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0/0 = 1?

  1. Jan 10, 2007 #1
    I did a search first but didnt really find anything, this is something ive been wondering for a bit:

    Both my calculator and math teacher tell me I cant ever divide by zero, but what if you had 0/0 ? couldnt that work, and equal 1? or is there a case in which this would not work?
  2. jcsd
  3. Jan 10, 2007 #2


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    Division by 0 is not defined. It's tempting to say that since every other number divided by itself is 1, so should 0. But this would have some strange consequences. Consider the function
    If x is not 0, then y=1. So it seems natural that y should also be 1 at x=0. But now consider this function:
    When x=0, we also get y=0/0. But when x gets close to 0, y gets larger and larger and approaches infinity. It doesn't seem reasonable to say y=1 when x is zero. This is one reason division by 0 is left behind. Functions that approch 0/0 are important in the study of calculus.
  4. Jan 10, 2007 #3


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    In fact, given any real number r, we can find fractions f(x)/g(x) where both f and g have limit 0 (as x goes to a) but f(x)/g(x) goes to r.

    By the way, while we say that a/0, for a not 0, is "undefined", it is common to say that 0/0 is "undetermined". If we try to set a/0= x then we must have a= 0(x) which is impossible. On the other hand, if 0/0= x then we must have 0= 0(x) for which is true for all values of x.
  5. Jan 10, 2007 #4


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    Remember that EVERY fraction a/b can be seen as the product a*(1/b)

    The number (1/b) is that number which multiplied with b yields 1, i.e, b*(1/b)=1, by definition of 1/b.
    1/b is called the reciprocal of b.

    However, without resorting to any idea of reciprocals at all, we may prove that for ANY number a, we have 0*a=0.
    But that means that THERE CANNOT EXIST A NUMBER 1/0!!!

    Therefore, the reciprocal 1/b can only be defined for numbers not equal to zero.

    Thus, the expression 0/0=0*1/0 tries to do the impossible thing, namely multiplying together something that IS a number (0), and something that ISN't a number (1/0). But multiplication requires that both factors are, indeed, numbers..
  6. Jan 10, 2007 #5
    Thanx, i guess i was thinking too simply. I managed to dissprove myself:


    0/0 = (0*3)/0 = (0/0)*3

    And if 0/0 = 1, Then:

    1 = 1*3

  7. Jan 11, 2007 #6
    I never understood a thing... isn't right to say that:


    When i can't solve a problem, can i go to the prof saying:" the solution is 0/0!!!!"?
  8. Jan 11, 2007 #7

    matt grime

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    R is a set. Are you asserting that, what ever 0/0 is, it is a set?
  9. Jan 11, 2007 #8
    yes. for example, the limits in the indeterminate form, if it can be removed, they can assume any result in R you wish. or not?
  10. Jan 11, 2007 #9


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    In this case, you have devised a fantasy function having two real arguments going to some set of sets. Nothing wrong with that of course, except it hasn't anything to do with a BINARY OPERATION, like multiplication or division.
  11. Jan 11, 2007 #10

    Gib Z

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    Sure you could choose to have a function where you choose whatever value from the real numbers you want, but you will find picking the right number is pretty hard :P
  12. Jan 12, 2007 #11
    There's a problem in your second step =).

    It'd be (0*3)/(0*3). Even then, it'd be 0 in the end.

    But it's just fantasy math with a lot of problems/flaws in the reasoning heh.
    Last edited: Jan 13, 2007
  13. Jan 13, 2007 #12
    It doesn't matter if they are equivalent, since it is obviously the case that 0*3=0.
  14. Jan 13, 2007 #13
    Then you have the case of saying 0/0 = 1; when before you stated 0 = 0*3... Where in you have 0/0 = 3 :p. A lot of flaws.

    You can't switch things up like that just because you feel like it.
  15. Jan 13, 2007 #14
    Obviously there are flaws. Was that unexpected? Division by zero is not defined. 0=0*3 is a common fact that follows from axioms for the real numbers, nowhere did we say 0/0=3, but this "disproof" is based on an assumption that 0/0=1.

    If they are EQUAL you certainly can. You seem to be saying something along the lines of I can't interchange 22 and 4. despite the fact that they are the same thing.
  16. Jan 14, 2007 #15


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    The thread has seem to run its course, so I'm closing it.
    Last edited: Jan 15, 2007
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