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What is meant by ∞ ? 
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#1
Oct2713, 06:59 AM

P: 32

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
Oct2713, 08:29 AM

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P: 5,794

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##. 


#3
Oct2713, 08:51 AM

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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. 


#4
Oct2713, 11:09 AM

P: 85

What is meant by ∞ ?
Infinity represents a value as big as you want it to be. That's the way I think of it. However, infinity cannot be a number because of Aleph numbers. http://en.wikipedia.org/wiki/Aleph_number



#5
Oct2713, 11:13 AM

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Oct2713, 02:15 PM

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#7
Oct2913, 10:38 AM

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#8
Oct3013, 09:41 AM

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#9
Oct3013, 01:10 PM

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#10
Oct3013, 01:18 PM

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#11
Oct3013, 01:25 PM

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#12
Oct3013, 02:05 PM

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I couldn't tell whether you were asking seriously or were attempting to be humorous...



#13
Oct3013, 02:16 PM

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


#14
Oct3013, 02:41 PM

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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
Oct3013, 03:01 PM

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Nothing more to be said really. 


#16
Oct3013, 03:06 PM

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