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Multiplication and Addition

  1. Jan 17, 2004 #1
    Hi,

    Can someone please show some example of Qualitative difference between Addition and Multiplication, when using Boolean Logic?

    (by Qualitative difference I mean that Quantity remains unchanged
    during addition or multiplication operations between n positive integers).
     
    Last edited: Jan 17, 2004
  2. jcsd
  3. Jan 17, 2004 #2

    matt grime

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    Multiplication and addition of what?


    You have started this thread instead of answering any questions coherently in some of your other threads. None of the questions was to do with this; once more you are changing the subject and obfuscating the lack of intelligibility in your articles.

    If you wish to get some answers, you are going to have to restate the question with some more information. Exactly what do you mean by different? Is it sufficient to demonstrate there exist numbers x and y with x*y not equal to x+y for x and y natural numbers? How far back into the basics must we go? Do you wish for us to talk about cardinals instead of numerals? Must we deal with Peano's axioms?
     
  4. Jan 17, 2004 #3

    matt grime

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    And why are you posting this in physics/general theory?
     
  5. Jan 17, 2004 #4
    Last edited: Jan 17, 2004
  6. Jan 17, 2004 #5

    Hurkyl

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    This is a less inappropriate location than the math forums.
     
  7. Jan 17, 2004 #6

    Hurkyl

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    It might be instructive to point out that, group theoretically, the only difference between the axioms of addition and multiplication is the symbol used.

    And there is at least one "normal" example of addition and multiplication having the same form:

    Addition on the real numbers is isomorphic to multiplication on the positive real numbers.

    Or to put it in layman's terms, through the proper renaming of numbers and symbols, one can convert addition of real numbers into multiplication on positive real numbers. (and this scheme is reversible)

    One example of such a renaming scheme is to replace any number [itex]x[/itex] with the number [itex]2^x[/itex], and replace [itex]+[/itex] with [itex]*[/itex]. So, for example, using this scheme, we convert the equation

    [tex]1 + 3 = 4[/tex]

    into the equation

    [tex]2 * 8 = 16[/tex]
     
    Last edited: Jan 17, 2004
  8. Jan 17, 2004 #7
    Hi Hurkyl,

    Thank you for your answer.

    But please try to answer to my question that can be found in my first post.

    Thank you.


    Organic
     
  9. Jan 17, 2004 #8
    Are you asking for the difference between multiplication and addition when they both give the same result (such as how 2+2=4 and 2*2=4)?
     
  10. Jan 17, 2004 #9

    Hurkyl

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    Your question is too vague. Matt grime asked you to specify better what you mean. ("Multiplication and addition of what?") Until you answer it, I have to rely on my psychic powers to figure out what sort of response you will find interesting. (Apparently, they failed on my first attempt)
     
  11. Jan 18, 2004 #10
    And I wrote muliplication and addition between n positive interegs.

    What is so vague in it?

    Here my question again:

     
    Last edited: Jan 18, 2004
  12. Jan 18, 2004 #11

    Hurkyl

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    I was unaware you had edited your original post to provide your response, instead of saying it in a new post.


    There are several qualitative differences between addition and multiplication of positive integers (at least in the English sense of "qualitative"); for instance:

    there is no additive identity, but there is a multiplicative identity.

    Using addition, the positive integers can be generated by a single element: 1. (in layman's terms, every positive integer can be written using as a formula involving only "1" and "+")

    Using multiplication, the positive integers cannot be generated by any finite set of generators; any set of generators must contain every prime number.1


    There is an isomorphism between the positive integers under addition and a subset of the positive integers under multiplication. (Lots, actualy) However, the reverse is false.


    I'm just not really sure what you mean by this, though... what quantity remains unchanged? Why did you capitalize "Quantity" and "Qualitative"? By an "addition operation between n positive integers", do you mean something like [itex]a_1+a_2+\ldots +a_n[/itex] where each [itex]a_i[/itex] is a positive integer?


    1: (A technical note: due to the way "generate" is defined, any set of generators will generate "1"; even the empty set)
     
  13. Jan 18, 2004 #12
  14. Jan 18, 2004 #13
    "Addition is repeated multiplication???"

    A + B = C

    A = C - B

    B = C - A

    [C - B] + [C - A] = C

    For example:

    [C - B] = [C^[1/2] - B^[1/2]]*[C^[1/2] + B^[1/2]]

    = [C^[1/4] - B^[1/4]]*[C^[1/4] + B^[1/4]]*[C^[1/2] + B^[1/2]]

    = [C^[1/8] - B^[1/8]]*[C^[1/8] + B^[1/8]]*[C^[1/4] + B^[1/4]]*[C^[1/2] + B^[1/2]]

    = [C^[1/16] - B^[1/16]]*[C^[1/16] + B^[1/16]]*[C^[1/8] + B^[1/8]]*[C^[1/4] + B^[1/4]]*[C^[1/2] + B^[1/2]]

    etc... [C^[1/2^n] - B^[1/2^n]] = infinite product...

    [C - [C - B]] + [C - [C - A]] = C

    etc...

    = Infinite composition...
     
    Last edited: Jan 18, 2004
  15. Jan 18, 2004 #14

    Hurkyl

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    Unbounded, not infinite.


    It is true that:

    [tex]
    C - B = (C^{2^{-n}} - B^{2^{-n}}) \prod_{i=1}^n (C^{2^{-i}} + B^{2^{-i}})
    [/tex]

    where [itex]n[/itex] can be any positive integer. However, there is not an infinite version of this product;

    [tex]\prod_{i=1}^{\infty} (C^{2^{-i}} + B^{2^{-i}})[/tex]

    diverges to infinity. Also note that

    [tex](C^{2^{-n}} - B^{2^{-n}})[/tex]

    approaches zero as [itex]n[/itex] approaches infinity.


    The lesson: just because you can repeat a construction, process, or whatever an indefinite number of times does not guarantee that you can directly translate it into an infinite version of that construction, process, or whatever.
     
    Last edited: Jan 18, 2004
  16. Jan 18, 2004 #15
    [A^[1/2^n] - B^[1/2^n]]*[A^[1/2^n] + B^[1/2^n]] ...

    As n--->oo

    X^[1/2^n] = 1

    The expression approches [1 - 1]*[1 + 1] ---> 0*2

    n-->oo [A^[1/2^n] - B^[1/2^n]] = 1 - 1 = 0

    n-->oo [A^[1/2^n] + B^[1/2^n]] = 1 + 1 = 2


    At unbounded[infinite?] number, does mathematical existence become a self similar pattern?

    0202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202
     
  17. Jan 18, 2004 #16

    Hurkyl

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    I'm not really sure how this relates either to your previous text, your previous post, nor to Organic's post.
     
    Last edited: Jan 18, 2004
  18. Jan 18, 2004 #17
    [C - B] = [C^[1/2] - B^[1/2]]*[C^[1/2] + B^[1/2]]

    = [C^[1/4] - B^[1/4]]*[C^[1/4] + B^[1/4]]*[C^[1/2] + B^[1/2]]

    = [C^[1/8] - B^[1/8]]*[C^[1/8] + B^[1/8]]*[C^[1/4] + B^[1/4]]*[C^[1/2] + B^[1/2]]

    = [C^[1/16] - B^[1/16]]*[C^[1/16] + B^[1/16]]*[C^[1/8] + B^[1/8]]*[C^[1/4] + B^[1/4]]*[C^[1/2] + B^[1/2]]

    n--->oo

    A^[1/2]^n + B^[1/2^n] = 2

    n--->oo

    A^[1/2^n] - B^[1/2^n] = 0

    Product of [A^[1/2^n + B^[1/2^n]] becomes 2*2*2*2*2*2...

    [A^[1/2^n - B^[1/2^n]] becomes 0.

    0*2*2*2*2*2... = 0*2*0*2*0*2... = TFTFTFTFTFTFTFTF ?
     
  19. Jan 19, 2004 #18

    matt grime

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    Does this thread in anyway *explain* what you meant by qualitative, quantiative, change, or for that matter multiplication and addition.

    Why isn't division complementary to multiplication?
     
  20. Jan 19, 2004 #19
    Dear Matt,

    This is a good question.

    I did not research it yet, but in general there opposites that destroy each other results, like *,/ or +,- .

    In the case of *,+ I have found a very simple model where they complement each other, as you can see here:

    http://www.geocities.com/complementarytheory/ET.pdf

    It would be a great help if you research my model from *,/ point of view, and show us that my complementary model of *,+ can't hold.

    Can you do that?

    Thank you.

    Organic
     
  21. Jan 19, 2004 #20

    matt grime

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    I can read more of your stuff in an attempt to
    *understand* it (in fact I think I've read this particular one), but I will not. As I've explained to you at length, I do not agree with your statements about boolean logic in NewDiagonal.pdf: they are obviously wrong. It is from these fallacious statements you make there that you seemingly decided to write a whole slew of incoherent pieces without ever bothering to define any of the terms you use prefering instead to rely on the 'inuitive'. You have not answered any of the criticisms of your work and refused to explain the meaning of any of the rhetorical (possibly mendacious) figures you introduce. I have no professional interest in the foundations of set theory, and my personal interest is not in the correctness of *your* theory (one without rigour, or proper definitions) but in attempting to correct the mistakes you make in proper mathematics, i.e. one with rigour, and in trying to make you define *any* of the words you use. At the moment your pieces smack of personal philospophy, and not mathematics.
     
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