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What does 1 / infinity =

  1. Mar 26, 2003 #1
    1 / infinity = ?


    is there anything that we can reduce this to?
    (i doubt there is, but you never know)


    well, it's REALLY close to 0

    for all practical purposes 1 / infinity = 0

    ================

    just thought of something!

    ok, start with
    1 / infinity = x

    1 = x * infinity

    ~~~~~ analyze this ^^

    x must be a REALLY small number to get infinity back down to 1

    i.e. x = .0000000000000000000001

    aka an infinite number of 0's and then a 1???

    or an infinite number of 0's and then a 2?

    or an infinite number of 0's and then a --
    3?
    47?
    31415926535897932?
    infinity? (hah .00000000000000000infinitity)

    ==================
    i suppose you can't really get anywhere when dealing with these "infinities"

    im stuck
     
  2. jcsd
  3. Mar 26, 2003 #2
    you're trying to make too much sense out of something which you can't do it with. The only way you can really think about it as is the limit of 1/x as x approaches infinity, not this crazy other stuff that you are floating around.
     
  4. Mar 26, 2003 #3

    Integral

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    1/infinity =0

    There is no need to play games with it.

    Infinity is a carefully defined extension of the real number system. since it is not a real number all operations involving infinity and the real number system are defined.
     
  5. Mar 27, 2003 #4
    1/infinity=infinitesimal

    you were right about this, an infinitesimal is an infinite number of zeros after the decimal point with a one at the end, but of course there is no end but ah well!

    infinitesimals are very interesting, archemedes used them to find the volume of the sphere, newton also used them to work out velocity and related stuff.

    now days it is used to explain bronwian motion and other nasty topics.
     
  6. Mar 27, 2003 #5
    1/infinity =0

    No argument.

    Where infinity is an element of the extended real line (real numbers with -infinity and infinity added).
     
  7. Mar 27, 2003 #6

    Hurkyl

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    Infinity isn't a uniquely defined concept. There are many ways to include infinity (infinities) into a number system, each with their own properties.

    The most common one is the example plus gave of the extended real numbers; in that system 1 / infinity is defined simply to be zero, because that's the choice that gives the right answer for the applications for which the extended real numbers were invented.

    Other number systems behave differently. 1 / infinity is undefined for any infinite quantity in the ordinal number system. In other transfinite systems each infinite value has a unique nonzero reciprical (an infinitessimal).


    In the real numbers or other nice systems (like division rings and fields!) that step is fine, but in general number systems you cannot make that step. That step is patently illegal in the extended real numbers when infinity is involved.


    And as I mentioned, you're no longer working in the real numbers, so you cannot generally use decimal expansions to describe the numbers you're using (since decimal expansions always represent real numbers). There are, though, number systems which generalize decimal expansions. For example in my favorite transfinite system:

    1 / infinity = 0 + 1 * infinitessimal
    = ... ; ...000.000... ; ...0000.00000... | ...00001.00000... ; ...000.000... ; ...
    = 0 | 1

    This is kind of a "nested" decimal representation. The numbers to the left of the pipe (|) are the ordinary real digit of the transfinite number. The next grouping left are multiplied by infinity, the next grouping left are multiplied by infinity^2, etc. The grouping to the right of the pipe is divided by infinity, the next to the right is divided by infinity^2, etc.

    This system will let you add, subtract, multiply, and divide infinities, but more advanced operations are not always allowed (e.g. sqrt(infinity) is not defined)

    Hurkyl
     
  8. Mar 27, 2003 #7
    1/infinity is Not equal to zero

    1/infinity is not a number

    however

    limitx->inf1/x = 0

    there's a key difference

    &infinity;
    &inf;
    fty &fty;
     
  9. Mar 27, 2003 #8

    Integral

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    From
    Principles of Mathematical Analysis (Thrid Edition) by Walter Rudin
    on page 12 in the section entitled The Extented Real Number system He states

    "The extented real number system does not form a field, but it is customary to make the following conventions:
    (a) If x is real then

    x + oo = oo, x - oo = -oo, x/+oo = x/-oo =0

    (b) If x>0 then x * (+oo) = +oo, x * (-oo) = -oo
    (c) If x<0 then x * (+oo) = -oo, x * (-oo) = +oo.
    When it is desired to make the distinction between real numbers on the one hand and the symbols +oo and -oo on the other quite explicit, the former are called finite."

    I have used oo to mean the sideways figure eight symbol.


    Notice that this is a matter of convention and is DEFINED to be true. There is no need of limiting processes.
     
    Last edited: Mar 27, 2003
  10. Mar 27, 2003 #9

    arivero

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    Not so easy

    The real quests are how to do a Taylor expansion of f(1/x) around x=infinity. Or also expansions of g(x) with x=infinity, or around h(1/x) with x=0... Poincare did a good deal of mathematics about this kind of questions.
     
  11. Mar 27, 2003 #10
    If you put the oo's between brackets, you'll get the infinity symbol, but you got the point across, and that's what's important.
     
  12. Mar 28, 2003 #11

    Integral

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    Mentat,
    New board, new rules. While we appear to have the greek alphbet, the brackets did not give me the infinty symbol

    &inf;
     
  13. Mar 28, 2003 #12
    infinity is "not limited by any succeeding number"

    1/ infinity is "infinitely near 0" ("not limited by any decreasing number")

    so ..infinity or 1/infinity are not numbers but terms describing limitlessness, just as numbers describe entities (or distinct parts of).
     
  14. Mar 29, 2003 #13
    You have to be careful with the assumption of 1/oo = 0 because that would sort of imply that 0(oo) = 1.
     
  15. Mar 29, 2003 #14

    Integral

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    If you will look at my quote from the Real Analysis text, ALL results of mathematical operations involving infinity are defined. If you stick to the definitons all will be well. Since infinity is NOT a real number it does not follow the same rules as the rest of the reals. It has its own set and they are definied right along with (mathematical) infinity.

    Note that earlier in the section I quoted oo was definined as

    oo>x for all real x>0 and -oo<x for all real x>0
    in this case our x = 0 the definition of infinity does not provide the answer, we must rely on the definition of the 0 element such that 0 * anything = 0, by this definition we have 0 * oo = 0 since.
     
  16. Mar 30, 2003 #15

    Integral

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    Please read my posts. To use infinity with real numbers you must carefully adhere to the definitions.

    0 * oo <> 1

    0 * oo = 0
     
  17. Mar 30, 2003 #16

    Hurkyl

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    0 * oo is indeterminate, not 0.


    (and for those who don't know, indeterminate does not mean "we're not sure"; it has a precise technical meaning related to the contexts where the extended real numbers are used)

    Hurkyl
     
  18. Aug 18, 2009 #17
    AAAHHH!!! 1/infinity=0!!! 1/infinity does not=0!!! why can't this forum make up its mind!!!

    I used to think 1/infinity=0, but now im not so sure thanks to what this guy said.
    great point, i hadn't thought of that.
     
  19. Aug 18, 2009 #18
    What is oo -oo ?
    Since none is real it cannot be oo or -oo.
    Is cannot be zero as well
    How is it that you solve Hilbert's paradox?
     
    Last edited: Aug 18, 2009
  20. Aug 24, 2009 #19

    HallsofIvy

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    This forum "can't make up its mind" because this forum is not one single person and so has many minds. Also whether or not "1/infinity= 0" or not depends upon which "infinity" you are talking about. There is NO "infinity" in the standard real number system but you can have other, more general systems, with different kinds of "infinity".
     
  21. Aug 28, 2009 #20
    i think everyone is mistaking the math of this. 1/∞≠0, however, lim x→∞ (1/x)=0. which is to say that 1/∞ is the closest one can get to zero without being zero.
     
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