Context: I am currently reading Rudin's real analysis and in chapter $1$ definition $1.23$ the extended real number system $\mathbb{\bar{R}}=\mathbb{R} \cup \{ -\infty , \infty \} $ was introduced, his definition raised a lot of questions in my mind, such as: How is this useful at all? I can see that any sequence of real numbers has a "convergent" subsequence in this system, $\infty , -\infty $ mean in this case ?Rudin refer to them as numbers (in definition 3.16 for example ) then what is even a number? How to define a "number" in this case? Another question that came to mind is how many "number" that belong to the extended real number system that don't belong to $\mathbb{R}$ in other words: Is $|\mathbb{\bar{R}}|=|\mathbb{R}|$?
The definition of extended real number suggest that $\mathbb{\bar{R}}$ has only two "numbers" more than $\mathbb{R}$ but does that mean any two infinite digit "numbers" are equivalent ? for example does $\dots 11111111= \dots 1212121121212$ ? if they are equal then what does equal means in the new real number system? Another question is why $\frac{x}{\infty} =0$ in this system. in other words does the number $10000000 \dots$ have an inverse that is $0.0000 \dots 000001$ which is the infinitesimal?
Another question is if $|\mathbb{\bar{R}}|>|\mathbb{R}|$ does than mean $|\mathbb{\bar{R}}|=|\aleph_2|$? or $|\mathbb{\bar{R}}|$ is even bigger than that
