A motivation for the definition of the "extended real line"

Question

Recently I came across the definition of the extended real line $\overline{\mathbb{R}}=\mathbb{R}\cup\{-\infty,\infty\}$. My question is: what is the motivation to define $\overline{\mathbb{R}}$? Any attempt to define operation with infinity (so it will work out with our intuition) will be doomed to failure. Does this definition simplify some of our theorems formulation in calculus? If so, I will be happy to see examples. Thanks!

Answer 1

Defining arithmetic on the extended real line is not doomed to failure; it's a well-understood and standard construction.

And it really is fairly intuitive; introductory calculus students learn about it every year without realizing it. You recognize, I assume, that

$$ \lim_{x \to +\infty} 5x = +\infty $$

How did you determine that? It's almost certainly going to be from the fact that $\lim_{x \to +\infty} x = +\infty$ together with the fact that $5 \cdot (+\infty) = +\infty$, except you will probably express everything in more long-winded and roundabout fashion that avoids acknowledging $+\infty$ as a number.

Other statements like $\arctan(+\infty) = \pi/2$ are easily understood, and much easier to work with than if you tried to avoid the extended real numbers.

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The extended real numbers, together with the topological definition of limit, immediately resolves all 9 of the different kinds of limits you learned in calculus (depending on whether the limit is $\pm \infty$ or a real number, and whether $x$ is approaching $\pm \infty$ or a real number) into a single kind of limit.

Other special cases in various definitions and theorems can be eliminated as well. For example:

• Every monotonic sequence has a limit in the extended real numbers — no need to restrict to bounded sequences.
• Every set of extended real numbers has a least upper bound. No need to require that the set is bounded. No need to require that the set is nonempty.

The extended real numbers are better behaved than the real numbers in almost every topological manner.

If you're familiar with the term, one major aspect of this is that the extended real numbers form a compact topological space.

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The main "counter-intuitive" aspect of arithmetic on extended real numbers isn't actually about the extended real numbers at all — it's simply that, to date, nearly every arithmetic structure you've studied has been closely modeled on the arithmetic of the natural numbers and/or real or complex numbers.

Since the extended real numbers are not based on that — their primary motivation is geometric/topological in nature — you will run afoul of your expectations on arithmetic structures.

E.g. a nonzero solution to $2x = x$ is surprising when you hold the expectation that all arithmetic is just like that of the real numbers. It's completely unexpected, though, when understood from a topological viewpoint; e.g. you obviously have

$$ \lim_{x \to +\infty} 2x = \lim_{x \to +\infty} x $$

so naturally you would expect

$$ 2 \cdot (+\infty) = +\infty$$

Answer 2

A typical application would be the formulation of results concerning the Lebesgue measure $\mu$ of subsets of $\mathbb R$. Instead of saying that $\mu$ is undefined for $\mathbb R$ itself for instance, one writes $\mu(\mathbb{R})=\infty$.