Real Numbers vs Extended Real Numbers

[srfriggen]

Today I watched a lecture that talked about the extended reals, ℝ U {+infinity, -infinity}.

Every course I've taken so far (differential calculus through multivariable calculus, linear algebra, a proofs writing course, etc) always defined the Reals with just they symbol ℝ and we always included infinity and negative infinity.

Were all of my teachers just using ℝ as shorthand for ℝ U {+infinity, -infinity} or is there some fundamental difference?

Actually, I shouldn't ask on this forum what my teachers were doing, but rather, is it commonplace to use ℝ as shorthand for the extended reals?

 

[lurflurf]

It is not common and a bad ideal to use R to denote extended reals as it might be confused with the reals better to use $$\mathbb{\overline{R}}$$ or something.

 

[srfriggen]

So do the reals, R, have infinity? How would you define the boundaries?

 

[micromass]

So do the reals, R, have infinity? How would you define the boundaries?

By its very definition, the reals do not have infinity. And the reals do not have a boundary (i.e. the boundary is empty).

 

[srfriggen]

okay thanks.

one last question: is 1/∞ an indeterminate form or does it equal zero exactly. I know the limit as n approached infinity of 1/∞ = zero, but what about the equation.

I ask because I've read posts where people say it equals exactly zero and even two professors (one in my class, the other on a youtube lecture) differed in the answer.

 

[micromass]

okay thanks.

one last question: is 1/∞ an indeterminate form or does it equal zero exactly. I know the limit as n approached infinity of 1/∞ = zero, but what about the equation.

I ask because I've read posts where people say it equals exactly zero and even two professors (one in my class, the other on a youtube lecture) differed in the answer.

This depends on the situation:

- In the real numbers, there is no thing as $\infty$. So $1/\infty$ is nonsense.

- In the extended reals, we do define $1/\infty =0$. But things like 1/0 are still undefined.

- When working with limits, if we encounter a limit of the type $1/\infty$, then the limit is 0. For example

$$\lim_{x\rightarrow +\infty}{\frac{1}{x}}=0$$

Also see the following FAQ: https://www.physicsforums.com/showthread.php?t=507003