What number system are you discussing? Are you talking about the nonnegative extended reals? The Riemann Sphere? Otherwise, the answer is, a number divided by zero does not equal infinity. Infinity is not a number in most number systems. It seems like you are discussing geometry. So, try this. Find the slope of the lines between the following points:
1. and
2. and
3. and
4. and
5. and
Given a fraction where the numerator is greater than zero and the denominator is positive, but very close to zero, the number resulting from the division is larger than the numerator. The closer you get the denominator to zero, the larger the number becomes. I can keep making the denominator closer and closer to zero, and the closer I get, the larger the number becomes after division.
So, using the real numbers, division by zero does NOT equal infinity. It does not equal anything else, either. It is simply undefined.
I understand your note as if you are saying the limit is infinity but nothing is equal to infinity, but you concluded corretly infinity is undefined. Your example of getting the denominator smaller and smalser the result of the division is a very large number that approches infinity. This is the intuitive mathematical argument that plunged philosophy into mathematics. at that level abstraction mathematics, as well as phyisics become the realm of philosophi. The notion of infinity is more a philosopy question than it is mathamatical. The reason we cannot devide by zero is simply axiomatic as Plato pointed out. The underlying reason for the axiom is because sero is nothing and deviding something by nothing is undefined. That axiom agrees with the notion of limit infinity, i.e. undefined. There are more phiplosphy books and thoughts about infinity in philosophy books than than there are discussions on infinity in math books.
Actually, not all infinities are considered equal. Georg Cantor developed a theory on this Georg Cantor - Wikipedia, the free encyclopedia
As grillage mentioned, Georg Cantor proved that there are different infinite cardinalities. In the example you mention, line segments AB and PQ considered as sets of points have the same infinite cardinality. On the other hand, has a smaller infinite cardinality than line segment AB.
If you are looking at the set of ordinal numbers, there are infinite distinct infinities.
If you are looking at the extended reals, there are two infinities: and . If you are looking at the projective reals or the Riemann Sphere , there is only one infinity. In each of these contexts, however, the number system is not a field.