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Infinity and number system

  1. Jul 14, 2004 #1
    Hi all !

    I came across an article (link: http://www.math.toronto.edu/mathnet/answers/infinity.html#2)

    The article states that in the context of a number system, infinity does not exist.

    Now, being the dimwit that I am, I just wanted to ask/confirm this:

    I THINK the article is saying that one cannot perform arithmetic operations like addition, subtraction etc. using infinity (plz do correct me if I'm wrong).

    If so, this would mean one simply cannot perform operations like:

    infinity + infinity = infinity

    OR

    infinity - infinity = ?

    OR

    3 * infinity = infinity

    Wouldn't all such operations be "illegal"? (this would of'course depend on whether I understood that article correctly).

    Thanking you in advance,
    Asif.
     
  2. jcsd
  3. Jul 14, 2004 #2
    [tex]\infty[/tex] is not a number. I prefer to think of it as a limit. Say you have some function which limit in point a is [tex]\infty[/tex], then it means that for each M>0, there is a [tex]\delta[/tex] such that when [tex]|x-a|<\delta[/tex] then f(x)>M.
     
  4. Jul 14, 2004 #3
    That's an interesting article.

    I think that what the author means by "usual rules of addition" is that the "number system forms a/an (abelian) group under addition"? Infinity cannot be regarded as "a number" because +ve/-ve infinity by definition is an element which is greater/smaller than any number. Thus,

    infinity + a = infinity (where a is any finite number)

    But if we regard infinity as "a number" for which the "usual rules of addition/subtraction" hold, then we may subtract infinity from both sides to obtain

    a = 0

    a contradiction.

    As to whether the operations you mention (i.e. (+ve)infinity + (+ve)infinity = (+ve)infinity) are legal, I think one may *define* them to be that way. Note that such relation is consistent with the ordering of elements (that +ve infinity is greater than any other elements), but at the expense that "the usual rules of addition/subtraction" do not hold for expressions involving "infinities".
     
  5. Jul 14, 2004 #4

    Zurtex

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    As said, infinity is not a number.

    Often it is just used as a mathematical short hand meaning slightly different things in different contexts. For example:

    We could have some function of x, y = f(x). Some properties of this function maybe:

    As [tex]x \rightarrow \infty[/tex]
    Then [tex]y \rightarrow \infty[/tex]

    This simple means as x gets bigger, y does not converge on a number and just keeps getting bigger.

    There are mathematical systems for arithmetical operations on transfinite numbers, perhaps someone else will post about those.
     
  6. Jul 14, 2004 #5
    [tex]+\infty[/tex] and [tex]-\infty[/tex] are elements in the extended reals arn't they?
     
  7. Jul 14, 2004 #6

    HallsofIvy

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    Yes, but in the originally quoted text they specifically referred to a number system with "rules of arithmetic" similar to standard arithmetic. As noted before, the "extended reals" don't follow that. The extended reals are geometric (strictly speaking, topological) extension, not algebraic.
     
  8. Jul 14, 2004 #7
    oh indeed.
     
  9. Jul 14, 2004 #8
    Guys, y'all lost me by the 2nd post...lol

    Lets try to make things simpler for idiots like myself.

    Thus far, what I could gather is that my understanding of what the article was trying to say is not entirely (if at all) correct.

    We can carry out operations like (+ve) infinity + (+ve) infinity, "but at the expense that "the usual rules of addition/subtraction" do not hold for expressions involving "infinities"."

    If the "usual rules" don't apply, then what does apply? And if the usual rules do not apply, then how can one say infinity + infinity = infinity (isn't that making use of the usual rules?)

    Pardon my ignorance,
    bye,
    Asif.
     
    Last edited: Jul 14, 2004
  10. Jul 14, 2004 #9

    JD

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    Might it be the case that infinite numbers exist in theory only, rather than in actuality?
    Do we need to count 'things' (whatever they might be)? For all I know, however, I could be completely wrong.
     
  11. Jul 14, 2004 #10

    Zurtex

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    :uhh: Can you give me an example under the usual rules of addition where:

    a is positive

    and

    a + a = a.
     
  12. Jul 14, 2004 #11

    matt grime

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    philosophically speaking, do any numbers exist in actuality? mathematically the existence of numbers in some physicaly sense is neither here nor there: if all snooks are green, and boojum's blue, then i can reason no boojum is a snook; that i've not told you they are, or aren't, things that exist is not important.

    As to what these alleged infinite numbers are, then we may offer the following interpretation (it is not the only one):

    we mean cardinal numbers, and by addition, we mean that the result of adding |X|+|Y| is the cardinality of XuY, where X and Y are some disjoint sets in the equivalence classes of cardinal numbers. Then Aleph-0+aleph-0=aleph-0
     
    Last edited: Jul 14, 2004
  13. Jul 14, 2004 #12

    JD

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    Given that no cap need be put (if a finite number of things is not being counted) on a number series (as you can always add one to the end of the series) and if numbers do not actually exist (whatever existence might exactly mean in this context), is there a valid argument against infinity?
     
    Last edited: Jul 14, 2004
  14. Jul 14, 2004 #13
    Hi Zurtex !

    You said: "Can you give me an example under the usual rules of addition where:

    a is positive

    and

    a + a = a."

    :smile: You've given my rusty brain cells a jump start...hehe.....so I guess this means (+ve) infinity + (+ve) infinity = (+ve) infinity isn't really making use of the "normal"/"usual" rules of addition.

    <sigh> Math !

    Bye,
    Asif.
     
  15. Jul 14, 2004 #14

    matt grime

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    JD, you'd have to say what you mean by infinity. As has been pointed out, the term infinity means a multitude of things, all linked by the idea of being 'not finite'. It would alse depend on your personal view point of what it means for something to 'exist' mathematically.
     
  16. Jul 14, 2004 #15

    JD

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    That's the difficulty I suppose. Infinity is impossible to visualise, to understand in a rational sense. At the same time, though, it is impossible to visualise a number series (which is not counting a definite number of objects) that cannot be added to indefinitely. So this is impossible in both senses.
    Existence wise, the number would not necessarily have to be applied to anything. I suggested that infinity might be merely theoretical, but that suggests a limitation.
     
    Last edited: Jul 14, 2004
  17. Jul 14, 2004 #16

    matt grime

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    No, that's not what i meant about indicating what you mean by infinity. Infinity's meaning is clearly understood dependent on context, though it is always better use some compound adjective form involving the word infinite when that is possible.

    For instance we know fully what we mean when we say "sum from 1 to infinity", and that is better written as "sum of n in N"

    We know what we mean when we say 1/x tends to infinity as x tends to zero (from above), we mean that for all L>0, there exists d>0 such that 0<x<d implies 1/x >L

    The "point at infinity" is the point we add to the complex plane, say, as a topological object such that it is compact in the norm topology (sorry, that's probably too high-faluting for this discussion), and can be visualized in many ways.
     
  18. Jul 14, 2004 #17

    JD

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    The context that I am interested in is the application of infinite energy to bring slower-than-light-speed objects up to light speed or slow FTL objects down to light speed.
     
    Last edited: Jul 14, 2004
  19. Jul 14, 2004 #18
    actually it really depends on what you're doing. You folks would probably be interested in Abraham Robinson's Nonstandard Analysis (usually abbreviated NSA) in which there are infinitely large numbers and infinitely small numbers. In NSA there are no (or very few) limits. You just "use" infinities and infinitesimals.

    http://members.tripod.com/PhilipApps/nonstandard.html

    is a link with some stuff and many more links for the interested parties.

    Kevin
     
  20. Jul 14, 2004 #19
    Oh I should add that Robinson also proved that a theorem true in NSA is also true in ordinary analysis and vice versa.

    Kevin
     
  21. Jul 14, 2004 #20

    matt grime

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    in that case i will be moving on from the discussion.
     
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