Division by zero. Go ahead and laugh.

Discussion in 'Undergraduate Math' started by Lefty, Oct 7, 2004.

  1. Mathamtics may be hard at times and subtle at times, but it is NOT a
    mystery.

    Bob Kolker
     
    robert j. kolker, Oct 8, 2004
    #61
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  2. Yes. There is no such real number as "infinity".

    If you had said you can make |1/x| larger than any arbitrarily large
    (finite) number by choosing x small enough, that would be correct.


    Bob Kolker
     
    robert j. kolker, Oct 8, 2004
    #62
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  3. What are the algebraic properties of the compactification point. Can you
    do arithmentic on them. Is the compactification point a memeber of the
    additive group of complex numbers? Does the compactifier satisfy any
    cancellation laws? Is it a member of the multiplicative group of the
    complex number - {0}? Tell us how to do complex arithmetic with this
    added compactification point. The compactifier is a topological artifact
    to assure the convergence of all sequences. It does not fit in with the
    complex -field-.



    It's truly amazing that
    Learn the difference between algebraic properties and topological
    properties.

    Bob Kolker
     
    robert j. kolker, Oct 8, 2004
    #63
  4. So what is 0 * infinity? If you consider infinity a number you loose
    closure under multiplication. In short the augment complexes do not
    constitute a field.

    Bob Kolker
     
    robert j. kolker, Oct 8, 2004
    #64
  5. Lefty

    Keckman Guest

    Hah. x is still allways <>0. Then for every x>0 there is
    finite number that is 1/x, so 1/x can not make larger than any
    arbitrarily large finite number, because 1/x is finite number,
    if x>0.

    And math today is incorrect nonsense.
     
    Keckman, Oct 8, 2004
    #65
  6. Lefty

    Lefty Guest


    I understand what you are saying exactly.And, at the risk of sounding like
    an argumentative imbecile, I continue to press -

    - I am not saying that division by zero should be allowed. What I am saying
    is that if it were allowed, then you would have problems. Again, this is
    known. Division by zero must be isolated somehow so that arithmetic will
    remain intact. BUT - defining it to "not exist" is incorrect. When you
    divide by zero you are _not_ creating a "non-existence". What you are
    creating is something else, possibly a contradiction, paradox, or
    inconsistency. I think that this is due to topology of space/time where
    mathematician lives.

    There is currently no algebraic symbology to express this, and if there were
    it might lead to new understandings.
     
    Lefty, Oct 8, 2004
    #66
  7. Lefty

    Lefty Guest


    I'm going to take that and run with it.
     
    Lefty, Oct 8, 2004
    #67
  8. You might enjoy taking a look at:

    Wheels – On Division by Zero (thesis for the licentiate degree),
    Mathematical Structures in Computer Science 14(1):143–184, 2004
    by Jesper Carlström
     
    Anders Göransson, Oct 8, 2004
    #68
  9. I suspect those are merely rhetorical "questions". (Indeed, I suspect that
    you already knew that 1/0 is defined in C* and _chose_ to ignore that fact
    when you said, without qualification, that 1/0 cannot be defined.) For the
    answers to those questions, you may look at texts which discuss C*. You
    should find something like

    -(oo) = oo

    oo + z = z + oo = oo for all z in C

    oo * z = z * oo = oo for all nonzero z in C*

    z/oo = 0 for all z in C

    z/0 = oo for all nonzero z in C*

    [Note that oo+oo, 0*oo and oo*0 are often undefined in C*.]
    Of course not, because oo is not a member of C itself. But I think that
    what you intended to ask is whether C* is a group under addition. Well,
    of course it isn't, just as C is not a group under multiplication.
    No. Similarly, if z*0 = w*0, we cannot conclude that z = w.
    Like C itself, neither C* nor C* - {0} is a group under multiplication.
    But of course C* - {0, oo} is.
    I've already covered that. See above.

    If it disturbs you that some operations are often undefined in C*, so be
    it. The important thing for the purpose of this thread is that z/0 _is_
    defined for nonzero z. And if you knew that already but chose to ignore it
    when proclaiming, without qulaification, that 1/0 cannot be defined, then
    that tells us a good bit about you. (Are you perhaps thinking about a new
    career in politics?)
    Granted, I was the one who used the term "compactification" first in this
    thread. But topological properties need not be discussed at all. Perhaps
    you would have been happier if I had called C* merely the one-point
    extension of C.
    Right, in the same sense that 0 does not fit in with a multiplicative
    group.
    I already know the difference.

    David Cantrell
     
    David W. Cantrell, Oct 8, 2004
    #69
  10. Didn't seem worth my time, seeing how you have already been answered,
    but ignored the reply in order to repeat your inane and ignorant
    assertion. And here you did it yet again.
    You are saying words that do not mean anything.
    The expression "a/b=c" means one thing, and one thing only: it means
    that b*c = a. The reason "1/0" is undefined is that the equation
    0*c = 1 has no solutions. Simple. Nothing to do with laziness. Nothing
    to do with "things you don't want to do", nothing to do with "defining
    out of existence", nothing to do with "inconsistency", nothing to do
    with any of the big words you insist on using without knowing what
    they mean.
    0 is the additive identity of R. It is the ONLY additive identity of
    R. In any ring, the additive identity has the peculiar property that,
    multiplied by anything, it yields itself. So in any ring, the equation
    0*c = a has no solutions for a different from 0.

    In the integers, there are plenty of other divisions that are
    undefined: 1/2 is not defined in the integers.
    Of course I can. Are you capable of learning something from the
    responses, or are you just interested in making noise by saying stupid
    things?


    --
    ======================================================================
    "It's not denial. I'm just very selective about
    what I accept as reality."
    --- Calvin ("Calvin and Hobbes")
    ======================================================================

    Arturo Magidin
     
    Arturo Magidin, Oct 8, 2004
    #70
  11. [...]

    Hmmm... I think this is not quite the proper historical frame.

    Have a loook at the book

    Charls Seife: Zero: The Biography of a Dangerous Idea, Penguin Books,
    2000, ISBN: 0140296476

    http://www.powells.com/cgi-bin/biblio?show=trade+paper:sale+books:0140296476:5.98

    Hope you find something to think about there.
    Matthias Kläy
     
    Matthias Klaey, Oct 8, 2004
    #71
  12. Whether one chooses to call infinity "a number" seems largely immaterial.

    Although 0 * infinity is often left undefined, there are other
    possibilities, such as 0 * infinity = 0, which allow us to have closure
    under multiplication.
    Right. (And of course that's still true even if 0 * infinity is defined.)

    David Cantrell
     
    David W. Cantrell, Oct 8, 2004
    #72
  13. Lefty

    Randy Poe Guest

    Perhaps it is the quantifiers that confuse you.

    Pick an arbitrary finite number, say 10^100. "Make |1/x|
    larger than your choice" means that I can find x such
    that 1/x > 10^100. Do you believe that?

    Pick another finite number. I can again find 1/x
    larger than that.

    Nobody is claiming that EVERY 1/x is greater than
    EVERY finite number, or however you're interpreting that.
    Merely that that any particular specified, fixed finite
    value is exceeded by infinitely many choices of 1/x.

    Again, I don't know whether it is the English or the
    mathematical reasoning that is confusing you.

    - Randy
     
    Randy Poe, Oct 8, 2004
    #73
  14. Lefty

    Keckman Guest

    Error, error, should be:

    so 1/x can not make larger than any arbitrarily large finite number,
    because
    (1/x)+1 is finite number that is larger.
     
    Keckman, Oct 8, 2004
    #74
  15. Lefty

    Keckman Guest

    Yeah. You right. I was stupidly wrong.
     
    Keckman, Oct 8, 2004
    #75
  16. Lefty

    Keckman Guest

    An error again. Damned.
     
    Keckman, Oct 8, 2004
    #76
  17. Lots to think about! For example: How could such a poor book have sold so
    many copies?

    Better, read some of the reviews written by _mathematicians_ first, such as
    <http://www.ams.org/notices/200009/rev-gray.pdf>. Having done that, I hope
    that you will then not waste your time and money on Seife's book. (Alas, it
    was one of the books used by the court in making its ruling in a case cited
    earlier in this thread.)

    David Cantrell
     
    David W. Cantrell, Oct 8, 2004
    #77
  18. It's not necessarily undefined. See below.
    That's true in R and also true in its two-point extension. But in R*, the
    one-point extension of R, the sequence converges to oo, unsigned infinity.
    [Indeed, in constructing R* as the set of equivalence classes of
    appropriate rational sequences, oo is simply the equivalence class of those
    rational sequences which increase without bound in absolute value.]

    David Cantrell
     
    David W. Cantrell, Oct 8, 2004
    #78
  19. No it isn't. No one has found a contradiction in the theory of real or
    complex variables.

    Bob Kolker
     
    robert j. kolker, Oct 8, 2004
    #79
  20. If it were allowed the system would be -inconsistent- therefore useless.
    In an inconsistent mathematical system -anything- can be proved.

    Again, this is
    Impossible. When you allow something you also allow the consequence.
    Dividing by zero is meaningless.

    You are pissing up a rope. There is no way of making a meaningless
    comvibnation of symbols mean something in the context the theory.

    If you want an "arithmetic" in which zero division is allowed you must
    give up all the other things you like about arithmentic in order to be
    consistent.
    No it wouldn't. It would still be nonsens.

    This is not like trying to find the square root of minus one. This can
    be done by finding a broader algebraic structure in which negative
    numbers have square roots. There is no such extension to allow division
    by zero.

    Bob Kolker
     
    robert j. kolker, Oct 8, 2004
    #80
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