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:
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.
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.