Divide by Zero/Trivia

Full disclosure: Mathematics has more than one meaning of zero. While you can't divide by zero in real numbers, you can use a process called "limits" to do something similar. For example, when you take the limit of dividing a positive number n by "positive zero" (the limit of x as x approaches zero from the positive side), you get another limit called "positive infinity," which basically means that as the number you divide by gets smaller, so long as it's above zero, the result gets larger. Taking the limit of dividing the same number by "negative zero" produces "negative infinity", and dividing by "two-sided zero" produces "projective infinity". But the physical consequences of these infinities, on the other hand, lead to things like black holes.

Also, there are various sets of numbers that include "infintesimals" which are smaller in magnitude than any finite number other than zero. It's commonly possible to divide by an infinitesimal, which would produce an infinite number. You still can't divide by zero.

The real projective line (real numbers with infinity thrown in) does allow division by zero, but "division" doesn't have any real meaning. It is a useful construct in analysis, but it contradicts the layman's notion of how numbers are supposed to work since it's almost a field, but not quite.

Related to the above, there's also a rather weird kind of mathematical structure called a wheel where division is always possible. Some of the rules of algebra have to be sacrificed for this; for example, x/x=1 is not always true, and nor are 0x=0 or x-x=0. But it is a real thing.

All of this leads to fun mathematics mind-benders where you can mathematically prove that 1=2 and all sorts of other nonsense by neatly leaving out the important bit about equations of the form x/x=1 when x does not equal 0.