Is 0 the absence of something?

Question

In computer programming, null and 0 values are two different things. I was wondering if the same applied to just mathematics in general.

The reason I ask this question is because I don't understand why dividing by zero is thought of as chaotic. There is no value in the divisor and so the dividend is not applicable.

Answer 1

Do you extend the same reasoning to e.g. addition: would you say that "$2+0$" shouldn't be evaluated since there is no value in the second summand? The case with division is the same: there is a divisor in "$2\over 0$," namely $0$.

It's worth pointing out that division means something: "${a\over b}=c$" means the same thing as "$bc=a$." We can say "$2\over 0$ isn't defined" without mentioning division at all, as: "there is no real number which, when multiplied by $0$, equals $2$."

$0$ is not an absence - it is a thing itself. For example, the set of positive numbers is different from the set of non-negative numbers; this is because the second set contains $0$ but the first doesn't. (Incidentally, thinking about sets it's worth pointing out the parallel fact that the emptyset is not the same as nothing: in particular, $\{\emptyset\}\not=\emptyset$.)

Answer 2

Dividing by $0$ is not "thought of as chaotic". Dividing by $0$ is undefined.

What does $6/2=3$ mean? It means that $6=2(3)$. So there cannot be any such thing as $6/0$, because saying $6/0=x$ means $6=0x$, and no matter what $x$ is, $6\ne 0x$.

Answer 3

Dividing by zero is not chaotic (which is a word with a particular meaning in mathematics), it is an undefined operation. One can define a division by zero, and extend the real number system in order to accommodate some 'infinite' values, but this destroys the algebraic structure of the reals.

Incidentally, at a low level, languages such as C actually do define NULL and 0 as the same thing. At a higher level of abstraction, NULL is supposed to mean something different to 0 (namely a pointer). There is a similar idea in mathematics, where many constructions of the natural numbers define the number 0 as the empty set, while in other contexts, it doesn't make sense to interpret the empty set as 0, and there are other ways one could define the number 0. This is something that model theory takes to all sorts of interesting places.