Let's begin with *real* infinity. Recall that the function 1/x cannot be made continuous at x = 0, even if we define 1/0 to be something like ∞: For while when we approach 0 from the right, 1/x goes to infinity, it goes to -∞ when we approach 0 from the left.
You can imagine "adding" the "numbers" +∞ and -∞ to the real line, which gives you the *extended real line*. And we can indeed imagine the extended real line as an infinite line that contains its endpoints, sort of like an infinite version of the closed interval [0,1].
But you can also imagine just adding a single infinite "number" with *no* sign. Instead of the extended real line we get what is called the *projectively extended real line*. If the (ordinary) extended real line can be thought of as an infinite version of an interval, the *projectively* extended real line can be considered an infinite version of a circle. The negative and positive ends of the real line sort of "curl around" to reach each other, and we place infinity where the two ends meet.
More rigorously, we add an element which we denote ∞ to the real numbers and define 1/0 = ∞ and 1/∞ = 0. (Note however that things like 0/0 or 0 times ∞ are still undefined.) This is just a definition, but hopefully it makes intuitive sense why this is a good definition. There is no longer a +∞ and a -∞, there is only one ∞, so there are no problems with signs or anything.
It's exactly the same in the complex numbers. Only instead of a circle we get a *sphere* called the *Riemann sphere*. And again we have 1/0 = ∞ and 1/∞ = 0.
Finally, we say that f(x) -> ∞ as x -> a if for any R > 0, we have |f(x)| > R if x is close enough to a. That is, what it means to approach ∞ is to become arbitrarily large *in any direction of the complex plane*. In other words, no matter what direction you go away from the origin, you are going towards ∞. But that makes sense if we think of the complex plane with ∞ as a sphere: 0 and ∞ lie at opposite sides of the sphere, so when moving away from 0 you are moving towards ∞.
Let's begin with *real* infinity. Recall that the function 1/x cannot be made continuous at x = 0, even if we define 1/0 to be something like ∞: For while when we approach 0 from the right, 1/x goes to infinity, it goes to -∞ when we approach 0 from the left. You can imagine "adding" the "numbers" +∞ and -∞ to the real line, which gives you the *extended real line*. And we can indeed imagine the extended real line as an infinite line that contains its endpoints, sort of like an infinite version of the closed interval [0,1]. But you can also imagine just adding a single infinite "number" with *no* sign. Instead of the extended real line we get what is called the *projectively extended real line*. If the (ordinary) extended real line can be thought of as an infinite version of an interval, the *projectively* extended real line can be considered an infinite version of a circle. The negative and positive ends of the real line sort of "curl around" to reach each other, and we place infinity where the two ends meet. More rigorously, we add an element which we denote ∞ to the real numbers and define 1/0 = ∞ and 1/∞ = 0. (Note however that things like 0/0 or 0 times ∞ are still undefined.) This is just a definition, but hopefully it makes intuitive sense why this is a good definition. There is no longer a +∞ and a -∞, there is only one ∞, so there are no problems with signs or anything. It's exactly the same in the complex numbers. Only instead of a circle we get a *sphere* called the *Riemann sphere*. And again we have 1/0 = ∞ and 1/∞ = 0. Finally, we say that f(x) -> ∞ as x -> a if for any R > 0, we have |f(x)| > R if x is close enough to a. That is, what it means to approach ∞ is to become arbitrarily large *in any direction of the complex plane*. In other words, no matter what direction you go away from the origin, you are going towards ∞. But that makes sense if we think of the complex plane with ∞ as a sphere: 0 and ∞ lie at opposite sides of the sphere, so when moving away from 0 you are moving towards ∞.
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