Can anyone give me a rigorous explanation, why one needs only one number "$\infty$", when dealing with complex numbers, instead of $2$ numbers $+\infty, \ -\infty$ like in the case, when dealing with real numbers?
I was told, that by adjoining a point $\{\infty\}$ to $\mathbb{C}$, the newly obtained set becomes a compact one (with respect to the euclidean topology, when $\mathbb{C}$ is viewed as $\mathbb{C}=\mathbb{R}^2$, I presume, though I am not sure), so I would assume, that in the case of the reals, just using one "$\infty$" wouldn't suffices to make it compact ?
(Please note, that my knowledge of topology is very limited.)
Are there also other reasons for just using just one "$\infty$"?
What would happen/would it make sense, if we decided to use multiple $\infty$-type numbers, when dealung with complex numbers?
Best Answer
There is such a thing as a "one-point compactification" of $\mathbb{R}$; you end up with something that is "like" a circle (where "like" can be made very precise) with a point corresponding to $\infty$.
This is in exactly analogy to viewing the complex numbers with $\infty$ added as a sphere: you do it by stereographic projection. Take a unit circle and put it on the plane so that its center is at $(0,1)$. It is tangent to the $x$-axis at $(0,0)$. To each real number $r$ there corresponds one and only one point in this circle, obtained by taking the line through $(r,0)$ and $(0,1)$, and identifying $(r,0)$ with the second point of intersection of the line with the circle (the first point being $(0,1)$). Every point on the circle except for $(0,1)$ itself corresponds to a real number, so you can think of the point $(0,1)$ as corresponding to "the point at infinity". In fact, this is a natural way of constructing the real projective line.
The counterpart of this is that you can "add infinities" to the complex plane the same way that we "add infinities" to the real line, one for every direction. In the real line, we have the "positive direction" and the "negative direction", which lead to a $+\infty$ and a $-\infty$, respectively. In the complex plane, you would want to add an "infinity" in every direction, so you would have to add an $+\infty_m$ for every real number $m$ that corresponds to the direction from $0$ to $1+mi$, and a $-\infty_m$ corresponding to the direction from $0$ to $-1-mi$ (in the "opposite direction" of $+\infty_m$); plus a $+\infty_v$ corresponding to the direction from $0$ to $i$, and yet another $-\infty_v$ for the direction from $0$ to $-i$. Doing this essentially gives you a closed disc.
Yet a third way is to consider adding a point for every slope, adding an $\infty_m$ corresponding to a line of slope $m$, plus an $\infty_v$ for the vertical lines on the complex plane. In this case, what you get is essentially the real projective plane.
You can do any of the three, but each gives a different kind of structure. Just like the extended reals (with $+\infty$ and $-\infty$) is (usually) the "right" setting to do a lot of calculus, rather than trying to do it in the projective real line, so the Riemann sphere (obtained by adding a single $\infty$ to $\mathbb{C}$) is (usually) the "right" setting to do complex analysis, rather than trying to do it in the closed disc or the projective plane. That is, you "can" any of them, but the "one $\infty$" completion of the complex numbers is more useful for analysis (and in other settings) than the "one $\infty$ per direction" completion or the "one $\infty$ per slope" completion. If you are doing other things (like hyperbolic geometry or projective geometry), then one of the other completions may be more useful.