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Zero divided by Zero

Discussion in 'Mathematics' started by AnotherMathsHoD, Mar 19, 2011.

  1. One way to deal with 0/0 which would probably make sense to Y5 student is to look at it like this:

    Let 0/0=a
    => 0=a * 0

    But we know that 0 times anything is zero so we can conclude that 'a' can be anything and therefore 0/0 zero can be any number, which makes no sense so instead we say it is undefined...
  2. Vince_Ulam

    Vince_Ulam Star commenter

    Gattengno was there first with pink and yellow marzipan confectionery. There’s a movie about it on YouTube.
  3. Nazard

    Nazard New commenter

    This is a nice thread. It raises an interesting question. There are some interesting responses too.
    The off-topic stuff means someone will reply. Then it will all get nasty. Then the thread will be unreadable because a couple of people will try to post on-topic but will get drowned out by the rest. And after that it will be deleted. And then there will be recriminations about why it was deleted.
    Please can this one stay on-topic?
  4. Vince_Ulam

    Vince_Ulam Star commenter

    Certainly, as you ask so nicely, but this is the thin end of the facilitation wedge.
  5. If you want to be more precise with this you can consider the limit of a*x/x as x tends to zero (which by L'Hopitals Rule is a). But as we didn't define 'a' we are again in the position of concluding that the limit of 0/0 is again any number...
    It's important here to avoid just looking at lim x/x as x tends to zero as it may lead to falsly assuming that 0/0 is 1, but that is just a special case!
  6. So
    2xy = 4x
    divide through by 2x
    y = 2
    is actually false, because x could be zero. That's how you construct proofs that two twos are five.

  7. Indeed the trick to avoiding this is to factorise instead so:
    => 2xy-4x=0
    => 2x(y-2)=0
    so either y=2 or x=0
  8. Vince_Ulam

    Vince_Ulam Star commenter

    The algebraic and analytical reasoning given so far is instrumentally valid but doesn’t address the necessary radical issues which may be touched upon in simple terms.
    Show your student the justifications given above but also tell them that their question is badly defined. They are treating “nothing” as zero but there’s no way of representing “no groups”. Trying to work with no-group containing zero is like trying to drink no-tea from a non-existent cup. Cups exist and have handles by which they can be picked up but a no-cup has no handle and so cannot be picked up. In the same way a no-group has no qualities which may be subject to arithmetical operations.
    Your student may not be ready for ontology itself but, keeping things in the classroom, they have a useful question. It provides you an opportunity to teach them that arithmetic is only a subset of rules about representation and is subject to other rules. Take this as far as your student‘s brain has legs

  9. wrldtrvlr123

    wrldtrvlr123 Occasional commenter

    There's a joke floating around here somewhere. How about a politician and a football player (pick your favorite groups to kick around) are last in line to be issued their brains. When it's their turn, god looks embarrassed and says, "Sorry, I have none left, so you lot will have to divide it between you".
    Apologies in advance.
  10. Vince_Ulam

    Vince_Ulam Star commenter

    Careful now, vicar; the post police are watching.

    Slice of Battenberg?
  11. Are we on the same lines as solving the following? (between 0<x<180 degrees):
    sin(x)cos(x) = cos(x)
    dividing through by cos(x)
    sin(x) = 1
    x = 90
    cos(90) = 0
  12. Yes! Exactly the same technique:
    sin(x)cos(x) = cos(x)
    =>sin(x)cos(x) - cos(x) = 0
    =>cos(x) (sin(x) - 1) = 0
    => cos(x)=0 or sin(x) =1
  13. Just wanted to say "Thanks" for the nice comment.
  14. I've said it before, but I'll say it again. On the early computers, division was done by repeated subtraction. In that sense division by zero was OK, but took rather a long time!
    If I remember rightly, the computer got into a state described as a 'dynamic stop'.
  15. Just pondering.....
    For some reason I wondered about powers with zero. ..what is zero to the power 0 ~ I think it's one.

  16. 0^x=0, but x^0 =1, so 0^0 = 0 or 1'
    Discuss amongst yourselves.
  17. if we divide zero by zero then it goes to undifinde similarly if we divide any number by zero then its become infinite
    => 0/0=infinity and 8/0=infinity, a/0=infinity
  18. This isn't really true, division by zero is much more subtle then that.
    Let 8/0 = k => 8 = k*0 but there is no k for which this is true so we say 8/0 is undefined
    However the limit of 8/x as x tends to 0 from above is +ve infinity; and the limit as x tends to zero from below is -ve infinity so generally it much safer (and more accurate) to say it is undefined
    However 0/0 is different. Let 0/0=k => 0 = k*0, which is true for any k, so you can show that 0/0 is any number you can think of, and again to avoid the confusion this would cause we say that 0/0 is undefined...

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