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Is 0 (zero) a convention?

  1. Nov 18, 2011 #1
    Hi, not sure if this is the right forum.. pls move if not.

    Almost 30 years ago, I was studying engineering and my math tutor spoke about 0 (zero) being special. We talked about how nothing can be divided by zero, and sure enough, enter 1/0 into a calculator produces an error. (I vaguely recall learning this in high school).

    Something occurred to me and I asked him about it. Here's the steps I put in front of him:

    0/0 = 1
    0/1 = 0
    1/1 = 1
    1/0= error

    But 1/0 x 0/1 = 1

    I asked, how can this be. He could not give me an answer. I've asked a few math teachers since then and none could give me an answer.

    Is it simply a matter of convention, or is something else going on?
     
    Last edited: Nov 18, 2011
  2. jcsd
  3. Nov 18, 2011 #2

    D H

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    0/0 is not one. It has no meaning. It is undefined.
     
  4. Nov 18, 2011 #3

    micromass

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    In order to see why division by zero is undefined, you must consider the very definition of division. We say that m is divisible by n if and only if there is a unique integer k such that k*n=m. We then call k=m/n.

    But if n=0 and m is nonzero, then this reduces to k*n=0=m. This can never be satisfied. So there is no such thing as division by zero.

    What if m is zero? Certainly the equation holds then: if m=0 and n=0 then k*n=m reduces to 0=0. The problem is that k is not unique. So k=1,2,... all satisfies the equation. So we could potentially say that 0/0=1 or 0/0=2. Every possible value can be given to 0/0!! This is why we do not define 0/0.

    As for your equation

    [tex]\frac{1}{0}*\frac{0}{1}=1[/tex]

    Well it just isn't true. 1/0 is not defined, so the left hand side is undefined.
     
  5. Nov 18, 2011 #4
    Let us first consider what an inverse is. The inverse
    is a number with a certain property, namely if you multiply
    a number with its inverse you will get 1.

    Example 1:
    number=2
    inverse=1/2
    Proof: Multiplication yields
    number*inverse = 2*(1/2) = 1

    Example 2:
    number=37
    inverse=1/37
    Proof: number*inverse = 37*(1/37) = 1

    2 and 37 have inverses, which is equivalent to saying that 1/2 and 1/37 exist.

    ------


    Now, consider the number 0.
    Assumption: 0 has an inverse.
    This assumption is equivalent to saying that 1/0 exists.

    Then we have:

    Example 3:
    number=0
    inverse=1/0
    Proof: number*inverse = 0*(1/0) = 1

    But this is a contradiction to the fact that 0 multiplied by
    a number always equals 0. Therefore, our assumption that
    the number 1/0 exists is wrong.
     
  6. Nov 18, 2011 #5
    Thanks guys, that's actually helped and both explanations were easy to follow - the unique integer explanation and the inverse explanation make sense. As for 0/0, that explanation clarified my long held misconception, that anything divided by itself = 1, as taught in high school, because it needs to be qualified in either or both senses as you've explained, or more simply applied to numbers other than zero.
    :smile:

    hmm.. Followup question.. Is ∞/∞ = 1 true?
     
  7. Nov 18, 2011 #6

    D H

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    Undefined, for much the same reason. In one sense it can be any value, but in another sense saying that this has any specific value will lead to a contradiction.
     
  8. Nov 18, 2011 #7
    Makes sense :)
    thanks all
     
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