# Real Numbers vs Extended Real Numbers

• srfriggen
In summary, the conversation discussed the concept of extended reals, which includes +infinity and -infinity, and whether it is common to use the symbol ℝ to represent it. It was noted that while some may use ℝ as shorthand, it is not a good idea as it can cause confusion. Additionally, it was mentioned that in the reals, there is no concept of infinity or a boundary, but in the extended reals, 1/infinity is defined as 0. However, in certain situations, such as limits, 1/infinity can be considered an indeterminate form and equal to 0.

#### 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?

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?

srfriggen said:
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.

srfriggen said:
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]

Last edited by a moderator: