Division by zero. Go ahead and laugh.

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

  1. Lefty

    Lefty Guest

    No, I do not "divide by zero". But it did occur to me to mention it here as
    an illustration of mathematical culture.

    I have said before that mathematicians "do not like" paradoxes and
    inconsistencies. It's all quite natural that they wouldn't, and I don't
    blame them. But there are many who have voiced their disagreement with this
    opinion. Please let me qualify my belief.

    Division by zero creates an inconsistency in arithmetic, and any result can
    be derived if such a thing is allowed. Now then, instead of always
    stipulating for example, 1/x, x not 0, we simply state that 1/0 is
    "undefined", and avoid it completely. We avoid it like the plague. The only
    place I could find these things being treated seriously is in singularity
    theory. But in general, people do not even acknowledge that this is a
    singularity. They just state flatly that 1/0 "does not exist". And other
    indeterminate forms are treated pretty much the same.

    The fact that arithmetic falls apart when you let 1/0 is fascinating and
    amazing. There is no reason to relegate this operation to the trash can. It
    would be better to acknowledge the inconsistency, the fact that we don't
    really understand why it exists, and to simply work around it.


    The problem of division by zero is that it creates many paradoxes.
    Mathematicians hate this, and therefore they have defined division by zero
    to be "non-existent". Why didn't you guys just throw away zero altogether ?
    I'll tell you why. Because you want to keep zero around because it's a handy
    thing to have, but those damned paradoxes have got to go, so we'll just
    "define them away".

    It's the biggest cover up since Watergate.

    There are no paradoxes or inconsistencies in mathematics, unless you remove
    all the ad hoc stipulations and band aids which hold it all together.

    Confess.
     
    Lefty, Oct 7, 2004
    #1
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  2. Lefty

    Zodness A Guest

    Wrong. 1/0 = infinity

    Computers have problems with it, and they "hate" it.

    What does 0/0 = ? infinity or 1 or some number n ?

    What about (-1)^(1/2) ?
     
    Zodness A, Oct 7, 2004
    #2
    in2infinity likes this.
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  3. Lefty

    Lefty Guest

    Wrong. 1/0 = infinity



    .....and I thougth that _I_ was picking a fight .....(!) heh heh -
     
    Lefty, Oct 7, 2004
    #3
  4. Lefty

    bh Guest


    I know this is not a reasonable thing to say, but I will say it
    anyway... Why is x/0 considered 'undefined' instead of 'infinity'...
    or worse, it is considered to mean 0... ie 1/0=0, 2/0=0,
    400y^2/0=0.... If I divide 4 by 1, I take 1 out and that leaves 3,
    then I take it out again, that leaves 2, then again 1, and finally a
    4th time I get zero, and that means that 1 goes into 4 exactly 4
    times... By the same method, if I divide 4 by 0, I take 0 out once and
    have 4, twice 4, three times 4, to infinity... It is counter-intuitive
    to work toward a lower and lower divisor and get a higher and higher
    answer, only to finally reach 0 and it becomes 'undefined' or 0... how
    annoying... 4/1=4, 4/.1=40, 4/.01=400, and on and on... Oh well, I
    guess this rambling probably belonged in a philosophy forum instead...
     
    bh, Oct 7, 2004
    #4
  5. Lefty

    Brock Nash Guest

    I know the people who argue about it believe me
    1/0 is in fact defined! It is infinite. Think about it logically and
    you'll understand.
    Here's two reasons to support why Real#/0=Infinite
    1: If you graph the equation y=2/x, when x = 0 y falls at infinite!
    2: If you have 2 donuts and are trying to split them up into groups of
    0, you can make infinite groups of 0. Here's your group of zero
    donuts, here's yours....

    It seems so obvious, I hate when people say it is undefined!!!!
     
    Brock Nash, Oct 7, 2004
    #5
  6. in <in
    sci.math,alt.math.undergrad,alt.psychology:

    [...]
    This is simply incorrect. There is no inconsistency; see
    below.
    No. What you apparently don't quite understand is that
    division is a *defined* operation, defined in terms of
    multiplication. The statement that a/b = c means _by_
    _definition_ that bc = a. If the equation bx = a has no
    solution in the domain of the multiplication operation, then
    there simply is no c for which bc = a, and a/b is a fortiori
    undefined. There's nothing at all mysterious about this; it
    just happens that when the domain is the rational, real, or
    complex numbers, 0 is the only choice of b for which this
    happens. If you define division similarly in the ring of
    integers mod 10, say, you'll find that 5/2 is similarly
    undefined, because there is no solution to the congrence
    2x = 5 (mod 10).

    [...]

    Brian
     
    Brian M. Scott, Oct 7, 2004
    #6
  7. This was an interesting thing when reading about the great
    mathematician Euler. Evidently, Euler allowed 0/0 to equal anything
    he wanted. In fact, I read that he once said that 0/0 = 2 because
    "the first is twice the second". I thought that was quite amusing,
    and I am wondering how much of his prolific writings used 0/0.

    Brian
     
    Brian VanPelt, Oct 7, 2004
    #7
    in2infinity likes this.
  8. A couple of analogies come to mind.

    A singularity is a like hole in a function. Asking what the value of the
    function is in the hole is like asking what color a hole in a piece of paper
    is. There's no color there; it's undefined.

    1/0 is the answer to the question "How many trips to the well will it take
    to get 1 gallon if my bucket holds 0 gallons?" You can go there as many
    times as you wish but you won't get any closer to 1 gallon. In that sense,
    the answer does not exist.

    --Keith Lewis klewis {at} mitre.org
    The above may not (yet) represent the opinions of my employer.
     
    Keith A. Lewis, Oct 7, 2004
    #8
  9. Lefty

    Lefty Guest


    The inconsistency which I refer to is division by zero, itself. It has been
    defined to not exist, out of laziness.



    No kidding. In fact, maybe it _is_ multiplication - eh ? And we'd all be in
    trouble if it were'nt defined. But the fact that your definition must
    contort itself to avoid a catastrophic breakdown of arithmetic, this is very
    suspicious.

    You really have to bend the definition quite a bit to say that 1/x is ok
    everywhere in R, except at 0. We're waving a magic wand and simply claiming
    that it dosent exist, and so just please pretend that you didn't see it
    because it dosen't exist.

    Hmmmmmmmm.....
     
    Lefty, Oct 7, 2004
    #9
  10. Lefty

    Lefty Guest

    dated Thu, 07 Oct 2004 01:18:44 GMT:


    Very true. Or you could say that you were a billionaire for zero minutes.
    There are lots of wierd things like that.
     
    Lefty, Oct 7, 2004
    #10
  11. Division can be -- and of course very often is -- defined in that way.
    But in a system (such as that of the extended complex numbers, for example)
    in which division of nonzero quantities by zero _is_ defined, division is
    not defined in that way.

    Division of nonzero quantities by zero being defined in such a system does
    not lead to "paradoxes". But of course, if one were mistakenly to assume
    that 1/0 and 0 were multiplicative inverses, then one would indeed be in
    trouble quickly.

    David Cantrell
     
    David W. Cantrell, Oct 7, 2004
    #11
  12. This is a false statement, as I explained in the previous
    post. Evidently you did not understand the explanation.
    Did you even bother to read it?
    No, it is not.
    No, you do not. That is an immediate and obvious
    consequence of the definition. It's a very straightforward
    definition; the fact that you understand it is rather
    telling.
    No. Division has no existence independent of its definition
    as an inverse to multiplication. The symbol a/b is given
    meaning only by its definition. Its definition says that
    the statement 'a/b = c' is true if and only if a = bc. To
    put it even more simply, 'a/b = c' is by definition
    synonymous with 'a = bc', an expression whose meaning is
    already known. Consequently, if the equation a = bx has no
    solution, then there is no value of c for which the
    statement 'a/b = c' is true.

    In the ring of real numbers the equation a = bx has no
    solution if b is zero and a is not, and as a result there is
    no real number c that makes the statement '2/0 = c' true,
    *because* there is no real number c that makes the statement
    '2 = 0*c' true.

    In other rings, like Z_10, the integers modulo 10, there are
    other pairs of a and b for which a = bx has no solutions. I
    already gave the example of the congruence 5 = 2x (mod 10),
    which has no solutions, pointing out that for this reason
    the expression 5/2 is undefined in Z_10.

    But I'm wasting my time, since you obviously have no desire
    to be confused by the facts.

    [...]

    Oh, by the way: if you want at least to pretend to be
    serious, learn to mark your snips.

    Followups set.
     
    Brian M. Scott, Oct 7, 2004
    #12
  13. Brian, he's a troll.
     
    The World Wide Wade, Oct 7, 2004
    #13
  14. Have you met James Harris, by any chance?
     
    Richard Henry, Oct 7, 2004
    #14
  15. One must be careful in dealing with analogies. Consider the following:

    1/2 is the answer to the question "How many trips to the well will it
    take to get 1 gallon if my bucket holds 2 gallons?"

    Of course, that's absurd. It will take 1 trip, and yet we certainly don't
    want 1/2 = 1 !

    What I said above was largely in jest. But seriously, if one chooses a
    better scenario (similar in spirit to what you had in mind), it's easy to
    see why 1/0 is indeed infinite. Below my signature is a copy of a response
    of mine from an old thread.

    David Cantrell

    -------------------------------

    [Copy]
    It is true that the notion of division is normally
    linked with the notion of multiplicative inversion. That works beautifully
    if you restrict yourself to, say, the real number system and avoid
    division by zero. However, if we wish to be able to speak reasonably of
    dividing a nonzero value by zero, then we cannot restrict ourselves to
    the notion of multiplicative inversion (since zero cannot have a
    multiplicative inverse).

    Consider this very simple idea. Take a nondegenerate interval A on the
    real line. Say, A = [0, 36].
    How many intervals of length 18 are contained in A? 2

    N.B. Here and below, I am thinking of _distinct_ closed intervals,
    pairs of which would have at most one point in common. So, for the
    above example, the two intervals are [0, 18] and [18, 36].

    How many intervals of length 12 are contained in A? 3
    How many intervals of length 1 are contained in A? 36
    How many intervals of length 1/2 are contained in A? 72
    How many intervals of length 1/100 are contained in A? 3600
    How many intervals of length 1/1000000 are contained in A? 36000000
    How many intervals of length 0 (which are, of course then, degenerate
    intervals) are contained in A? This is essentially the same as asking
    how many points are in A. The answer, clearly, is that there are
    infinitely many, oo.

    Just as the second example above corresponds with 36/12 = 3,
    the last example corresponds with 36/0 = oo. Very simple, and not
    based on the notion of multiplicative inversion. (But I wouldn't call it
    "mathematical magic" either. :)

    Note that, despite what is often said, taking x/0 = oo for nonzero x
    does not lead to contradictions -- that is, _if_ you know what you're
    doing. Of course, for those who don't know what they're doing, all sorts
    of things can seem to lead to paradoxes.
    [End copy]
     
    David W. Cantrell, Oct 7, 2004
    #15
  16. Lefty

    Lewis Mammel Guest

    I think you've got a problem here.
     
    Lewis Mammel, Oct 7, 2004
    #16
  17. Lefty

    Jon Haugsand Guest

    *
    What color has number 10? What is the least sad number? There are a
    lot of operations that the mathematicans "avoid like the plague".
     
    Jon Haugsand, Oct 7, 2004
    #17
  18. Lefty

    Norm Dresner Guest

    A properly programmed modern computer that uses the right representation
    system for numbers can produce the results
    1.0 / 0.0 = "infinity"
    1.0 / "infinity" = 0.0 IIRC
    and it will also give NAN (not-a-number) in most appropriate situations
    too.

    It all depends on how it's programmed. The hardware could care less.


    Norm
     
    Norm Dresner, Oct 7, 2004
    #18
  19. --
    Richard.


    "I have yet to see any problem, however complicated, which when looked at in
    the right way, did not become still more complicated"
    Poul Anderson


    In this case the calculation should be carried out in the ring of Intigers,
    trips can only be whole numbers, and the rules of division should be framed
    accordingly, i.e. that odd / even = (odd + 1) /even.
     
    Richard Wrigley, Oct 7, 2004
    #19
  20. Lefty

    Justin Guest

    : It's the biggest cover up since Watergate.
    ....
    : Confess.

    You're either stupid or trolling. I would venture the latter, but you
    are free to suggest it may be the former.

    Best,
    Justin
     
    Justin, Oct 7, 2004
    #20
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