# Real Numbers vs Extended Real Numbers

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
Homework Helper
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.

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

micromass
Staff Emeritus
Homework Helper
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).

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
Staff Emeritus
Homework Helper
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 [Broken]

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