Division by zero. Go ahead and laugh.

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

  1. If 1/0 is defined a logical contradiction follows, to wit, 1 = 0. If you
    like your systems to be consistent then forget about dividing by zero.

    The only place were dividing by zero makes any sense is in an algebra
    who only element is 0. Which is kind of useless.


    Bob Kolker
     
    robert j. kolker, Oct 8, 2004
    #81
    1. Advertisements

  2. How to make 1/x larger than N. Set pick x so that |x| < 1/N.

    For any large N there exists an x for which |1/x| > N. Obviously the x
    you choose is related to the given N.

    Bob Kolker
     
    robert j. kolker, Oct 8, 2004
    #82
    1. Advertisements

  3. The compactifier is not an element of R so it is the case that lim 1/x x
    goes to 0 does not exist in R, which was all that was asserted.
    Compactification of a system is a topological matter, not an algebraic
    matter. The compactifiers do not have the same algebraic properties as
    the normal members of the space. You can't have it both ways. Either you
    make things nice topologically and loose algebraically or vice verss.
    Either , but not both.

    Bob Kolker
     
    robert j. kolker, Oct 8, 2004
    #83
  4. No contradiction follows. But please, by all means, feel free to exhibit
    here how you think that 1 = 0 could be deduced after defining 1/0 in, say,
    C*. If you have in mind what I suspect you do, then you will have made
    an unwarranted assumption in order to get your contradiction. Namely, you
    will have incorrectly assumed at some point that 0 has a multiplicative
    inverse, thereby allowing factors of 0 to be cancelled. But of course
    that's nonsense; 0 cannot have a multiplicative inverse, and so cancelling
    factors of 0 would be unjustifiable.
    Wrong.

    David Cantrell
     
    David W. Cantrell, Oct 8, 2004
    #84
  5. Right. That's what I said at the beginning: "That's true in R ..."
    You snipped the part supplying possible context. But IIRC no particular
    system was specified, thereby making the assertion vague.
    Nor does 0, for that matter!
    You're entitled to your opinion about that. I don't share it.

    David Cantrell
     
    David W. Cantrell, Oct 8, 2004
    #85
  6. Hi David

    There is at least one other very different review of this book by a
    professional mathematician:

    http://www.maa.org/reviews/zero2.html

    The point I wanted to make is that the discussions around Zero and
    Infinity have a long history, perhaps as long as the history of
    humankind itself.

    I think it is legitimate to take up this discussion today, but one
    should at least be aware of the existing background. Even if Seifes
    book deserves the criticism by Gray, it could show the OP that perhaps
    s/he is not the first person in the world to be troubled about these
    issues.

    Also, I feel that there are two kinds of answers that are possible:

    - the answer that a professional mathematician gives (many examples
    in this thread)

    - a more historical and philosophical approach to the questions of
    infinity and zero

    Greetings
    Matthias Kläy
     
    Matthias Klaey, Oct 8, 2004
    #86
  7. Definition a/b = c if and only a = b*c. That is what division is.


    assume 1/0 = x. This means 1 = 0*x (definition of division).
    0*x = 0 (well known theorem for rings)
    hence 1 = 0.

    QED



    If you have in mind what I suspect you do, then you will have made
    Division means precisely multiplicative inverse. That is what it is.

    My proof stands.

    Bob Kolker
     
    robert j. kolker, Oct 8, 2004
    #87
  8. Not so. 0 + x = x for all x. 0 is the additive identity for the ring.

    1 is the multiplicative identity for the group consisting of all
    elements of the ring except 0.

    0 is just what it ought to be.
    I stated a mathematical fact, not a mathematical opinion.

    Bob Kolker
     
    robert j. kolker, Oct 8, 2004
    #88
  9. Hi Matthias

    Yes, that is indeed a very different review. I may have read it before,
    and forgot about it. In any event, my opinion is much closer to Gray's.
    If one wishes to read a book about zero, then I would suggest that
    Kaplan's _The Nothing That Is_ is substantially better than Seife's
    book.

    I agree with everything you said below.

    Best wishes,
    David Cantrell
     
    David W. Cantrell, Oct 8, 2004
    #89
  10. I agree with that, of course. But by your saying "Not so." you are claiming
    that 0 has "the same algebraic properties as the normal members of the
    space".

    Let's look at two examples:

    (1) x*y = x*z implies y = z
    That's true for all "normal members of the space"; it is not true if x is 0.

    (2) x has a multiplicative inverse
    That's true for all "normal members of the space"; it is not true if x is 0.

    So I can't imagine why you think that 0 has "the same algebraic properties
    as the normal members of the space". 0 is absorptive under multiplication;
    that's not "normal".
    Fascinating. I must have missed the math course in which "nice" and "loose"
    were rigourously defined.

    David Cantrell
     
    David W. Cantrell, Oct 9, 2004
    #90
  11. Lefty

    Lefty Guest

    If 1/0 is defined a logical contradiction follows, to wit, 1 = 0. If you

    Precisely.

    So what do we do ? We declare it to be "undefined", or "does not exist".

    What is really happening here is some type of disintegration of logical
    consistency at the singularity which is created by attempting to divide 1/0.

    Dont you find this amazing ? I am perplexed by this. I find it really
    incredible. Yes, you can define things so that the whole question just goes
    away. But, if you dare to attack it, there could be something to learn from
    it.
     
    Lefty, Oct 9, 2004
    #91
  12. Lefty

    Lefty Guest


    You did'nt even bother to read what I said. Or if you did, you certainly
    did'nt understand what I was saying.

    Where did I say that one must allow division by zero ? I did'nt. I said that
    it needs to be understood in the proper context instead of burying your head
    in the sand and pretending that it "does not exist". Clearly it DOES exist,
    otherwise we would not be having this conversation. Dont you agree ?

    Undoubtedly. And I also qualified that by stating that it must be "isolated"
    to prevent your consequences.

    Seriously, this conversation is just a little bit more than highschool level
    stuff.

    You can, if you wish, define an operator, number of other object which could
    be added to R such that arithmetic could be consistent even if you allow
    division by zero.

    For example, let @ be the object which represents inconsistency. Maybe @ is
    a number. Maybe it's not. I dont know. Maybe you should get creative, figure
    it out, then YOU can tell ME what it is.

    If @ was contained in R, or maybe an R' or something, then you could say
    that division is wellbehaved everywhere except that 1/0 = @. This can all be
    done by definition. Sure seems more realistic that just stating that things
    "do not exist" - as if -.





    You are being very closed minded to this. I am not going to pursue this for
    the next 5 years. I need quality feedback. If you want to criticize
    professionally then please continue. If you want to ridicule then please
    move on.
     
    Lefty, Oct 9, 2004
    #92
  13. Lefty

    Lefty Guest



    Now that was a good read !

    Pretty funny too !
     
    Lefty, Oct 9, 2004
    #93
  14. Lefty

    Lefty Guest


    Very hilarious stuff !
     
    Lefty, Oct 9, 2004
    #94
  15. Lefty

    Lefty Guest



    I'm irritated by a physics which consists of hodgepodge mathematical models
    which are basically things which are just pasted together from algebra.

    You have two separate worlds. The abstract world of math, and the physical
    universe. Math is traditionally exported from the abstract world into the
    real world in the form of models. I think that physics is suffering from
    this. Physics should be a mathematics in it's own right. And in thinking
    about this, it seems that the best place to start is some type of
    understanding of what a number is in the physical universe. Are there
    numbers which really exist and are not merely abstractions ? I think so. And
    I think that those numbers would be 1 and 0. You can have something, and
    then you can have the abscence of something. You must be able to define
    non-existence otherwise "everything" would exist. Clearly there are things
    which do not exist.

    If something exists you have 1 of that thing. You cannot have two, because
    no two things can be identical. You can only have one of any item.

    You can have zero of an item. This merely means that something does not
    exist in the physical universe.

    It is problematic to say that you have 1.5 inches, or 1/2 mile, or .75
    liters. These things are quite impossible. Maybe you can come close, but
    it's always just an illusion. You will never be able to obtain 2.5 inches of
    material in the physical universe, it is physically impossible to obtain
    _exactly_ 2.5 inches of material, for example.

    So you have 1 and zero. Two numbers which seem to "exist very precisely" in
    both the physical world and the abstract world. And this whole thing hinges
    on the fact that no two physical objects can ever be identical.

    If you acknowledge this as a fundamental axiom, you may be able to
    reconstruct physics as an axiomatized system instead of a hodge podge of
    algebraic contraptions.
     
    Lefty, Oct 9, 2004
    #95
  16. Lefty

    Chan-Ho Suh Guest

    It's not offered at all universities.
     
    Chan-Ho Suh, Oct 9, 2004
    #96
  17. Fascinating. So according to _your_ definition:

    Since 0 = 0*0, we must have 0/0 = 0. And
    since 0 = 0*1, we must also have 0/0 = 1.
    Therefore, since 0/0 equals both 0 and 1,
    by transitivity, we have 0 = 1.

    Seriously now: Of course I know the definition you intended to give. And
    there's clearly nothing wrong with using that definition of division in
    some systems. But you were responding to a statement concerning C*, for
    heaven's sake! The definition you intended to give is therefore
    inappropriate, not only to C* but to any system in which division of
    nonzero quantities by 0 is defined.
    No, that's obviously _not_ what it means in C*.
    No, in C* and other systems in which division of nonzero quantities by 0 is
    defined, division does not mean that.
    It never even got off the floor!

    David Cantrell
     
    David W. Cantrell, Oct 9, 2004
    #97
  18. 0 is quite normal. It is excluded from the multiplicative group.
    0 has a property that other numbers does not have. 0*x = 0 for all x.
    This follows from the distributive law of multiplication over addition.
    It is the distributive law that "pastes" together the additive group and
    the multiplicative group. 0*x = 0 essentially follows from the fact that

    (a - a)* x = ax - ax = 0. But a - a = 0 (definition of additive inverse)
    hence 0*x = 0. So blame zero's peculiarities on the distributive law.

    This is the way the numbers we use work. If you want to deal with things
    other than numbers, go ahead.

    If you find having consistency a burden go play with systems that are
    inconsistent.

    Bob Kolker
     
    robert j. kolker, Oct 9, 2004
    #98
  19. We declare -division by 0- as undefined. 0 itself is a perfectly nice
    number and it is special being the identity of the additive group.

    Bob Kolker
    There is no inconsistency. If division by 0 is allowed then 1 = 0 which
    means everything is equal to 0.
    The lesson to be learned is not to permit the rules of the system to
    lead to inconsistency. That is not the least bit astounding. It is just
    plain good sense.

    Bob Kolker
     
    robert j. kolker, Oct 9, 2004
    #99
  20. In order to do that you have to eliminated logical implication. Which
    would render mathematics totally useless. Do you really want to get rid
    of implication and inference? Mathematics is all about implication and
    infernce. It is all about drawing conclusions from the axioms of the
    system. What you propose is the elimination of mathematics in the entirety.

    I have told you why that is not possible if you want to do any
    mathematics. Why bother with axioms if you can't draw conclusions from
    the. Just write down isolated facts and forget any sort of logically
    hierarchical system.
    It isn't a number. It is isn't anything that makes logical sense. You
    want a symbolic expression of inconsistency? Here it is: A and not-A
    where A is any propositon. There is your @.

    Bob Kolker
     
    robert j. kolker, Oct 9, 2004
    1. Advertisements

Ask a Question

Want to reply to this thread or ask your own question?

You'll need to choose a username for the site, which only take a couple of moments (here). After that, you can post your question and our members will help you out.