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Infinity - Wizards, please help me out

  1. Feb 28, 2005 #1
    I am a Civil Engineer so I have some college education in mathematics and Calculus. But I am also an old fart so it has been quite a few years since my head had to use this stuff seriously (my work never required it). I tell you this so that you know I am a partial dummy and not a total dummy. As a believer in God, I am asking this question to relate to a theological question (i.e. Creation Ex Nihilo, creation out of nothing).

    As you know, the smaller you make a denominator/divisor "B", the larger the answer "A":

    C/B = A ---->Value A increases as value B decreases

    Hence as B approachs "zero," value A becomes massively large, relatively speaking.

    IF B = "0" is to create an "undefined" answer on your calculator.

    However, is the "undefinable result" not actually the concept of "Positive Infinity" given C is a positive value, since "Infinity" is indeed an "undefinable" thing.

    ...since 0 is an "infinitely small," value 'A' must be infinitely "big."

    1. Is this correct?

    X/0 = +INFINITY (where X is a positive value).

    2. If so, is this then correct?

    IF X/0 = Infinity

    THEN, cannot we also say:

    (INFINITY) x (0) = X

    In such a case, X can be ANY finite value.

    This simple equation would therefore describe something Infinite creating something which is anything out of nothing.

    If this simple mathematical "logic" is not correct (as I suspect), help me understand why this is not correct.

    Thank you.
     
    Last edited: Feb 28, 2005
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  3. Feb 28, 2005 #2

    HallsofIvy

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    It is correct to say that lim (as b-> 0) of A/b "is infinity" (which most mathematicians think of as short hand for "there is no limit because it just keeps getting bigger).

    If you want to say that X/0= infinity, then you are going to have to very precise about which definition you are using for "infinity"- infinity is NOT a regular real number and there are several different ways of extending the real numbers to include a notion of "infinity". In none of them do the regular laws of arithmetic work with infinity: if you do have infinity defined in such a way that X/0= infinity, it does NOT follow that X= 0*infinity!
     
  4. Feb 28, 2005 #3

    Yes, I am not defining infinity as a regular real number (which I perceive is quite impossible anyway if one remains true to the concept of infinity as a never ending unlimited unbounded continuum).

    However, if a "correct" idea of infinity is used, and if X/0 = INFINITY, then can I also correctly say that:

    INFINITY x 0 = X.
     
    Last edited: Feb 28, 2005
  5. Feb 28, 2005 #4

    matt grime

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    No, you cannot since you'v not defined the arithemetic of your non-real object "infinity". In none of the arithmetic extensions (that I am aware of) of the reals are is infinity*0 defined.
     
  6. Feb 28, 2005 #5

    DaveC426913

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    I am not an expert, but it is my understanding that x/0 does not equal infinity, it is literally undefined. The distinction is deliberate.

    i.e "there is no number that can be assigned to the right side of the equation that uniquely equals the left side of the equation".

    This why is proper mathematical rules do not apply. You cannot rearrange the equations as you see fit, because the two sides of your equation (x/0=infinity) are not, in fact, equivalent.

    For a more demonstrable answer, try the manipulation longhand:
    Start here:
    X/0 = Infinity
    End here:
    (INFINITY) x (0) = X
    Write out each step (whatever you want, divide both sides by X, or whatever)
    Don't break any rules.
     
  7. Feb 28, 2005 #6
    I do understand what you are saying and this has occurred to me. However, one wonders if one should be bound to the "rules of arithmetic" here anymore than one would drag his mathematical and algebraic rules into his first Calculus class in order to govern the the methods of Calculus (we all know what happens when you try that one and can't get past that). What I mean is this:

    It does seem to me that it is conceptually "provable" that X/0 = Infinity by realizing that increasingly smaller denominators result in increasingly large results. We also know that going beyond zero into the negative zone results in negative results. In other words, a mirror image of the positive begins to result.

    1/2 = 0.5
    1/1 = 1
    1/0.5 = 2
    1/0 = Infinity
    1/-1 = -1
    1/-2 = -2
    1/-2 = - 0.5

    .... 1/ - Infinity = 0

    and

    1/+Infinity = 0

    So if 1/0 = Infinity

    and

    1/Infinity = 0

    where 1 could be any other value X...

    Is not the concept of Infinity X 0 = X at least "provable" in theory?

    Let me put it another way:

    Multiplication and division are opposite concepts, reverse mirror images of each other (not to be confused with neg vs pos mirror images).

    That being said, would not this fixed concept thereby create a "rule" (that you are asking for) that

    IF X/0 = Infinity

    THEN Infinity x 0 = X

    ???

    Put another way, it appears to me the "rules of arithmetic" do not apply to X/0 in the first place since you cannot arithmetically divide by zero and arrive at a result since standard arithmetic is quite incapable of it. Why then would we need to apply arithmetic rules when you multiply Infinity by X? It sounds to me like we are asking for a rule to govern an equation when it should not. Does the equation reside "under the law" of arithmetic in the first place?

    It appears to me that you are asking whether or not the fixed arithemetic concept of multiplication mirroring division, as one being the reverse image of the other, and as an arithmetic rule with Real numbers/values, actually does indeed apply to such an equation containing "infinity" and hence just because X/0 = Infinity, does it really mean the "reverse" is true (Infinity x 0 = X)? Would you agree?

    In other words, can we not ask, "Is not the one equation (division) the reverse mirror image of the other (multiplication)?"

    X/0 = INFINITY<----|---->INFINITY x 0 = X

    And at least conceptually conclude that it must be so unless we can demonstrate otherwise?
     
    Last edited: Feb 28, 2005
  8. Feb 28, 2005 #7
    Hmm, what "rules of arithmetic" (the field axioms) have been broken in calculus ? Have you taken a course in real analysis (non-math majors are usually not required to take this course) ?
    This is only if you define the symbol "infinity" to mean "the limit of a sequence which increases without bound". In order to speak sensibly about logical concepts we must define precisely what you mean by your use of the symbol "infinity". Otherwise, we will be going around in circles with opinions about what your symbol "infinity" is instead of answering the actual question. You cannot rigorously prove an ambiguous statement. Whittling ambiguous sweeping statements down to precise terms is the first step on the way to understanding and using them.
    Actually, you have shown that the term "infinity * 0" is indeterminate. Read your sentences again and read the definition of indeterminate. You have shown that your product can be taken to be any value of x, and thus it is indeterminate.
     
  9. Mar 1, 2005 #8

    matt grime

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    Ben,
    If you cannot tell me what *you* mean when you talk about multiplying two objects together then that is a fault in *your* argument. Multiplication is an algebraic operation, so feel free to define [tex]\infty*0=X[/tex] but don't expect anyone else to bother with *your* system if you can't state what the symbols mean.

    Mathematical symbols do not have innate properties that exist independently of us. They are symbols to which assign properties, and those properties are what determines the objects behaviour. (Some may argue they exist in some other platonic realm, but that is neither here nor there.)

    In the extended complex plane z/0 for z not zero or infinity is defined to be infinity, where infinity is the topological/geometric point that when added to the plane makes it compact. z/infinity for z not zero or infinity is 0. But that is only something that makes sense in the Riemann sphere.
     
  10. Mar 1, 2005 #9

    HallsofIvy

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    "one wonders if one should be bound to the "rules of arithmetic" here"

    It's not that you should or even can be "bound" to rules of arithmetic but, as has been pointed out repeatedly, you have not said what rules you are using. Until you say that, your question makes no sense.

    By the way, what "mathematical and algebraic rules" do you feel do NOT work in Calculus class?
     
  11. Mar 1, 2005 #10

    I generally understand what everyone is saying here and I appreciate your patience. But what I am trying to get at is the idea that discovery of a concept should not be guided by preconceived rules which could thereby prevent that discovery. It is like saying, "Don't look here until you have a road map to prove you should look there and can show us how to get there."

    I am not here demanding the veracity of a concept but exploring the possibility of a concept. In order to do this it seems to me one must think outside the box and ask ourselves what we are actually doing with the symbols themselves.

    It has been pointed out that X is indeterminate in the expression (I hope I understand this correctly). To me this is just another way of saying, "We have no systematic mathematical way of discovering what X should or should not be." So then does that mean we should not explore without a road map? It seems to me that everything said so far is insisting that this is in fact what we should do.

    Forgive me if I am going off in a direction without seeing all the problems associated with it. Like I said, I am not a wizard at this and am just trying to come to grips with a concept. I am viewing INFINITY is an increasingly growing continuum or series without end whether one dimensional or multi-dimensional.

    What IF there is a way to discover that the expression:

    INFINITY * 0 = X

    ...is a way of indicating that X is indeed determinate by INFINITY itself but is not determinate by the finite mind viewing the expression from a finite perspective that is thereby unable to determine X due to these limitations? And what IF there is a means to establish that while we cannot find X to be determinate we can discover that INFINITY can/does find X to be determinate?

    In other words, we have rules that determine A in the expression:

    C * B = A

    and these rules dictate to C and B what A shall be and C and B themselves as fixed finite entities are themselves governed by these rules.

    And what IF the same does not apply to INFINITY and we can establish this by allowing ourselves to set aside the normal rules for one moment and discover a new rule for INFINITY? What if the factor that determines X is not a set of known rules or concepts we know but a factor residing in INFINITY itself? And what if this is discoverable?

    And what IF the solution to the question is a function of relativity? While something may not be determinate from one perspective perhaps it is quite determinate from another perspective. What if X is determinate from the perspective of INFINITY but indeterminate from the perspective of the finite mind viewing the expression?

    Let me put it to you another way. As I am writing this post, there is no poem "X" so far in this box. The box contains "0" poem up to this point in time. But in just one moment Poem "X" will indeed exist. Let us suppose I have an infinite mind with endless creative and poem writing possibilities. And let us say that "X" is the poem in its final state.

    INFINITY * 0 = X

    My Creative Mind takes nothing and produces something which could be anything "X" and "X" is determinate by my the infinite possibilities that my mind can create.

    The "X" Poem

    Roses are red and violets are blue
    Infinity is creative and so are you
    I began with nothing but what my mind will do
    And the result is Poem "X" and that is true

    Hence, there was absolutely NO WAY for YOU to determine X but if my mind is infinity, having infinite poem creating possibilities, then X is determined by me and quite determinate on my own account. X was indeed determinate and it was a function of relativity, that is, it was a function of seeing X from a different spatial perspective than your own and a function of time.

    Hence, the expression is not 'governed by rules' but INFINITY in the expression 'makes the rules.'
     
    Last edited: Mar 1, 2005
  12. Mar 1, 2005 #11
    The fact that B never fully reaches zero would be a start. Further C/B never fully reaches infinity, since it doesn't have an actual value. Check this out:
    http://www.geocities.com/jsfhome/Think4d/infinity.html
     
  13. Mar 2, 2005 #12

    matt grime

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    Ben, you're into the realms of philosophy and not mathematics anymore.
     
  14. Mar 2, 2005 #13
    Is there such a thing you wizards know about that looks something like this and that is solvable?

    (1,2,3,4......) * A = B

    And

    (1,2,3,4......) * 0 = "X"
     
  15. Mar 27, 2008 #14
    Rearranging

    I think the whole thing comes down to the rearrangement of your equation.
    say I had the equation
    A/B=C
    To rearrange this to make A the subject I would have to multiply both sides by B to remove the denominator

    (A/B)*B = C*B
    (A*B)/B = CB
    A= CB
    In this case you have

    (X/0)*0 = Infinity*0
    (X*0)/0 = Infinity*0
    0/0 = Infinity*0
    And this poses a new question. what does 0/0 equal?
     
  16. Mar 27, 2008 #15
    The problem is here, this is not correct.

    Your issue is that you think that, X / 0 = +infinity

    Is the same thing as saying:
    The Limit as C approaches 0 (from the right) of B/C (where B is a positive Real) is equal to +infinity.

    These are two VERY different statements, however you are treating them as the same.
     
  17. Mar 27, 2008 #16

    CRGreathouse

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    Ben, you can define whatever terms you like. Mathematicians can then look at the rules you set up and see if they're consistent and if they meet certain 'important' properties like being a field or a ring over + and *.

    Let's start like this. Let the set [itex]\mathcal{B}[/itex] be the real numbers, [itex]\mathbb{R}[/itex], along with [itex]+\infty[/itex] and [itex]-\infty[/itex].

    Then define [itex]b\times0=0=0\times b[/itex] for all [itex]b\in\mathcal{B}[/itex], [itex]+\infty\cdot b=+\infty[/itex] for all [itex]0<b\in\mathcal{B}[/itex], and the 'usual' ordering <.

    Then the inverse [itex]\div[/itex] of [itex]\times[/itex] does not exist, since [itex]1\times+\infty=2\times+\infty[/itex] so there's no way to choose 'which one' works. That is, it's not even sensible to talk about X/0.
     
  18. Mar 28, 2008 #17
    Forgive me if I am just way of base in replying to this, but ...

    If I understand math correctly (which is open for debate), then

    x/0 = undefined. Division by zero is undefined, it has no answer, let alone infinity.

    What Ben has been describing (as I understand it) is a different question.

    Lim( x/y ) as y approaches zero. (sorry, not up on my Latex formating).

    While the Limit certainly may exist as positive infinity, the actual y = 0, is asymptotic, which means that it is undefined at zero.

    Algebraically, I do not know of a way to do the algebra to take the "0" out of the Limit, but that is what Ben has done.
     
  19. Mar 29, 2008 #18

    arildno

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    Here is your basic misunderstanding.
    True, a set of rules can be said to determine a class of discoverable concepts or theorems, namely those that follows logically from those rules. Also, that set of discoverable concepts are NOT immediately obvious to us, we'll need to ponder them out!

    Furthermore, it is certainly true that the same set of rules will determine two other classes of concepts, namely those that stand in condradiction to logical consequences of our set of rules (i.e, "false" concepts) and those concepts whose "truth" status cannot be determined by the original set of rules (let's call them meaningless or indeterminate staments, shall we?)


    Now, what YOU bemoan is that it might be concepts in those latter two classes which might be interesting, but that our set of rules "prevent" them from being elucidated or "discovered".

    Nothing of this is true.
    It is, in fact very SIMPLE to discover such cases, namely by changing the set of rules you are working with!

    Nobody is preventing you from doing that, and you can make your own maths, according to your premises as much you like.

    Numerous such "alternative" maths have already been made.

    Such systems are necessarily logically incommensurable to each other, since they disagree among their first principles. I.e, the one system is not "truer" than any of the others.

    Truth in math is always relative to the basic premises made; there are no premiseless absolute truths in maths.
     
  20. Mar 29, 2008 #19

    Hurkyl

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    Since this thread is three years old, and our necromancer doesn't seem to be paying attention anymore, I'm going to close this thread.
     
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