A question on transfinite quantities.

Discussion in 'Math' started by ErnieM, Sep 8, 2013.

  1. ErnieM

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    A simple question perplexing us on another sub forum:

    Can infinity times zero ever equal a constant?
     
  2. WBahn

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    Yes and no.

    The product itself is indeterminate. Period.

    But it can be part of a function that limits, from both sides, to the same constant value no matter how close you get to it, so you can define the value of the function to be that constant at that point.

    Consider, for example, the function:

    f(x) = sin(x)/x = sin(x) * (1/x)

    What is f(0)?

    You have sin(0), which is 0, and (1/0), which is ∞. So you have 0*∞.

    But if you ask about the value of f(x) as x approaches zero from either the left or the right, you discover that as x approach zero f(x) gets arbitrarily close to a value of 1.

    Now, we can't do anything about f(x) at x=0. The function is indeterminate at that point and nothing we can say changes that. But we can define a new function, g(x), as follows:

    g(x) = 1 for x=0 and f(x) otherwise

    We call g(x) the sinc() function.

    Notice that f(x) is NOT the definitoin of the sinc() function, because it has a singularity while the sinc() function does not.

    When the value of the function limits to a fixed and finite value as you approach a singularity from any direction, the singularity is said to be removable. But the singularity still exist in that function. You remove it by defining a new function that defines the value at all removable singularities to be the limiting values as you approach that point.
     
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  3. studiot

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    You normally use de l'Hospital's rule to evaluate indeterminate forms such as

    ∞ * 0 or 0/0 or ∞/∞

    This rule applies where the functions converge to a limit.

    On occasion the functions do not converge and the rule fails.

    For example

    \frac{{{e^x}}}{{{x^n}}} \to \infty \quad as\quad x \to \infty

    This ratio has no limit and WBahn's singularity is not removable.
     
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  4. WBahn

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    And one of the easy to forget subtleties of using L'Hospital's rule is that you can't apply it to just any singularity, but only an indeterminate one.

    For instance, if your function evaluates to ∞/0, it is a natural reflect to grab for L'Hospital's rule and you might find that it converges to a constant. But doing this is invalid because the starting form was not indeterminate, it was infinite on steroids!

    Also, L'Hospital's can't be applied directly to the product of two functions, but only to the quotient. But this is easy to accommodate since you just use the recipricol of one of the functions, though which one you pick can make the difference between getting a limit and chasing a rabbit down a hole.

    I think there are some other fine points, too.
     
    Last edited: Sep 8, 2013
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  5. THE_RB

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    Like zero being a real thing? Infinity being impossible in the real world, at best it is just a vague math concept that people try to fudge into a formula to make it balance.

    Zero is real, and understood and well defined, so it overrules infinity.

    So X*0=0 for any value of X.

    If you were to continue increasing X over time, the result would ALWAYS remain zero. That is proven. You cannot prove that if X gets large enough the result suddenly becomes >0.
     
  6. WBahn

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    So then what is 0/0?

    What is sin(x)/x for x=0?
     
  7. THE_RB

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    x/x = 1
    ie; "how many dogs are there in a dog". Always 1.

    Math is imperfect and will rely on creating a vague value for "infinity" to get the sums to balance.

    But there is no actual infinity. So when the concept of infinity is actually HURTING the solution to the problem then the fact that zero is a real number and a real world implementation means that it outranks infinity, and the result calced based on zero.

    If you want to argue about infity vs zero, my favorite example is this;
    A. For any real world number, you can subtract (or add) 1 repeatedly and it WILL reach the real world number of zero.
    B. For any real world number, you can add or subtract 1 from it and it CAN NEVER reach the theoretical value of "infinity".

    "Will" far outranks "can never".

    Mathematicians like the idea they can solve everything with pure math, but the reality is that pure math is quite lacking, which is why it cannot solve ?*0 and why the solution requires going beyond pure math and bringing in other tools.

    The solution to ?*0 is not possible for a mathematician but is fairly easy for a realist or a lawyer. ;)
     
  8. studiot

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    You need to move your studies on about 1800 years from the ancient Greeks to the 17th centuryAD.

    Zeno's paradoxes have a solution.

    Take the example I posted, it is the same argument to show that the top function (which is not about linear addition) grows faster than the bottom.

    You need to understand limits properly.

    You also need to understand the development of number systems properly. Every number is the solution to some equation. We have introduced new numbers every time we have found equations that cannot be solved in the existing system. The process has occurred several times over the centuries, and it is thought murder was once committed over it.

    You have two choices,

    You can listen, enquire further and learn, or you can close your mind and carry on.
     
  9. WBahn

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    And I knew that would be your answer.

    So you have just given two contradictory answers.

    Here you say that x/x=1. Always 1

    Yet in your prior post you said, "So X*0=0 for any value of X."

    Which is it? You can't have it both ways.

    To explore this a bit more, what is the following function evaluated at x=2.5?

    <br />
f(x) \ = \ \frac{2x^2-4.9x-0.25}{x^2-2.9x+1}<br />

    The numerator and denominator are both 0 at x=2.5. So is the answer 0 because 0 times anything, including 1/0 = ∞, is zero? Or is the answer 1 because anything divided by itself is 1?

    What is the answer for x=2.49?

    What is the answer for x = 2.51?

    What is the answer for x=2.4999999?

    What is the answer for x = 2.5000001?

    How does it compare to 49/29?

    EDIT: Actually, the answer to that last question is, "It doesn't," or some other equally meaningless observation. I made the mistake of evaluating the derivatives at x=0 and not x=2.5, I literally realized my mistake while getting oiut of bed this morning. I guess it does pay to "sleep on it".

    So how about this one:

    How does it compare to 51/21?


    So what would the lawyer say? Would they pick one of the two different answers you have given, or something else entirely? ;)
     
    Last edited: Sep 9, 2013
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  10. THE_RB

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    And so you have highlighted the problems pure math has in working with infinity. No big surprises there.

    1/0 = failure. Just ask any simple computer chip that didn't go to advanced math calss. ;) If you insist on giving that failure a name you can call it "infinity" if you like, but you should be aware that infinity does not exist.

    There are not two different answers to the initial question. Pure math, within its limitations gives an equal credibility to the concepts of zero and infinity, so "infinity * 0" cannot be resolved. It is "the irresistable force vs the immovable object" where both have been defined to be equally credible.

    Limiting yourself to only math in your toolbox will leave you impotent with the problem unsolved. I will reach for other tools.

    The lawyer would very quickly realise zero has a very high credibility and can exist in the real world, where "infinity" is a contrived concept that very low credibility outside of the small arena of pure math. So it becomes "a low credibility irresistable force meets a high credibility immovable object" and the answer to that is the immovable object wins. infinity*0 = 0

    It's obvious both yourself and Studiot are intent on using pure math alone to solve this problem, so I fully expect you will come to the result expected in pure math, that it is unsolvable. I welcome any proofs you can give outside the scope of math, as we already know what pure math says.

    No problems at this end, I fully understand why a mathematician will think "infinity*0" is unsolvable. Can you open YOUR mind, and allow attempts at solutions that extent BEYOND math to use some of the other tools at our disposal? Math is only a subset of reality, although the arrogance of mathematicians is legendary so few will ever admit maths "dirty little made-up secrets".
     
  11. studiot

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    Instead of mocking what you do not understand, I offered you the chance of explanation.

    You are partly correct. Infinity does not have a place in all number systems.

    It is a shame that you do not wish to extend your knowledge and understanding beyond that.
     
  12. WBahn

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    Math has no problem with this problem, but I notice that you have chosen not to use your "tools outside pure math" to even attempt an answer.

    And you once again contradict yourself.

    Is 0/0 equal to 1 because "how many dogs are in a dog"?

    Or is 0/0 equal to 0 because anything multiplied by 0 is equal to zero?

    You have made both claims and don't seem willing to recognize it or deal with it.

    I'm waiting for you to show us how those solutions that extend beyond math will answer the question I posed, which has a very definite finite solution that is equal to neither 1 nor 0.

    What do those tools tell you the value of that function is at x=2.5?
     
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  13. kokkie_d

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    Can I just add to all this (mind I am only beginner):It is my understanding that when people write about infinity they generally write: approaches infinity. Much like they write: Approaches zero. That's why they talk about limits and not about actual values.

    I just love the argument:
    \frac{0}{0}=0*0
    Where:
    \frac{0}{0}=0*0^{-1}

    Can the theoretical value of infinity outrun zero?
    I like to believe my teachers old rule in this: I have zero apples in my hand no matter how far into infinity I still have to take them from the tree when I am hungry. ;-)
     
  14. studiot

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    Yes this is exactly the point, we are talking about functions (or series) not single points as with your old teacher's adage

     
  15. WBahn

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    I'm not following. The second line is clear, but what is the basis for the first line?
     
  16. kokkie_d

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    It was just my interpretation of the previous discussion and my attempt of being philosophical using math:

    \frac{0}{0}\neq 0*0

    where
    \frac{0}{0} = 0*0^{-1}

    It doesn't matter from my perspective since
    0*0 = 0
    and thus
    0*0^{-1} = 0

    Which would imply:
    \frac{0}{0}= 0*0

    And the basis for the final line in my basic understanding would be: something times 0 remains 0. I can not multiply what is not there and how computers solve this (matlab uses NaN and Inf for possible outcomes depending on what you are doing).

    My2p
     
  17. studiot

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    I noted this earlier.

    I should perhaps have said also that zero does not have a place in all number systems since it is a number without an inverse.

    Your old teacher probably also forbade division by zero on pain of cookie confiscation or worse.

    :)
     
  18. Tesla23

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    It's probably better avoiding dividing by zero and leaving 0/0 undefined, otherwise:

    Assume
    \frac{0}{0}=1

    so as clearly 0 = 0

    then
    1*0 = 2*0

    and dividing by 0

    1*\frac{0}{0} = 2*\frac{0}{0}

    but if
    \frac{0}{0}=1
    then

    1*1 = 2*1 or

    1 = 2

    and in case this is not disconcerting enough, you can use this to prove that Bertrand Russell (or anyone else) is the pope!
    http://www.nku.edu/~longa/classes/mat385_resources/docs/russellpope.html
    (or you can prove any other fallacy you like).
     
  19. studiot

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    But it's not undefined.

    And sometimes we need to know the answer as WBahn asked in post# 9

    The rest of your post is based upon not understanding number systems sufficiently.


    Edit
    Most people are quite relaxed with the idea that there is more than one solution to

    √2 = ?

    So why does there have to be only one solution to 0/0 = ?

    In fact there can be infinitely many solutions or no solutions, depending upon the circumstances and number system. That is the difficulty that elementary theory seeks to avoid by discounting it.
     
    Last edited: Sep 13, 2013
  20. WBahn

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    If by "philosophical" you mean trying to explain something based on what you would like to be true or what would be nice if it were true, then you have taken a step many thousands of years backward when the explanation for why objects speed up as they fall was based on equating a falling object to a weary traveller whose pace picks up the closer to home that they get and since an object's home is clearly on the ground and not in the sky, it will go faster the closer it gets to the ground.

    And this is based on your assertion that 0 times absolutely anything is 0. You then use this assertion to basically prove this assertion. Well, I could assert that 1000=12 and "prove" all kinds of things, including that 1000=12. But your only basis for this assertion is that it makes you feel good and you then choose to ignore the myriad of problems and contradictions that result if this is the case, presumably because facing those would make you feel not so good.

    This is absolutely a horrible example to hang your hat on. First off, you are talking about a particular set of floating point representations, all of which are fixed width. The NaN and Inf values are merely flags used to identify particular situations in which the value cannot successfully be represented within the limitations of the basic representation. But there are an infinite number of perfectly fine values that cannot be represented. For instance, in a 32-bit IEEE floating point representation, multiplying 1x10^20 by 1x10^20 will produce one Inf. But the answer is clearly 1x10^40 -- a well-defined number that is certainly not infinite.

    But even if you want to hang your hat on it, what is 0*Inf or 0*NaN?

    What is

    f(x) = sin(x)/x at x=0?

    By your reasoning, it must be zero. Yet for x close to zero (and by "close to" I mean just as close as you want to get), sin(x)/x is close to 1. The closer you get get zero, the closer the result gets to 1. Does it make sense that, somehow, this function is going to magically pop down to zero at x=0 when you can get just as close as you want to it, say |x|=1/(10^(10^(1000^1000)), and you get 1 to whatever level of precision you want?
     
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