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Yes, I know, can't be answered, blah, blah, blah.... but here are a few of my theories. I know, plenty of other questions like this, but before marking this as a duplicate, consider this, my mathematical friends:

  1. We know that $ x/x = 1 $.
  2. We also know that $ 0/x = 0 $.

Then, considering these to facts, $$ 0/0 $$ could be

  1. $ 1 $, because $ x/x = 1 $
  2. or $ 0 $, because $ 0/x = 0 $

Then, of course we can consider $ x/0 $, where $ x $ can be any real number. Then $ 0/0 $ can be $ 0 $ or $ 1 $, and then the rest according to theory, is $ \infty $ or $ \text {undefined} $. Then how come we say $ \dfrac {x}{0} = \infty $? Or is this only true being $ x \neq 0 $?

Or is it that just for practical sense, we just say that $ x/0 = \infty $? Is it because it is just because it is not computationally possible? Or is it just because $$ \dfrac {5}{\dfrac {1}{2}} = 5(\dfrac {2}{1}) = 10 $$ and then $$ \dfrac {5}{\dfrac {1}{100}} = 500 $$ and then $$ \dfrac {5}{\dfrac {1}{100000}} = 500000 $$ and so the numbers keep on going to $ \infty $ as we get closer to $ 0 $?

I know there are lots of possible answers but then the theory that $ \dfrac {x}{0} = \infty $ even though $ \dfrac {0}{0} $ could be 1 or 0, that just does not make sense to me. I will appreciate any answers / at lease possible answers, because I understand that this is just a very controversial topic of mathematics.

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    $x/x=1$ only if $x\ne 0$. $0/x=0$ only if $x\ne 0$. Nobody defines $x/0=\infty $ for any value of $x$. Beside, $\infty $ is not a number. – Ittay Weiss Aug 17 '14 at 08:24
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    This is not a controversial question--it's not a question at all, except, perhaps, of armchair philosophy. It says on your profile that you program in C. If someone said to you "I am here today to ask the following controversial question about operators in C. What is $\pi % 7$?" It's not controversial, it just doesn't make sense. You could claim that that $\pi\text{%}7=4$, and perhaps even make a good argument. But, that's not answering the question, it's creating a new system whose usefulness/validity need to be argued. – Alex Youcis Aug 17 '14 at 08:26
  • I know that $ \infty $ is not a number, but it is just a representation of the first number in the set of transfinite numbers. It's just that it is a convenient way to represent some very big unknown value. –  Aug 17 '14 at 09:30
  • Maybe because it can be defined in many different ways and none of this definitions would be useful. Unlike $1+2+\cdots=-1/12$, where multiple definitions yields to the same result and have a practical results. – m0nhawk Aug 17 '14 at 10:04
  • I just had this thought: if you say that $$ \lim_{x \to 0} \dfrac {1}{x} = \infty $$ going by the same analogy, since $ \dfrac {1}{0} = \infty $ then $ \dfrac {2}{0} = \infty $. And then you would get nonsense like $ 1 = 2 $, which is obviously not the case. –  Aug 17 '14 at 10:27
  • But then, in the same limit: $$ \lim_{x \to 0} \dfrac {1}{x} = \infty $$ you could graph it (one axis $ x $ and the other $ \dfrac {1}{x} $), you approach 0 from the positive side, and it gets closer and closer to $ \infty $, which is what the limit said. But then if you approach 0 from the negative side of the number line, you would get closer and closer to... $ -\infty $? Then the limit would become: $$ \lim_{x \to 0} \dfrac {1}{x} = \infty \text {or} -\infty $$ which totally does not make sense. –  Aug 17 '14 at 10:34
  • Actually, $0/0$ could be any number, not just $0$ or $1$. As for $1/0$, there is no answer in the real numbers (or complex numbers for that matter). $\infty$ just isn't in $\mathbb{R}$ (that symbol means "the real numbers"). Even if it was, would $1/0$ be positive or negative? If it was positive infinity, then $-\infty=-(1/0)=1/(-0)=1/0=\infty$, which is a contradiction; similarly for if it was negative. – Akiva Weinberger Aug 17 '14 at 10:38
  • @columbus8myhw: So $ \dfrac {0}{0} $ is possible on the imaginary number line...??? –  Aug 17 '14 at 10:39
  • @SmallDeveloper No, not that either. (As I said, it doesn't work for complex numbers.) I was thinking of the Riemann sphere, which has $1/0=\infty$, but $\infty=-\infty$, and $\infty+\infty$ is undefined.

    See, there's no way to escape getting something "undefined."

     – Akiva Weinberger Aug 17 '14 at 10:41
  • But then, get this, let's say you wanted to graph $ \dfrac {x}{y} $, and of course at the origin, this would be $ \dfrac {0}{0} $. So approaching it from $ y = x $, this would be 1. But then, approaching from $ y = -x $, it would be -1. Approaching from the x-axis, in other words $ y = 0 $, this would be $ \pm \infty $. And then that's where things start to get confusing. –  Aug 17 '14 at 10:47
  • Then again, this would only be true for something like $ \dfrac {x}{y} $, only for ordered pairs. –  Aug 17 '14 at 10:47
  • @SmallDeveloper And approaching from $y=2x$, you'd get $2$. Approaching from $y=\pi x$, you'd get $\pi$. – Akiva Weinberger Aug 17 '14 at 10:48
  • Yes, and then that's where stuff starts mixing up. So then the question comes, what is $ \infty $ or undefined? What is it, a number, a representation, a concept, a limit....? –  Aug 17 '14 at 11:01
  • It's not that you cannot divide by 0, but you must define what division by 0 means first and there's no way to do that without breaking some properties of real numbers. If you work inside another algebraic structure (i.e. wheels) you can divide by 0 without problems – Alessandro Codenotti Aug 17 '14 at 12:08

4 Answers4

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This is the way I convinced myself :

$$any number \times 0 = 0\; \rightarrow \! \frac{0}{0}=any number$$

I thinks this is why it is unidentified.

Shabbeh
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  • This is relevant to calculus. Anynumber could equal 1 - so if the numerator and denominator are 'the same zero' we could state that it does and cancel. –  Aug 17 '14 at 09:15
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    That is actually... rather convincing. Short but great answer! :) –  Aug 17 '14 at 09:37
  • So then, $ \dfrac {0}{0} $ could be any number, but then why do we say that $$ \lim_{x \to 0} \dfrac {x}{0} = \infty $$ in other words $ \dfrac {x}{0} = \infty $, even though it could be... anything? –  Aug 18 '14 at 07:54
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You can find a few good illustrations on why we can't divide by $0$ here. I want to note that infinity is not the answer to $x/0$, what one can say however is that $\lim_{x\rightarrow 0^+} \frac{1}{x}= \infty$. Try it out yourself: compute $1/0.5$, then $1/0.3$, $1/0.1$ etc and you'll see that the answers keep increasing. Because you can keep picking a smaller number to divide by, the limit is infinity.

dreamer
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    Your limit is false. – Andrew Thompson Aug 17 '14 at 08:29
  • @AndrewThompson Yes, sorry. Forgot the $+$ (it's fixed now). – dreamer Aug 17 '14 at 08:31
  • That is better. If you add the fact that $\lim_{x \to 0^-}1/x = -\infty$ and argue that the two-sided limit is undefined, and use that as an intuitive perspective as to why division by zero is undefined, you'd be very close to what I would consider a good answer. (Shabbeh's answer might be the one most suited for OP, tho.) – Andrew Thompson Aug 17 '14 at 08:34
  • @AndrewThompson I wasn't trying to use it as an argument for why division by zero is undefined. The arguments in the link that I posted are quite convincing for that I think. I just wanted to point the OP to the fact that his assertion that $x/0=\infty$ is false (limits are needed). But you argument is nice too. – dreamer Aug 17 '14 at 08:36
  • Referring to the link you gave me, it used the analogy that you can't share anything among $ 0 $ people. But then when we say $ \dfrac {x}{0} $ couldn't the answer be $ 0 $ and the remainder be $ x $? That would be what would make sense if you were dividing among $ 0 $ people. –  Aug 17 '14 at 09:43
  • Hmmh, you could see it that way but the point there is more that it doesn't make sense to think about dividing something between 0 people. But the other arguments used there are probably more compelling. – dreamer Aug 17 '14 at 09:46
  • If you go to the limits page (on the same site, here), then it states that $$ \lim_{x \to 1} \dfrac {x^2 - 1}{x - 1} = 2 $$ But when $ x $ is 1, then the answer is $ \dfrac {0}{0} $. So can't we say that in any relation,where at any point, the output is $ \dfrac {0}{0} $, whatever the limit is, is also $ \dfrac {0}{0} $ ? –  Aug 17 '14 at 10:11
  • @SmallDeveloper (1/2) No. The fact that the answer to your original question is that you cannot divide by $0$ also means that you cannot always simply fill in a number (in this case $1$) in a limit if that would imply that you would divide by $0$ (because the statement is then pointless; division by $0$ is not defined). In this case filling in $1$ would result in division by $0$, so this is such a case in which we cannot simply fill in the value of $x$. Instead, we look at what happens when $x$ gets very close to $x$. Intuitively, you could do this by plugging in values close to $1$ – dreamer Aug 17 '14 at 10:17
  • @SmallDeveloper (2/2) but there are also more easy 'formal' methods to find limits such as these. The most well-known method is L' Hopitals rule, which says that you can take the derivative of both the numerator and denominator in such cases, which would imply that you have $\lim_{x\rightarrow 1} \frac{2x}{1}=2$. – dreamer Aug 17 '14 at 10:19
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If you want meaningful arithmetic (for $+$ and $\cdot$), then you want it to obey some rules, for example that $a+0=a$ for all numbers $a$ and the distributive property. These simple rules imply that $a\cdot 0=0$ for all numbers $a$. As division is defined as the inverse operation to multiplication it is clear at this point that you can't define what division by 0 is supposed to mean: $\frac b0$ would be the unique $c$ with $c \cdot 0=b$, but for no $b\neq 0$ is there such a $c$...

Frunobulax
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The binary operator, division, can be thought of as a statement which asks how many times can "b" go into "a", a/b. If b is zero, I.e. nothing, the operator then asks how many times can nothing go into something?; Well, an infinite number of times. If a and b are zero, then how many times can nothing go into nothing, an infinite number as well, however, this case is less defined then simple division by zero. And, infinity is not a number, it's a concept.

user170141
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