1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    LeonhardEuler

    User Avatar
    Gold Member

    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
    [tex]y=\frac{x}{x}[/tex]
    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:
    [tex]y=\frac{x}{x^3}[/tex]
    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

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    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

    arildno

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    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*3

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

    And if 0/0 = 1, Then:

    1 = 1*3

    1=3
     
  7. Jan 11, 2007 #6
    I never understood a thing... isn't right to say that:

    [tex]\frac{0}{0}=\mathbb{R}[/tex]

    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

    User Avatar
    Science Advisor
    Homework Helper

    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

    arildno

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    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

    User Avatar
    Homework Helper

    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

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    The thread has seem to run its course, so I'm closing it.
     
    Last edited: Jan 15, 2007
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: 0/0 = 1?
  1. Why 0^0 = 1 (Replies: 15)

  2. (x/0) / (x/0) = 1 (Replies: 42)

  3. 0! = 1 (Replies: 24)

  4. Does 0^0 = 1 (Replies: 106)

Loading...