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I once heard my mathematics teacher say that $\frac{0}{0} = \infty$ and she said proving this is difficult.

How would I prove this?

Then will $1 \times 0 = \infty$ as well or just $0$?

Benedict W. J. Irwin
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George O'connelly
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    Your math teacher is wrong. First off all $\frac{0}{0}$ is not defined, and $\infty$ is not a number. – Cornman Nov 17 '17 at 17:17
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    Definitely $1×0 = 0$. – MJD Nov 17 '17 at 17:21
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    "I once heard..." Well, depending on the audience and the math topic at the time, she may have been correct. My elementary math teacher said that "square roots of negative numbers as a RULE can not be done". We were too young to understand and we were all little rascals and our teacher liked strict rules to keep us in pace. Was he wrong? At that moment obviously not! – imranfat Nov 17 '17 at 17:24
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    @imranfat I don't think there is any context where it's appropriate to claim that ${0\over 0}=\infty$. – Noah Schweber Nov 17 '17 at 17:27
  • Actually, the correct term would be "indeterminate". When we say "undefined", it means if we were to assign a value to it, there is no value we can assign to it that would make sense (e.g. $1/0$ is undefined because if $1/0=k$, then $1=0\times k=0$ which is false). But when we say "indeterminate", it means that we cannot assign a unique value for it (if $0/0=k$, then $0=k\times 0=0$ which is true for any $k$) – Prasun Biswas Nov 17 '17 at 17:29
  • @NoahSchweber I certainly agree, but we don't know the entire context of the OP here and I don't like to discredit a teacher just on that basis either... – imranfat Nov 17 '17 at 17:31
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    @imranfat I mean, can you think of a single context in which that comment would be appropriate? I don't see any way in which that statement could possibly be appropriate (assuming the OP heard it correctly of course), and I think it's wrong to suggest to the OP that it has any validity. – Noah Schweber Nov 17 '17 at 17:32
  • @NoahSchweber True, but the assumption about the OP you mention is critical! – imranfat Nov 17 '17 at 17:33
  • @imranfat You wrote "Well, depending on the audience and the math topic at the time, she may have been correct." If you wanted to suggest that the OP may have misheard, you could have done so more clearly; your comment suggests that what the OP heard could have been correct for a certain situation, and I don't see any way for that to be true. – Noah Schweber Nov 17 '17 at 17:34
  • @PrasunBiswas. With Indian kids, "indeterminate" would probably work at the lower level. In other parts of the world, I am not so sure. Depending on the audience and level, I would use "prohibited", particularly if I would have to teach citykids... – imranfat Nov 17 '17 at 17:35
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    @NoahSchweber. Again, we do not know the entire context as the OP has not given us such information. If the context came from a limit $x$ to zero on $\frac{x}{x^3}$, then graphically the limit would be "infinity". How did the teacher "translate" this result verbally? Maybe like saying "so here we have a 0÷0 situation that results in infinity". How did the OP remember that? Well...... The thing is what I don't like is that people very easily discredit a teacher from the pieces and remnants they themselves only remember. Peace! – imranfat Nov 17 '17 at 17:40
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    @imranfat, AFAIK, the term "indeterminate" is standard terminology. Here's the MathWorld page on it and here's a SE post on the difference between the terms "undefined" and "indeterminate". Even Wolfram|Alpha uses it to classify expressions like $0/0,1^\infty$,etc. – Prasun Biswas Nov 17 '17 at 17:44
  • @PrasunBiswas Yes true, I get your point, but did you get mine? – imranfat Nov 17 '17 at 17:46
  • @imranfat, I was just addressing the "with Indian kids..." part of your comment. It seemed as if you were suggesting that it is not standard terminology and hence wouldn't work in other parts of the world but I think I get your point now. – Prasun Biswas Nov 17 '17 at 17:55

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I would never claim that $\frac00=\infty$. There are contexts in which we use the expression "$\frac00$" as a shorthand for certain types of limits in calculus, but it is never assigned a particular value on its own.

It is definitely true that $1\times 0=0$, and indeed $k\times 0=0$ for any real number $k$. Note that $\infty$ is not a real number. (By "real number", we mean a number with a location on the number line. There is no specific location on the line that corresponds to $\infty$.)

Sometimes, people say that $\frac10=\infty$, but this is a fairly sloppy way of writing something more meaningful. More commonly, we simply say that division by $0$ is undefined.

The more meaningful idea can be captured by this example: Consider the sequence $\frac1{.1},\frac1{.01},\frac1{.001},\ldots$. These quotients evaluate to $10,100,1000,\ldots$. As the denominator gets closer to $0$, the value of the quotient grows without bound. If you continue this sequence, the values will eventually be greater than any finite upper bound you can think of. Thus, we say that as the denominator "approaches" $0$, the value of the quotient "approaches" $\infty$.

In order to prove this rigorously, you really need the tools of calculus, in particular, you need a well-defined notion of "limits", which turns all claims about so-called "$\infty$" into concrete claims about real numbers.

G Tony Jacobs
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