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I asked my teacher what is the real meaning of $\cfrac{0}{0}$, and the answer I got was "nobody knows". I can't leave this subject "as is". I need a decent explanation, at least an explanation to why "nobody knows". I'm sure you'll come up with a few good answers.

Micah
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yoniyes
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    These might be interesting to you: http://math.stackexchange.com/questions/332520/a-book-of-wheels, http://math.stackexchange.com/questions/26445/division-by-0, http://math.stackexchange.com/questions/144715/will-division-by-zero-be-defined-eventually, ... – Dejan Govc Oct 15 '13 at 20:44
  • Am I right when I say $\frac 0{0} = \frac0{1} * \frac1{0}$ and because $\frac1{0}$ is not defined, you cannot do the multiplication, so it is undefined? – zerosofthezeta Oct 15 '13 at 20:59
  • This answer might also be of interest. – robjohn Oct 15 '13 at 20:59
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    @zerosofthezeta Neither side of your equation is defined, which makes it meaningless. But you are right that $0/0$ being undefined is related to $1/0$ being undefined. – Trevor Wilson Oct 15 '13 at 21:02

4 Answers4

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I really hope your calculus teacher didn't say "nobody knows": $\frac{0}{0}$ is pretty well understood. It's just that it's undefined (without further context).

What I mean by "context" is: it's possible to give meaning to limits of the form $$\lim_{x \to a} \frac{f(x)}{g(x)}$$ where $\lim_{x \to a}f(x) = 0$ and $\lim_{x \to a} g(x) = 0$, but this depends on your choice of $f$ and $g$. For instance $$\lim_{x \to 0} \dfrac{\sin x}{x} = 1, \qquad \lim_{x \to 0} \dfrac{e^{kx}-1}{x} = k, \quad \lim_{x \to 0} \dfrac{x}{x^3} = \infty, \quad \cdots$$ These are all different, despite the fact they're all of the form $\frac{0}{0}$.

So it's not that nobody knows, it's just that it's meaningless! But we can assign meaning if we know how the $0$s in the fraction came to be.

Clive Newstead
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The expression $0/0$ is meaningless because the operation of division $(a,b) \mapsto a/b$ is not defined for pairs $(a,b)$ where $b$ is zero, just as it is not defined for pairs $(a,b)$ where $b$ is an elephant.

Sometimes you will hear that $0/0$ is an "indeterminate form" that can equal different things depending on the context. This is a horribly imprecise way of speaking. What it really means is that if you are evaluating a limit and you do the computation

$$\lim_{x \to 0} \frac{f(x)}{g(x)} = \frac{\lim_{x \to 0} f(x)}{\lim_{x \to 0} g(x)} = \frac{0}{0}$$

Then you have made a mistake in the first step and you have to evaluate the limit in a different way, e.g. with l'Hospital's rule, to get a valid answer (which could be anything depending on the particulars of the problem, hence "indeterminate.") This reason that the first step in the displayed calculation is a mistake is that the rule "the limit of a quotient is the quotient of the limits" is not true in general—it is only true when the limit of the denominator is nonzero.

Trevor Wilson
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Math is logic.

$12/3=4$ because if you have twelve oranges and three kids, each gets four.

However, dividing by zero is illogical; what does it mean? It is no longer a math question, but a philosophical one.

Everything you do in math needs to make sense!

wjmolina
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  • Exactly. So when I divide nothing to no one (as giving no-apples to nobody) is like I'm doing something (or nothing) infinitely. And as much as it doesn't make any sense, it kinda sounds like something that happens all the time. – yoniyes Oct 15 '13 at 20:50
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    I resent the implication that philosophy is not logical. – leonbloy Oct 15 '13 at 20:59
  • Correction: math is applied logic, in the same way physics is applied math, chemistry is applied physics, biology is applied chemistry...it goes on forever..! – zerosofthezeta Oct 15 '13 at 21:07
  • Some infinities being larger than others also doesn't make sense, yet is accepted as math. – Kaz Oct 16 '13 at 00:38
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    Does taking the square root of $-1$ make sense in terms of oranges? – Trevor Wilson Oct 16 '13 at 06:00
  • Yes, but you woll not survive the ensuing release of energy. A negative orange (a structure such that putting it with another orange causes both to disappear), must necessarily be made out of antimatter. Putting a piece of antimatter with a piece of antimatter, each being the size of an orange (0.4 kg total), will release $3.6\times10^{16}$ Joules, which is about 3 times the annual energy consumption of the entire country of Mongolia. Released in a matter of microseconds, the ensuing blast would remove the experimenter (as well as anyone in the area) from the gene pool immediately. – FundThmCalculus Dec 21 '14 at 12:28
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Well, if $0/0$ is well-defined then it should play nicely with our definition of division. Specifically, $a/b = c$ if $a = cb$. So there's gotta be some c such that $0c = 0$. Let's try 1. $0\cdot 1 = 0$, so $0/0 = 1$.

Not too bad. Now let's try two. $0\cdot 2 = 0$, so $0/0 = 2$. Since equality is transitive, $1=2$.

Aaaaaand we just broke math. We resolve this by saying that $0/0$ is not defined, which keeps everything running nicely with a minimum of fuss.

Hovercouch
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