While studying advanced complex analysis, I have finally encountered the extended complex numbers. The extended complex numbers is defined as the complex numbers $\mathbb{C}$ together with {$\infty$} such that $\mathbb{C} \cup${$\infty$}; which is also defined as the Riemann sphere. Where addition and multiplication are defined as for $z \in \mathbb{C}$: $$z + \infty = \infty$$ $$z \bullet \infty = \infty.$$ As I studied these operations on the set $\mathbb{C} \cup${$\infty$}, regrettably, I began to ask questions about the intuition of these operations. If infinity is not a number, how is it that $1 \in \mathbb{C}$ such that $1+ \infty = \infty$ if there are different sizes of infinity? Even worse, what does it mean to union an uncountably infinite set $\mathbb{C}$ with infinity?
I understand the scope of this question is quite large, however any response (intuitive or not) will be greatly appreciated.
[Made into an answer at OP's request]
I think the geometrical point of view is the most intuitive. Using stereographic projection, you map $\mathbb C$ to a sphere with its north pole missing. That missing point is what you label as $\infty$. You might say it's not really "infinity": it's a very concrete object, a point on a sphere. Then you see how the various operations can be extended to this new point.