Undefined

According to a Wikipedia article on the subject, in Herbert B. Enderton's book Computability: An Introduction to Recursion Theory (2011), even if nowhere else (no other reference is given, and I've never seen the usage):

If is a partial function on and is an element of , then this is written as and is read as " is defined."

If is not in the domain of , then this is written as and is read as " is undefined".

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, $\log(ab)\equiv\log a+\log b\pmod{2\pi i}$).