that {...-2 : /, -1 : -, 0 is successorship property, 1 : +... etc etc}; and let x belong to the set of Z.

(m, c) M(x) (n, d) = (m+n, c) iff (c = d = x)
eg; (4, 1:+) M(1) (3, 1) = (7, 1) = 4 + 3 = 7

(m, c) M(x) (n, d) = (m-n, c) iff (c =/= d, (c = |d| = x) and m>n)
eg; (6, 2) M(2) (3, -2) = (6-3, 2) = (3, 2) = (6, *) * (3, /) = (6-3, *) = (3, *)

(m, c) M(x) (n, d) = (m-n, d) iff (c =/= d, (c = |d| = x) and m<n)
eg; (6, 2) M(2) (8, -2) = (8-6, 2) = (-2, 2) = (2, -2) = (6, *) * (8, /) = (2, /)

Essentially the rule is that if the operator is of the same order as the properties (i.e: multiplicative/divisive property are of the same order as multiplication and division) we adopt the same rules as integers for operation.

I'm hoping this is beginning to form a picture? Once you get passed this you can start to reach really complicated things like:
(m, c) M(x) (n, d) where c =/= d =/= x, c =/= |d|
eg; (5, 3) M(5) (4, -2) = ?

And then you'd have to add the log operator when x >= (c + 2) >= (d + 2).

But let's not get into that yet, let's try to understand the most basic rules.
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January 9th, 2010, 08:25 PM   #10
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Re: Expanding the set of Integers to include more than pos/negs

So now you have:

J is the set of ordered pairs (m, c) with m in N = {0, 1, 2, ...} and c = {/, *}.
1. (m, c) = (n, d) iff m = n and c = d.
2. (m, c) * (n, c) = (m+n, c)
3. (m, c) * (n, d) = (m-n, c) iff (c ? d, and m > n)
4. (m, c) * (n, d) = (m-n, d) iff (c ? d, and m < n)

Comment 1: I notice that (m, *) * (m, /) is undefined. Is this intentional?

Quote:
Originally Posted by jamesuminator
(5, *) * (7, /) = (5-7, *) = (-2, *) = (2, /)
Comment 2: The first equality does not follow from your axioms; it should be
(5, *) * (7, /) = (-2, /)

Comment 3: (-2, /) is not a member of J, and thus * as you have defined it is not a map from J x J to J. Is this what you intended?

Comment 4: (-2, *) ? (2, /) by Axiom 1. You wrote the opposite; what did you intend?
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