undefined
adjective
- Lacking a definition or value
- (mathematics) That does not have a meaning and is thus not assigned an interpretation
Saying that 1 divided by 0 is undefined, does not mean that you can carry out the division and that the result is some strange entity with the property “undefined”, but simply that dividing 1 by 0 has no defined meaning. That is just like when you ask whether the number 1.9 is odd or even: That is not defined. Or when you ask what colour the number 7 has.
Answer from Carsten S on Stack ExchangeVideos
Probably a trivial question, but it's been bugging me lately.
Most of us learn in school that 1÷0 is "undefined", but usually the teacher doesn't go much further than that. I've taken it to mean that dividing by zero is an operation that is not defined in standard mathematics, i.e. the answer to 1÷0 is not defined. However, I've seen people (both students and teachers) talk about "undefined" as though it's a mathmatecal term in of itself. Is this proper usage of the term, or does it simply mean that something is not defined?
tl;dr -- When discussing division by zero, is "undefined" a noun or an adjetive?
Saying that 1 divided by 0 is undefined, does not mean that you can carry out the division and that the result is some strange entity with the property “undefined”, but simply that dividing 1 by 0 has no defined meaning. That is just like when you ask whether the number 1.9 is odd or even: That is not defined. Or when you ask what colour the number 7 has.
To put matters straight: Division is a function 
whereby 
is the unique number 
such that 
.
When we say that 
is undefined then this means no more and no less than that the pair 
is not in the domain of the function 
.
Now to your three ways of understanding "undefined" in the realm of division by 
:
If
could be any number, say 
, then this would enforce 
, which is wrong when $a\ne0$.This is even worse. Why should
be the Eiffel tower?There are circumstances where division by zero makes sense, e.g. in connection with maps of the Riemann sphere, or with meromorphic functions. There one has
as an additional point in the universe of discourse. But these circumstances require special exception handling measures, and the "usual rules of algebra" are not valid when dealing with 
.