# Why is there no negative infinity in the extended complex plane?

I'm reading Ravi Agarwal's "Introduction to Complex Analysis". He says this:

It is often convenient to add the element $$\infty$$ to $$\mathbb{C}$$. The enlarged set $$\mathbb{C} \cup \{\infty\}$$ is called the extended complex plane. Unlike the extended real line, there is no $$-\infty$$.

He uses infinity to discuss the idea of a "neighbourhood of infinity", but he defines them without the need for infinity as a set of complex numbers $$z$$ following $$\{z:|z-z_0|>r>0\}$$. What's the point?

Why is adding the element $$\infty$$ convenient? Is there not a better way to discuss the concept of "neighbourhoods of infinity"? And why not add $$-\infty$$?

• If you think there is a better way, feel free to propose one. What the book proposes is that the complex infinity is one single point which is infinite "in all directions". Imagine you put a unit sphere sitting on a complex plane, its "South Pole" touching the plane at zero. By connecting the "North Pole" via rays to the points on the plane, you can map those points to the (intersection) points on the sphere - except for the North Pole itself. The intuition is that you are "completing" the complex plane with $\infty$ just the same as the sphere wouldn't be complete without its North Pole.
– user700480
Commented Mar 16, 2023 at 10:49
• (and with that intuition, "you get 'close' to North Pole" means "you get far away from zero on the complex plane", thereby the strict definition with "neighbourhoods" of $\infty$ being sets containing the sets $|z|>r$.)
– user700480
Commented Mar 16, 2023 at 10:51
• Moving from $\mathbb{C}$ to $\overline{\mathbb{C}}$ is an example of canonical compactification. You only need one extra point to compactify the complex plane. Commented Mar 16, 2023 at 10:52
• Possibly helpful: math.stackexchange.com/q/3208492/42969 Commented Mar 16, 2023 at 13:55

## 1 Answer

The short answer is that there are many ways to compactify $$\mathbb{C}$$. Likewise for $$\mathbb{R}$$. With $$\mathbb{R}$$, the distinction between positive and negative looms so large, that it is often helpful to distinguish between going to infinity in these two directions. However, there are no "positive" and "negative" complex numbers. It turns out that adding a single point at infinity is the most useful way to compactify $$\mathbb{C}$$.

This can be seen in a couple of ways.

First, if you look at the Riemann sphere with its associated stereographic projection, you will see that adding a single point results in a structure with many nice properties. The geometry is very appealing: circles and lines in $$\mathbb{C}$$ all become circles on the Riemann sphere, and vice versa, with circles passing through the north pole corresponding to lines. There's much more to the geometry, which you'll find at the link.

Second, from the viewpoint of projective (or algebraic) geometry, we can identify $$\mathbb{C}$$ with the complex line (just as $$\mathbb{R}$$ is the real line; see below for more on the terminology). The complex projective line is one way to compactify $$\mathbb{C}$$. It is given by the set of all ratios $$(A:B)$$, where $$A$$ and $$B$$ are not both 0. A ratio $$(A:B)$$ is the same as $$(tA:tB)$$ for any $$t\neq 0$$. That means that $$(A:0)$$, with $$A\neq 0$$, is the same ratio whatever the value of $$A$$. This ratio represents the point $$\infty$$. On the other hand, a ratio $$(A:B)$$ with $$B\neq 0$$ is the same as $$(A/B:1)$$. Thus the complex projective line consists of $$\mathbb{C}\cup\{\infty\}$$.

I realize the preceding paragraph may not help much if you haven’t studied any projective geometry, but you don’t need to learn much about it to see a pattern emerging in which this compactification of $$\mathbb{C}$$ makes sense.

The two viewpoints come together when studying fractional linear transformations. These are of the form $$z\to\frac{az+b}{cz+d}$$ Algebraically, these are "induced" by linear mappings $$(X,Y)\to (aX+bY,cX+dY)$$ by setting $$z=(X:Y)$$. Insisting that $$Y\neq 0$$ and/or $$cX+dY\neq 0$$ would bring on an annoyance of special cases. Allowing all ratios--in other words, letting the transformation take the Riemann sphere to itself--simplifies matters. And these FLTs turn out to have very nice geometrical properties.

Addendum on terminlogy. If $$\mathbb{K}$$ is a field, then affine-$$n$$ space is $$\mathbb{K}^n$$. Projective $$n$$-space is the compatification, and consists of all ratios $$(X_1:\cdots:X_n:Z)$$, where not all the coordinates are zero.

Both affine and projective $$n$$-space have $$\mathbb{K}$$-dimension $$n$$.

For $$\mathbb{K}=\mathbb{R}$$, all is terminologically fine and dandy: the real affine line is just the usual real line, likewise the real affine plane. The real projective line is topologically a circle; the real projective plane plays an important role in two-dimensional topology.

But for $$\mathbb{K}=\mathbb{R}$$, we have a historical hiccup. The complex affine line is what everyone calls the complex plane, for obvious reasons. It has complex dimension 1 but real dimension 2. The same is true for the complex projective line. Topologically, it's a sphere, with real dimension 2, but complex dimension 1.

• instead of "complex projective line" did you mean plane? Commented Mar 17, 2023 at 5:28
• @Anixx No, the complex projective line is one-dimensional with respect to $\mathbb C$, even if it’s two-dimensional with respect to $\mathbb R$, so it’s somewhat confusingly called a line. Every field has its projective line, and behind these two examples the real dimension of course stops being available anyway, so “line” is really all there is to call it. Commented Mar 17, 2023 at 6:00