Why not to extend the set of natural numbers to make it closed under division by zero?

You can add division by zero to the rational numbers if you're careful. Let's say that a "number" is a pair of integers written in the form $a\over b$. Normally, we would also say that $b\not=0$, but today we'll omit that. Let's call numbers of the form $a\over 0$ warped. Numbers that aren't warped are straight.

We usually like to say that ${a\over b} = {c\over d}$ if $ad=bc$, but today we'll restrict that and say it holds only if neither $b$ nor $d$ is 0. Otherwise we'll get that ${1\over 0} = {2\over 0} = {-17\over 0}$, which isn't as interesting as it might be. But even with the restriction, we still have ${1\over 2}={2\over 4}$, so the straight numbers still behave as we expect. In particular, we still have the regular integers: the integer $m$ appears as the straight number $m\over 1$.

Addition is defined as usual: ${a\over b} + {c\over d} = {ad+bc\over bd}$. So is multiplication: ${a\over b} \cdot {c\over d} = {ac\over bd}$. Note that any sum or product that includes a warped number has a warped result, and any sum or product that includes $0\over 0$ has a the result $0\over 0$. The warped numbers are like a hole that you can fall into but you can't climb out of, and $0\over 0$ is a deeper hole inside the first hole.

Now, as Chris Eagle indicated, something must go wrong, but it's not as bad as it might seem at first. Addition and multiplication are still commutative and associative. You can't actually prove that $0=1$. Let's go through Chris Eagle's proof and see what goes wrong. Chris Eagle starts by writing $1/0 = x$ and then multiplying both sides by 0. 0 in our system is $0\over 1$, so we get ${1\over 0}{0\over 1} = x\cdot 0$, then ${0\over 0} = x\cdot 0$. Right away the proof fails, because it wants to have 1 on the left-hand side, but we have $0\over 0$ instead, which is different.

So what does go wrong? Not every number has a reciprocal. The reciprocal of $x$ is a number $y$ such that $xy = 1$. Warped numbers do not have reciprocals. You might want the reciprocal of $2\over 0$ to be $0\over 2$, but ${2\over 0}\cdot{0\over 2} = {0\over 0}$, not ${1\over 1}$. So any time you want to take the reciprocal of a number, you have to prove first that it's not warped.

Similarly, warped numbers do not have negatives. There is no number $x$ with ${1\over 0}+x = 0$. Usually $x-y$ is defined to be $x + (-y)$, and that no longer works, so if we want subtraction we have to find something else. We can work around that easily by defining ${a\over b} - {c\over d} = {ad-bc\over bd}$. But then we lose the property that $x - y + y = x$, which only holds for straight numbers. Similarly, we can define division, but if you want to simplify $xy÷y$ to $x$ you'll have to prove first that $y$ is straight.

What else goes wrong? We said we want ${a\over b} = {ka\over kb}$ when $a\over b$ is straight and $k\not=0$; for example we want ${1\over 2}={10\over 20}$. We would also like ${a\over b}+{c\over d} = {ka\over kb} + {c\over d}$ under the same conditions. If $c\over d$ is straight, this is fine, but if $d=0$ then we get ${bc\over 0} = {kbc\over 0}$. Since $bc$ could be 1, and $k$ can be any nonzero integer, we would have ${p\over 0} = {q\over 0}$ for all nonzero $p$ and $q$. In other words, all our warped numbers are equal, except for $0\over 0$. We have a choice about whether to accept this. The alternative is to say the law that $a + c = b + c$ whenever $a = b$ applies only when $c$ is straight.

At this point you should start to see why nobody does this. Adding a value $c$ to both sides of an equation is an essential technique. If we throw out techniques as important as that, we won't be able to solve any problems. On the other hand if we keep the techniques and make all the warped numbers equal, then they don't really tell us anything about the answer except that we must have used a warped number somewhere along the way. You never get any useful results from arithmetic on warped numbers: ${a\over 0} + {b\over 0} = {0\over 0}$ for all $a$ and $b$. And once you're into the warp zone you can't get back out; the answer to any question involving warped numbers is a warped number itself. So if you want a useful result out, you must avoid using warped numbers in your calculations.

So let's say that any calculation that includes a warped number anywhere is "spoiled", because we're not going to get any useful answer out of it at the end. At best we'll get a warped answer, and we're most likely to get $0\over 0$, which tells us nothing. We might like some assurance that a particular calculation is not going to be spoiled. How can we gain that assurance? By making sure we never use warped numbers. How can we avoid warped numbers? Oh... by forbidding division by zero!

I would also like to point out that the IEEE floating-point specification allows division by zero. It incorporates three nonstandard numbers, called infinity, negative infinity, and "NaN", which is short for "Not a Number". $a\over 0$ is infinity, negative infinity, or NaN according as whether $a$ is positive, negative, or zero, respectively.

Infinity represents an overflow condition. So you might add two very large, positive numbers and get a result of infinity. As a result, IEEE floating point numbers fail to obey many of the usual mathematical laws. For example, $(a+b)+c = a+(b+c)$ may fail: suppose $a = b = -c$ and all are large. If $a+b$ overflows, the left-hand side of the equality will be infinity, but the right-hand side will be the finite quantity $a$.

The behavior in some cases can be very subtle and counter-intuitive. But it is widely used in practice.

Here's a proof that $1/0$ can't exist:
Suppose $1/0=x$.
Then $1=0 \cdot x$.
So $1=(0+0) \cdot x$.
So $1=0\cdot x + 0 \cdot x=1+1$.
So $0=1$. Contradiction.

Thus, if we're going to extend our number systems so that dividing by $0$ makes sense, we need to change things so one step in this proof doesn't work. Let's have a look at what properties the proof used.

First I assumed that if $a/b=c$, then $a=b \cdot c$. This is the defining property of division. If we give it up, the resulting thing won't deserve to be called division at all.

Then I assumed that $0=0+0$. Making this false would be a pretty radical redefinition of addition: none of the extensions you mention involve redefining addition for the numbers we have already.

Next I used that $(a+b)\cdot c=a \cdot c + b \cdot c$. This is a key relationship between addition and multiplication. You could give this one up, but the result will be pretty weird.

Next I used that, if $a=a+b$, then $0=b$. Again, you could give up this key property of addition, but you'll get a weird structure as a result.

Finally I used that $1$ does not equal $0$. Again, an "extension" which messes with the numbers we have already like that is a pretty odd extension.

In short, while you can extend your number system to allow division by $0$, doing so requires breaking far more fundamental rules of logic and arithmetic than needed in constructing the rationals or the complexes. As a result, the structures you get are not very pleasant and not all that useful.