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Thread: Fundamentals

  1. #1 Fundamentals 
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    Hi,

    I have nowhere near the level of understanding that you guys have but I have a question that just won't go away. When looking below:

    oo- -9 -7 -5 -3 -1 0 +1 +3 +5 +7 +9 +oo

    They are the two number lines and each of them extends to infinity in both their relevant directions, (with all that comes with infinity).

    Is there any concept in mathematics that joins the two infinities together? Any concept that gives the two continually extending lines, which would be intentionally made to meet, a useful purpose?

    Something that shows a new relationship between positive and negative numbers?

    If so, could a new kind of maths be built on this concept and would it have any worth?

    I'd be happy for your thoughts,

    All the best,

    Ste.


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  3. #2  
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    AFAIK there is no such idea nor is there any use to it. In fact, I think it would only complicate some matters. The concept of and are simply to give directionality to the number line. It goes on in both directions without ever ending, meaning it's infinite in both directions, which is all those symbols mean.


    Wise men speak because they have something to say; Fools, because they have to say something.
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  4. #3  
    Forum Radioactive Isotope MagiMaster's Avatar
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    It's called the one point compactification of the real number line and it has already been well studied. It's also, as far as I can remember, not considered to be terribly useful since it's just a more complicated way of making a circle.
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  5. #4  
    Forum Ph.D. Leszek Luchowski's Avatar
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    You may want to check Projective Geometry and Homogeneous Coordinates. It's not really about numbers, but it does help make ends meet - I mean the two "ends" of a straight line .
    Leszek. Pronounced [LEH-sheck]. The wondering Slav.
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  6. #5  
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    Suppose I take a segment of the real line and join the two ends together. This we recognize as a circle; it's called the 1-sphere . Now since this is essentially the "same" as our line segment , we might expect to be able to find a one-to-one correspondence between them. This correspondence - an isomorphism on sets - will be given by an invertible set function from one to the other (this is called a bijection, btw).

    Of course no such function exists; the map sends the two ends of to same point of - our "join". And since well-behaved functions are not allowed to send a single point in the domain to distinct points in the co-domain, there can be no inverse .

    Now there is a theorem (due I think to Dedekind and Pierce) that states that an infinite set is always isomorphic to a proper subset of itself. Since and since isomorphism is an equivalence relation, by transitivity we might expect to be able to do something like the following.

    Place our circle anywhere on the real line and project each point of "down" to a point in in such a way that none of these projections intersect each other or .

    All goes well until we get to the North pole, where now our projections run parallel to and seem never to meet it.

    But we have the perfect construction specifically made for just this purpose, called the "projective real line". This is formed by adjoining what's called the "point at infinity" to to form .

    So now our point at the North pole projects to this point at infinity, and an isomorphism can be recovered, but at some cost; parallel lines meet!

    Worse still, since the two projections - "left" and "right" - from the North pole map to the same point in our projective line, this implies that, in , the "end-points" of the real line may be considered as the same point.
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  7. #6 Hello again. 
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    Hello again.

    You obviously have a far greater understanding of maths than me.

    I can only think conceptually, which may lead me to make stupid questions but I like to ask because you answer the questions so well and it's really interesting for me.

    I do have another question if I may. My understanding of infinity is that it 'goes on forever' as in the example real number line that I was talking about before.

    Is there any sense in saying that infinity is only possible because humans have an understanding of the concept finite?

    If that is so, could it be said that infinity can also be described as a not an ending, but at some point, changing. Would there ever be a use in maths to say that infinity 'ends here' only under a special and certain condition?

    If we take each of the 3 zero's below to donate a stop / change:

    -0 -9 -7 -5 -3 -1 0 +1 +3 +5 +7 +9 +0

    Could a new definition of infinity(I'm assuming that it already hasn't already been said) be the meeting of the two lines? Does this make any sense or have any useful purpose in maths?

    -0 0+
    -9 +9
    /\
    -7 | +7
    |
    -5 | +5
    |
    -3 | +3
    \/
    -1 +1
    0


    Also if it is useful in some way, what would be the relation between zero and the newly combined infinity(denoted by the central line above).

    Is there a(previously undefined) connection between zero and infinity. A link that is no longer separated by 'numbers'; 'space' or time. I think a lot about black holes, I have done since I was a kid.

    This could well be unstructured and highly non mathematical, but I would like to hear your views as this is a great site, with top people on it.

    All the best,

    Ste.
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  8. #7 And again. 
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    The numbers that are squashed to the side are basically the number line folded in a circle so the two ends( -0 and 0+ ) meet and combine to create a new(possibly?) definition of infinity. There is also then, a line that runs directly from zero to infinity. A direct link. 0 ---------> 00 which may be of some consequence.

    All the best,

    Ste.
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  9. #8 Re: Hello again. 
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    Quote Originally Posted by Stiggy20
    Hello again.

    You obviously have a far greater understanding of maths than me.

    ...Also if it is useful in some way, what would be the relation between zero and the newly combined infinity(denoted by the central line above).

    Is there a(previously undefined) connection between zero and infinity. A link that is no longer separated by 'numbers'; 'space' or time. I think a lot about black holes, I have done since I was a kid.

    This could well be unstructured and highly non mathematical, but I would like to hear your views as this is a great site, with top people on it.

    All the best,

    Ste.
    There is really no definition of "infinity" for good reason.

    There is a pretty clear definition of "finite". A finite set is one that can put into 1-1 correspondence with the set consisting of the first n natural numbers. An infinite set is a set that is not finite.

    But there are lots of different versions of "infinity" which depend on the issues at hand.

    There, is as previously noted the one-point-compcatification of the real line, and the point that is added is usually called "the point at infinity". You can do a similar thing for any non-compact topological space, and the point added is still called "the point at infinity". In the case of the complex plane this procedure yields the Riemann sphere.

    There are also notions of infinite cardinal and ordinal numbers. These make precise the notion of "size" for sets, including infinite sets. And here you find that there are different sizes of infinity. The real numbers are properly larger than the integers, though both are infinite.

    About the only useful link between zero and infinite is that the map is an analyzic map that takes neighborhoods of zero to neighborhoods of "infiinity" on the Riemann sphere.

    In any case these issues have been rather thoroughly studied, and there is no pressing need for new concepts. What would be appropriate is to understand the mathematics that already exists.
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