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How can one tell the difference when the result is undefined or math just doesn't know how to provide a value for that particular equation? (the value still exists however)

For example, how could one prove that by definition division by zero is undefined; it's not that math doesnt' know the value, the value just doesn't exist.

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What does it mean for “math” to know a value? Anyway, you can’t prove something is undefined. For something to be undefined just means that we haven’t defined what it should be. In the case of division by zero, if we consider the set $\mathbb{R}$ of real numbers, there is no “natural” value in $\mathbb{R}$ that we could define $1/0$ to be. – Adeel Jun 21 '12 at 21:16

Whether something is defined or not is a matter of, well, definition. Division by zero is undefined because we explicitly exclude it from the definition of division. The reasons we exclude it from the definition are varied, of course, but it's not a matter of lack of knowledge.

It is not quite clear what you mean by "provide a value"; there are numbers which we can prove cannot be explicitly described in terms of a terminating algorithm (that is, there is no Turing Machine that will produce the number). But does that mean we do not provide a value?

We cannot write down exactly a number that solves the equation $x^2-2=0$. We cheat when we say the solutions are $\sqrt{2}$ and $-\sqrt{2}$ because... what does "$\sqrt{2}$" mean? It means "the positive real number that is a solution to $x^2-2=0$". Does that mean we "don't know how to provide a value"?

On the other hand, there are equations which we may genuinely not know whether they have solutions of a special kind or not. For a long time, it was unknown whether there were any positive integers $a$, $b$, and $c$, and a positive integer $n\gt 2$, such that $a^n+b^n=c^n$. Now we know there are none.

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We don't "prove" that such statements are undefined, we "choose" not to define them, because we believe that in some contexts it doesn't make sense to do so. For instance, over the real numbers, we choose not to divide by zero because there is no explicit interest in doing so. But over the complex numbers, it makes great sense to say that $1/0 = \infty$, and in some contexts it is very important to understand what it means.

For instance, over real numbers, if we would try to define $1/0$ by continuity, we would suggest $1/0 = \lim_{x \to 0} \frac 1x$, but this limit doesn't exist. However, if we take the same definition over $\overline{\mathbb C}$, it works! The limit exists and is worth $\infty$ in the compactification of the complex plane. We don't prove that something is undefined, we just don't define it when we don't want to. That's the big idea.

Hope that helps,

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In the real numbers we can prove that division by zero is not only undefined but cannot be defined at all. Why? Because we would like the real numbers to have certain properties which are not consistent with the idea of division by zero.

We do can prove the "undefinability" by showing that if division by zero were possible to define we could derive a contradiction (e.g. $1=2$), and contradictions are bad. So we avoid things which prove contradictions.

To see that indeed this is the case suppose that $\frac10$ was defined, then $0\times\frac10=0$ since everything times zero is zero; on the other hand $\frac10\times0=1$ because we multiply a number by its inverse. Therefore $0=1$... contradiction!

On the other hand, we can prove that a continuous function $f$ which satisfies: $$\lim_{x\to\infty} f(x)=\infty,\text{ and }\lim_{x\to-\infty}f(x)=-\infty$$ Has at least one root, that is there exists some $c\in\mathbb R$ such that $f(c)=0$. Even though we don't know what $f$ is or how to find this $c$.

Why can we prove that? If we assume that this is not the case then we can once again derive contradiction in one form or another.

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