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Undifined 1/0 ?

  1. Nov 14, 2003 #1
    undifined 1/0 ?!!!

    why is anything over zero undefined?
    This is a question I have faced for a while and have not found an answer.
    Could you please help me?
     
  2. jcsd
  3. Nov 14, 2003 #2

    NateTG

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    It depends on how you define division.

    Usually
    a/b=x
    means that
    x*b=a

    Now, let's take a look at the case where b=0 and a=0;
    x*0=0.

    Clearly any x works.

    And in the case where b=0 and a is not zero:
    x*0=a
    Clearly no x works.

    Either way, there is no unique x so that the equation works, so it remains undefined.
     
  4. Nov 15, 2003 #3

    HallsofIvy

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    By the way, because of the distinction that NateTG noted,
    any x satisfies x*0= 0 but no x satisfies x*0= b for b non-zero,

    it is common to say that 0/0 is "undetermined" while b/0, for b non-zero, is "undefined".
     
  5. Dec 21, 2003 #4
    i thought it was...

    because... 1/ .01 = 100 and 1/ .00000001 = 1000000000 and so forth... and thus as the numbers get smaller on the bottom, the answer becomes higher... so much that 1/0 = undefined
     
  6. Dec 21, 2003 #5

    HallsofIvy

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    You were mistaken. If I define f(x) to be 1/x if x is not 0 and 1 if x=0, then it is also true that f(.1), f(.01), f(.00000001), etc get larger and larger but f(0) is not "undefined" (f is merely "discontinuous" at 0). 1/0 is "undefined" because there is no way to define it that does not violate some basic property of the real numbers and the definition of "/".
     
  7. Dec 22, 2003 #6
    Re: i thought it was...

    At the same time, I can say that:
    because... 1/ -.01 = -100 and 1/ -.00000001 = -1000000000 and so forth... and thus as the numbers get smaller on the bottom, the answer becomes lower... so much that 1/0 = undefined


    More formally,
    [tex]\lim_{x\uparrow 0}\frac{1}{x}\rightarrow -\infty[/tex]
    but also
    [tex]\lim_{x\downarrow 0}\frac{1}{x}\rightarrow +\infty[/tex]

    That's bad news! For me, that's the simplest argument for saying that 1/0 is undefined. Note that I can also get other answers, if you'd like I can probably find a limit such that
    [tex]\lim_{x\rightarrow 0}\frac{1}{x}\rightarrow -\sqrt{\frac{i\pi}{e}}[/tex]
     
  8. Dec 22, 2003 #7

    HallsofIvy

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    Why in the world would that be simpler? The original question is about basic arithmetic and has nothing to do with limits.

    1/0 is undefined because 1/0= x is equivalent to 1= 0*x which is not true for any x.

    I don't see how using limits on some specific sequence would be simpler than that!
     
  9. Dec 22, 2003 #8
    I guess it's a question of mindset and what you're used to. I find my argument simpler / clearer than yours, HallsofIvy, though I directly acknowledge that (many) others might disagree. That is why I included this argument: if people might have problems with your or NateTG's argument, then maybe the one I provided will give some them insight. If not, then no harm done. However, if you think it's stupid, then please feel free to remove my posts.
     
  10. Dec 22, 2003 #9

    HallsofIvy

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    Well, if you find limits easier than multiplication, you have a remarkable mind!
     
  11. Dec 22, 2003 #10

    NateTG

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    In some sense, limits are a much simpler concept, even if the formalisms associated with it are a bit more involved.

    On a conceptual level, I would say that limits are much simpler than multiplication.

    In practice, multiplication is much more usefull.
     
    Last edited: Dec 22, 2003
  12. Jan 5, 2004 #11
    and 1/infinity is equal to 1/0
    1/-infinity is equal to 1/0
    if i am incorrect please correct me
     
  13. Jan 5, 2004 #12

    HallsofIvy

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    No, I wouldn't say you are incorrect but you are not very precise (and in mathematics precision is essential!).

    "infinity" is not a standard real number and if you are talking about the real numbers, "1/infinity= 0" and "1/(-infinity)= 0" are short hand for "the limit of 1/x as x goes to infinity is 0" and "the limit of 1/x as x goes to -infinity is 0".

    If you are using one of the several "extended number systems" in which infinity is defined, then you should say so.

    That is one reason why it is not very good mathematics to say "1/0= infinity".
     
  14. Jan 5, 2004 #13
    So if zero is the equivalent of infinity and -infinity then the numberline as a whole is not just a straight line, but a loop that resembles the sign for infinity. (KUNUNDRUM?)

    I am treating infinity as a variable, and the number system explained and described above.
     
  15. Jan 5, 2004 #14

    NateTG

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    It's quite possible to do math with [tex]\infty[/tex] but it lacks some of the properties that are normally associated with numbers.

    For example, [tex]\intfy-\infty[/tex] and [tex]\frac{\infty}{\infty}[/tex] will cause problems.
     
  16. Jan 7, 2004 #15
    I see no problem with using negative infinity, but i do see what you mean for the tricky infinity, because if you multiply infinity by its tricky one then it will turn out as the tricky one.
    infinity multiplied by itself commes out as infinity. This is the problem with working within the boudaries of infinities, you have no room to move. That is why infinities do not work to explain the world around us.
    I do understand this consept, as you can see, but i do not understand using negative infinity as a problem.
     
    Last edited: Jan 7, 2004
  17. Jan 7, 2004 #16

    HallsofIvy

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    Would you mind telling us what in the world you mean by the "tricky infinity"?

    By the way, you copied what I said about infinity not being in the standard real numbers and then said
    How in the world did you get that "zero is the equivalent of infinity and - infinity" from what I said? No, zero is not the equivalent of infinity in any sense!
     
  18. Jan 8, 2004 #17
    sorry-- I didn't explain

    infinity divided by infinity
    it is an algebraic function

    I was merely visualizing the numberline as a whole.

    I isolated the denomenators
     
  19. Jan 8, 2004 #18

    HallsofIvy

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    No, it not a function at all. If you are talking about the regular real numbers then it is just a shorthand method of talking about a the limit of a quotient in which numerator and denominator each go to infinity. The result might be any number depending on the specific form of the numerator and denominator functions.

    I wasn't complaining about your saying talking about the "figure eight", I was talking about you quoting ms saying that infinity and -infinity not being real numbers and then saying "So if zero is the equivalent of infinity and -infinity" which is pretty much the opposite of what I said in the quote.

    Actually there are 2 standard ways of "extending" the real numbers. One adds a single infinity that is lies at both "ends" of the number line. That (called the "one point compactification") makes the extended number line geometrically equivalent to a circle.
    The other (the "Stone-Cech" compactification), more common, method adds +infinity at one end and -infinity at the other and makes the extended number system geometrically equivalent to a close line segment.

    No, 1/infinity is not equal to 1/0 and -1/infinity is not equal to 1/0.

    Sometimes you will see "1/infinity= 0" (not 1/0 !) or "1/0= infinity" but those are again "shorthand" for more complicated limit statements (and which I prefer not to use because they are so easily misunderstood).
     
  20. Jan 9, 2004 #19
    I do understand the mathematical deffinition of infinity now, and I do realize that infinity can not be treated, in any way like a variable.

    Somewhere down the line the denomenator would become infinitely small, and never reach 1/0,and the solution would be infinitely large, and not be able to be defined by any logical means. That means that 1/0 is not able to be defined by any progressive means. Other than the limit statements you were quoting about.
    I think that is a competent solution to the information given.

    I still have to do more research.
    I need to see those limit statements.
     
  21. Jan 13, 2004 #20
    Lets try arguing in not mathematical terms but logical reasoning. if a number x is divided by any number n, it means giving every individual n a single part from x... if our number is zero how are we going to divide x to zero individuals...? we have a x objects to be divided among no one... either the number stays as it is or a the number is gone. but the number cannot be gone because it was distributed to no one.. how is this???

    Please do not insult my argument, im just a high school student sharing my views in life!!
     
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