What exactly does undefined mean? I'm currently taking a course on Calculus and Analytic Geometry with a quite rich insight on set theory and propositional, predicate logic. I've noticed that some values have not been numerically defined, hence the wording "undefined". Now, what exactly does undefined mean? Because I've also noticed that most mathematicians tend to use "undefined" when interpreting a value that is approaching both infinity and negative infinity. e.g: the value of the tangent ratio at 90 degrees is interpreted as "undefined", while on the graph we can see that it seems to be approaching infinity from the top and negative infinity from the bottom. Or anything (except zero) divided by zero is also "undefined" since it could be both infinity and negative infinity. Is there some technical definition that I am not aware of? I thought, intuitively perhaps, that it is some sort of special set, the undefined set. Much like the empty set is used to illustrate the fact that no value satisfies a given equation, the undefined set illustrates the fact that the equation is satisfied by exactly two values, i.e $\infty$ and $-\infty$. So by definition, this undefined set could be $\{\infty, -\infty\}$. I don't know if either infinity or negative infinity is recognized as a number, but perhaps it could be treated as simply a matter of notation. Hence $\infty$ simply means a perpetually increasing value, and $-\infty$ a perpetually decreasing value. Is this correct, or is there no other ultimate meaning behind "undefined" other than it simply hasn't been given a proper definition?
 A: The "first level" answer should be: "undefined" plainly means "not defined", i.e. the expression has no value. For example, $\frac{a}{0}$ is undefined for all $a\in\mathbb R$, $\tan(k\pi+\pi/2)$ is undefined for all $k\in\mathbb Z$, $\log(x)$ is undefined for $x\le 0$. This means: those expressions have no meaning, don't write them, do not divide by zero, don't calculate tangent of an odd multiple of $\pi/2$ or a logarithm of a negative number (or zero).
The "second level" answer should be that people have tried to assign some value to some of these expressions (i.e. it's not that we were lazy!), but have found that there is no universal answer. There are partial answers, depending on the context. Thus, the consensus is: at the "first level" say that those expressions are undefined, and then, at the second level, do define them in various contexts, but be aware of the limitations. This is an instance of "walk before you run".
This is also a story of compromise: you gain something (by defining something) but you also lose something (by sacrificing some of things that you are taking for granted).
For example, the infinity. Why don't we "just add" another number, call it "infinity", label it with $\infty$, and define $\frac{1}{0}:=\infty$? What would go wrong? As it happens, a lot. $\mathbb R$ is not just a set, it is a field, i.e. a very regular structure with addition, subtraction, multiplication and division. How do those extend to $\infty$? Is $\frac{2}{0}=\infty$ too? Is $\infty=\frac{1}{0}=\frac{2-1}{0}=\frac{2}{0}-\frac{1}{0}=\infty-\infty=0$? This "paradox" shows you that, whatever you decide $\frac{1}{0}$ to be, you cannot expect the ordinary arithmetic rules to stay valid. So you have to sacrifice something: if it is not the ability to calculate $\frac{1}{0}$, it is the ability to apply the laws of arithmetic!
The latter sacrifice seems bigger, but it isn't always, and it isn't in all contexts. We must still say "infinity is not a number" to remind ourselves to not use arithmetic operations on it, but we can extend the topology of $\mathbb R$ ("compactify it with one point") and talk about convergence. This is what you will be doing in Calculus. Except - as it happens, there is another, equally valid, and complementary, way to extend $\mathbb R$ with infinities: don't add one infinity $\infty$ but add two infinities: $+\infty$ and $-\infty$. (Again, don't do any arithmetic on them!)
Sometimes you can extend "undefined" expressions without any hassle: $\frac{\sin x}{x}$ is undefined for $x=0$, but not only that $\lim_{x\to 0}\frac{\sin x}{x}=1$ is well-defined; it is, in a way, a part of the function. Namely, when you "patch up" $\frac{\sin x}{x}$ to "become" $1$ for $x=0$, you are not patching up anything - you are discovering a new reality. The function "patched up" in such way is a very nicely behaved, in fact it is an analytic function on the whole $\mathbb R$ (and even on the whole $\mathbb C$). On the other hand, if you try to "patch up" $\frac{1}{x}$ at $x=0$, whatever you do you cannot get even a continuous function.
Somewhere "in-between" is the case of the logarithm. Expand $\mathbb R$ into $\mathbb C$, and suddenly you have a logarithm (except for $x=0$ - this one stays undefined) - but you get more than you've bargained for: you can solve $e^y=x$ for any $x\ne 0$ but $y$ is not unique. Thus, people usually restrict the range of $y$'s to a horizontal strip of width $2\pi$ in the complex plane. So again: you gain (can define logarithm) but you lose (which strip you've chosen is arbitrary, and some rules, e.g $\log(ab)=\log a+\log b$, aren't valid anymore: instead, $\log(ab)\equiv\log a+\log b\pmod{2\pi i}$).
A: Division is defined as the inverse of multiplication. Multiplication by zero is defined as always giving the answer zero. Given $b\times0=0$ there is not any number which you can multiply the right hand $0$ by and get $b$ back.
If we say that there is such a number $0^{-1}$ such that $0\times 0^{-1}=1$ then what about $2\times0=0$? What would be the inverse of that such that $0\times 0^{-1}=2$ as well as $0\times 0^{-1}=1$?
Suppose I define a function f with the domain $[0,1]$, then $f(-1)$ is undefined, just like $f(\text{banana})$ is undefined, and in exactly the same way $1/0$ is not defined. The tangent example is not defined because division by zero is not defined. It really has absolutely nothing to do with anything approaching infinity.
The resolution of the paradox "what happens when an unstoppable force meets an immovable object" is that you cannot have an unstoppable force in the same universe as an immovable object. The resolution of zero times everything equals zero is that you cannot define an inverse of multiplication by zero.
A: As noticed in te comments, literally, undefined means not defined and of course its use depends from the context.
For example, on the reals $\sqrt{-1}$ is undefined but on the complex we assign a meaning to it.
We can also more precisely use undefined when there are no solutions to a question or problem and in indeterminate, when there are infinitely many solutions.
Sometimes we use the term undefined as a synonymous of indeterminate as for example for the division by zero. In this case, in the algebra or calculus context, let consider $c \neq 0$ then $\forall b\in \mathbb R$ the ratio $a=\frac{b}{c}$ is (well) defined with $a$ uniquely determined such that
$$a=\frac{b}{c} \iff a\cdot c = b$$
but for $c=0$ we can't assign an unique value to $a$ indeed $\forall a\in \mathbb R$
$$a\cdot c = a\cdot 0 =0$$
and therefore in this case the ratio $a=\frac{b}{c}$ is undefined (indeterminate) whatever is the value assumed for $b$.
