What is meant by ∞ ?

  1. I know ∞ is not really a number. It represents 'greater than every real number'. And for any real number x, we can say
    -∞ < x < +∞

    Now, my questions are -
    (A) how come arithmetic operators interact with ∞ if it is not a number?
    (B) what are the results for the expressions below?
    (C) and also why we do have results for these if ∞ is not a number?

    01. (+∞) + (+∞) =
    02. (-∞) + (-∞) =
    03. (+∞) + (-∞) =
    04. (+∞) - (+∞) =
    05. (+∞) - (-∞) =
    06. (-∞) - (-∞) =
    07. (+∞) * (+∞) =
    08. (-∞) * (+∞) =
    09. (-∞) * (-∞) =
    10. (+∞) / (+∞) =
    11. (+∞) / (-∞) =
    12. (-∞) / (+∞) =
    13. (-∞) / (-∞) =
    14. (+∞) + any real number > 0 =
    15. (-∞) + any real number > 0 =
    16. (+∞) - any real number < 0 =
    17. (-∞) - any real number < 0 =
    18. (+∞) * (any real number > 0) =
    19. (+∞) * (any real number < 0) =
    20. (-∞) * (any real number > 0) =
    21. (-∞) * (any real number < 0) =
    22. (+∞) * 0 =
    23. (-∞) * 0 =
    24. (+∞) / (any real number > 0) =
    25. (+∞) / (any real number < 0) =
    26. (-∞) / (any real number > 0) =
    27. (-∞) / (any real number < 0) =
    28. (+∞) / 0 =
    29. (-∞) / 0 =
    30. (+∞) ^ (any real positive number except 0, 1) =
    31. (+∞) ^ (any real negative number except 0, -1) =
    32. (-∞) ^ (any real positive number except 0, 1) =
    33. (-∞) ^ (any real negative number except 0, -1) =
    34. (+∞) ^ 0 =
    35. (+∞) ^ 1 =
    36. (+∞) ^ -1 =
    37. (-∞) ^ 0 =
    38. (-∞) ^ 1 =
    39. (-∞) ^ -1 =
    40. (+∞) ^ (+∞) =
    41. (-∞) ^ (-∞) =
    42. (+∞) ^ (-∞) =
    43. (-∞) ^ (+∞) =
    44. 1 / (+∞) =
    45. 0 / (+∞) =
    46. 1 / (-∞) =
    47. 0 / (-∞) =
    48. 0 / 0 =
    49. 1 / 0 =
    50. 0 / (+∞) =
    51. 0 / (-∞) =
    52. 1 ^ (+∞) =
    53. 1 ^ (-∞) =
    54. 0 ^ (+∞) =
    55. 0 ^ (-∞) =

    I know that is long. But I will very much appreciate your help.
     
  2. jcsd
  3. Every expression in the list above should be interpreted as a shortened form of a problem involving limits. For example, #1 stands for $$ \lim_{x, \ y \ \to +\infty} (x + y) = +\infty $$ This can be proved rather trivially. ## x \to \infty ## means that for any given ## X > 0 ## there is ## x > X ##; same for ## y ##. Thus, if given any ## Z > 0 ##, let ## X = Y = Z/2 ##, then there are ## x > X = Z/2 ## and ## y > Y = Z/2 ##, so ## x + y > Z ##, which means ##(x + y) \to \infty##.

    #22 is trickier. It can mean two things: $$ \lim_{x \to +\infty} x \cdot 0 $$ and $$ \lim_{x \to +\infty, \ y \to 0 } x \cdot y $$ The first of these is zero. The second cannot be resolved unless some relationship between ## x ## and ## y ## is known. For example, if ## y = x^{-1} ##, then, obviously, the limit is 1. If ## y = x^{-2} ##, then the limit is zero. If ## y = x^{-1/2} ##, the limit is ##+\infty##. If ## y = -x^{-1/2} ##, the limit is ##-\infty##.
     
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  4. arildno

    arildno 12,015
    Science Advisor
    Homework Helper
    Gold Member

    1. "I know ∞ is not really a number"
    ----------------
    Incorrect.
    Infinity is not a real number, but might perfectly well be a number in another number system than the reals.
     
  5. arildno

    arildno 12,015
    Science Advisor
    Homework Helper
    Gold Member

    Am I allowed to prefer it to be equal to 5?
     
  6. You're equivocating here. The "infinity" referred to in set theory (i.e. the cardinal numbers) is not the same as the "infinity" referred to in analysis in the context of limits, or the extended real line. The word has many different meanings in different disciplines.
     
  7. HallsofIvy

    HallsofIvy 40,303
    Staff Emeritus
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    Of course- for sufficiently large values of 5!
     
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  8. arildno

    arildno 12,015
    Science Advisor
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    Gold Member

    :biggrin:
     
  9. Why bring 120 into this?
     
  10. Mark44

    Staff: Mentor

    The exclamation point is punctuation, not factorial. HoI is being facetious...
     
  11. You don't say... :rolleyes:
     
  12. Mark44

    Staff: Mentor

    I couldn't tell whether you were asking seriously or were attempting to be humorous...
     
  13. If you seriously thought I was talking about 5! being 120 then I think you have to check your funny bone.

    I'm not saying any of the recent posts were particularly funny, but I was just adding to the "comedy" which HoI started. o:)
     
  14. Mark44

    Staff: Mentor

    People on this forum write all sorts of stuff that seems ridiculous, but that they seriously mean. Since you gave no indication that you were asking with tongue firmly placed in cheek, how was I to know? In your later posts you included the smiley faces, so I could tell your intention.
     
  15. arildno obviously was on the same lines as me. ie we both knew HoI was having a joke.

    Nothing more to be said really.
     
  16. Mark44

    Staff: Mentor

    It's not about whether HoI was being facetious - that was clear to me as well. What I'm saying is that it wasn't clear to me whether you got that joke.
     
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