Answering Questions About ∞: Arithmetic Operators, Results & More

[jacket]

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.

 

[voko]

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 $-\infty$.

 

[arildno]

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.

 

[goldust]

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

 

[arildno]

Infinity represents a value as big as you want it to be.[/url]

Am I allowed to prefer it to be equal to 5?

 

[Number Nine]

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

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.

 

[HallsofIvy]

Am I allowed to prefer it to be equal to 5?

Of course- for sufficiently large values of 5!

 

[arildno]

Of course- for sufficiently large values of 5!

 

[skiller]

Of course- for sufficiently large values of 5!

Why bring 120 into this?

 

[Mark44]

Of course- for sufficiently large values of 5!

Why bring 120 into this?

The exclamation point is punctuation, not factorial. HoI is being facetious...

 

[skiller]

The exclamation point is punctuation, not factorial. HoI is being facetious...

You don't say...

 

[Mark44]

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

 

[skiller]

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

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.

 

[Mark44]

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.

 

[skiller]

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.

arildno obviously was on the same lines as me. ie we both knew HoI was having a joke.

Nothing more to be said really.

 

[Mark44]

arildno obviously was on the same lines as me. ie we both knew HoI was having a joke.

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.