Real Numbers vs Extended Real Numbers

  • Context: Undergrad 
  • Thread starter Thread starter srfriggen
  • Start date Start date
  • Tags Tags
    Numbers Real numbers
Click For Summary

Discussion Overview

The discussion revolves around the distinction between real numbers and extended real numbers, particularly in the context of mathematical notation and definitions. Participants explore whether it is common to use the symbol ℝ to represent the extended reals, and they delve into the implications of this notation in various mathematical contexts, including limits and indeterminate forms.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • One participant questions whether the use of ℝ in their courses included infinity, suggesting that it may have been shorthand for the extended reals.
  • Another participant argues that it is uncommon and potentially misleading to use ℝ to denote extended reals, recommending alternative notation such as \mathbb{\overline{R}}.
  • There is a query about the definition of boundaries for the reals, with one participant asserting that the reals do not include infinity and have no boundaries.
  • A participant raises the question of whether 1/∞ is an indeterminate form or equals zero, noting conflicting opinions from professors and the distinction between real and extended reals in this context.
  • Another participant clarifies that in the real numbers, 1/∞ is nonsensical, while in the extended reals, it is defined as zero, and emphasizes the importance of context when discussing limits.

Areas of Agreement / Disagreement

Participants express differing views on the notation of ℝ and its implications, particularly regarding the inclusion of infinity. There is no consensus on whether 1/∞ should be considered an indeterminate form or equal to zero, with multiple perspectives presented.

Contextual Notes

Participants highlight the limitations of definitions and notation in mathematics, particularly regarding the treatment of infinity and boundaries in the context of real and extended real numbers. The discussion remains open-ended with unresolved questions about notation and definitions.

srfriggen
Messages
304
Reaction score
7
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?
 
Physics news on Phys.org
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
 
Last edited by a moderator:

Similar threads

High School How can hyperreal numbers make infinitesimals logically sound in calculus?
  • · Replies 24 ·
Replies
24
Views
7K
Undergrad Countably Infinite Unions and the Real Numbers: Can They Really Be Uncountable?
  • · Replies 4 ·
Replies
4
Views
3K
Graduate How Do Complex Numbers Extend the Real Number System?
  • · Replies 2 ·
Replies
2
Views
4K
Intro Math Plan to revise my learned math & start learning pure math
  • · Replies 5 ·
Replies
5
Views
5K
Analysis Study plan for Functional Analysis - Recommendations and critique
  • · Replies 13 ·
Replies
13
Views
5K
Graduate Rational numbers and periodic decimal expansions
  • · Replies 1 ·
Replies
1
Views
5K
Undergrad Is Zero a Concept or a Real Number?
  • · Replies 34 ·
2
Replies
34
Views
9K
Undergrad Lorentz Transformation: Premises + Derivation
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Cantor's Diagonalization Proof of the uncountability of the real numbers
  • · Replies 93 ·
4
Replies
93
Views
21K
Undergrad What is the definition of greater than or less than in terms of real numbers?
  • · Replies 2 ·
Replies
2
Views
5K