division by different kind of infinities

Question

Is 1/countable infinity = 0 or not(or is it infinitisimal).I already know that countable infinity/uncountable infinity= 0 and 1/uncountable infinity is also = 0 .On this wiki link-https://en.m.wikipedia.org/wiki/Infinitesimal it is some what mentioned to be = infinitisimal but not clearly. Would be very thankful for any useful input or perspective along with the explanation.

Answer 1

Actually, one doesn't "divide by $\infty$", since $\infty$ is not a number. In mathematics, you should always understand division as multiplication by an inverse. In particular, you have to be within some ring.

Moreover, to say that an infinitesimal is $\frac{1}{\infty}$ is a blatant abuse, an oversimplification (unless you decided to name a specific invertible element in a ring $\infty$, but that's ugly). An infinitesimal is a number : you can multiply/add it with/to another number.