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Real Numbers vs Extended Real Numbers

  1. Oct 27, 2011 #1
    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?
     
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  3. Oct 27, 2011 #2

    lurflurf

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    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 [tex]\mathbb{\overline{R}}[/tex] or something.
     
  4. Oct 27, 2011 #3
    So do the reals, R, have infinity? How would you define the boundaries?
     
  5. Oct 27, 2011 #4

    micromass

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    By its very definition, the reals do not have infinity. And the reals do not have a boundary (i.e. the boundary is empty).
     
  6. Oct 27, 2011 #5
    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.
     
  7. Oct 27, 2011 #6

    micromass

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    This depends on the situation:

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

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

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

    [tex]\lim_{x\rightarrow +\infty}{\frac{1}{x}}=0[/tex]

    Also see the following FAQ: https://www.physicsforums.com/showthread.php?t=507003 [Broken]
     
    Last edited by a moderator: May 5, 2017
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