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  • HalationEffect
    replied
    Originally posted by Detonate View Post
    On another forum. where we discuss the weather a lot I find CTRL,SHIFT, U 00B0 useful.°
    There's an easier way for the degree symbol: (first configure which key will be your compose key, if you haven't already - I've set mine to the right ALT key). COMPOSE o o. Gives the same result °.

    (Edit) Useful list of compose key sequences: http://www.hermit.org/Linux/ComposeKeys.html
    Last edited by HalationEffect; Jul 30, 2013, 09:23 AM.
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  • HalationEffect
    replied
    When I was younger, it amused me greatly to say "for any real value of x, x/∞ = 0, and x/0 = ∞, therefore ∞0 = x, therefore all real numbers have the same value (because they can all be obtained by multiplying zero by infinity)".

    Even though I knew it was wrong, I loved trolling people with it! <evil laugh>

    I would completely neglect to mention that I knew full well that there are different infinities and that they aren't all the same (e.g. the infinite set of all positive integers contains within it the infinite set of all positive even integers and also the infinite set of all positive odd integers). One infinity is bigger than another?? <Head explodes>
    Last edited by HalationEffect; Jul 30, 2013, 09:07 AM.
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  • Detonate
    replied
    Originally posted by HalationEffect View Post
    (Edit) Typing a lemniscate "∞" is really tricky! Hold down CTRL, SHIFT and U, then release the U while continuing to hold down CTRL and SHIFT, then type 221E, then release the CTRL and SHIFT.
    On another forum. where we discuss the weather a lot I find CTRL,SHIFT, U 00B0 useful.°
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  • Qqmike
    replied
    I won't even touch the probability discussion as THAT was also one of my specialties for some years and my thesis was in applied prob.
    ;-)

    You are right about the "1/∞ = 0"! That hurts my eyes and my brain in ways you'll never know! But, of course, I know what it means as a shorthand. But ... oh jeez, here it comes ... how is the limit being approached? We are assuming it is obvious that we are letting x --> infinity through only positive values of x (and the zero is approached positively, 0+) . Not x --> MINUS infinity (and the zero is approached negatively, 0-)? Not the absolute value of x --> infinity? Or, more precisely, |x| --> + infinity? Which is it? My feeling is that if we want to talk about, to say, zero, just do it:
    0
    and avoid mentioning the specific computation of this zero. Or, do mention it and make it precise!
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  • HalationEffect
    replied
    Originally posted by Qqmike View Post
    I like your rimshot, HalationEffect. And, of course, you do know what I mean by not being "too" huffy, right? Like, just huffy enough (where "enough" shall remain undefined), it's clear, right?
    Crystal clear

    I was just reading an article on probability theory (because it curdles my noodle(*), and I like to keep banging my head against such things until my noodle uncurdles), and I may have had an insight as a result:

    As a non-mathematician, I would be quite happy to say that "1/∞ = 0", but a mathematician would be more likely to say something like "the limit of 1/x as x approaches ∞ = 0". Because I consider both statements to be semantically equal, I'd go with the former statement for brevity's sake.

    (*) It curdles my noodle when I read things such as "Some events with zero probability (p = 0) are impossible, but some aren't".

    (Edit) Typing a lemniscate "∞" is really tricky! Hold down CTRL, SHIFT and U, then release the U while continuing to hold down CTRL and SHIFT, then type 221E, then release the CTRL and SHIFT.
    Last edited by HalationEffect; Jul 30, 2013, 08:39 AM. Reason: Extra trivia
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  • kubicle
    replied
    Originally posted by Detonate View Post
    You guys are way over my head, but I could turn you into great artillery fire direction officers pretty fast with your knowledge of math.
    You sure about that, I think we'd still be discussing in the trenches when the bajonets start swinging.
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  • Qqmike
    replied
    Yeah, you have a point there, kubicle.

    The term "limit" is nothing more than the mathematician's (perhaps clumsy) way of saying, "it can be made arbitrarily close to."
    In our case, when we write
    0.999 ...
    with the repeating periods (ellipsis) and 9's, that right there is a tip-off that something's up, that we are dealing not with 0.9 but with some other animal, so
    1 = 0.999 ...
    works in SOME understood contexts or discussions as a shorthand.
    However, for some who might be thinking of the reals and an algebraic field with multiplicative identity symbolized by "1" OUCH to the eyes! And even more ouch if one were to even hint that we are "defining" 1 by this equation. However, we can define the limit (or the symbol "0.9999....") to be 1 by this equation
    0.9999.... = 1
    and it is OK.
    Semantic, precision in, as Halation said above. Or, maybe even, personal "style" preference for rigor. And when you are drunk (which I was not last night, by chance), it may all be different.
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  • Detonate
    replied
    You guys are way over my head, but I could turn you into great artillery fire direction officers pretty fast with your knowledge of math.
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  • kubicle
    replied
    Originally posted by Qqmike View Post
    But the compact notation, 0.999.... gives no clue to most people that it is actually a limit, the limit of a series, the value of that limit is 1, or the limit equals 1, or the series (represented here) converges to a limit and that limit is 1, or the limit of the series (represented here) equals 1, and so on.
    Does it really matter (honest question, my mathematics days are long gone ) other than metaphysically that it is a limit?
    I mean most people would say:
    1 - X = 1
    is true, when
    X = limit ( n --> infinity) (1/10^n)
    so
    X = 0 (even though it's just a limit with a value of 0)
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  • Qqmike
    replied
    I like your rimshot, HalationEffect. And, of course, you do know what I mean by not being "too" huffy, right? Like, just huffy enough (where "enough" shall remain undefined), it's clear, right?
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  • Qqmike
    replied
    Yes, Halation, I completely agree, and I don't think either of us meant to be (too) huffy ;-)


    If I may have a little technical aside with kubicle ... :-)

    kublicle, actually, your 1 - X argument is insightful. It is true that

    limit (as n --> infinity) (9*(10^(-1) + 10^(-2) + 10^(-3) + ... 10^(-n))) EQUALS the real number 1.

    We can make the sums as close to 1 as we wish, arbitrarily close, by choosing enough summands, i.e., by choosing n large enough. How close do you want to represent 1 by a summand? Let's say by X units. Your X is my epsilon (Greek letter) in my (several) posts above.
    Choose, for example, X = epsilon = 0.0001.
    You could easily figure out an n large enough so the difference between

    9*(10^(-1) + 10^(-2) + 10^(-3) + ... 10^(-n))

    and the number 1

    is less than 0.0001. Similarly, we could represent 1 by a sum to within an "error" (= X = epsilon) of 0.0000000001 by choosing enough summands, and so on.

    THAT is a verbal description of the true mathematical proof that
    limit (as n --> infinity) (9*(10^(-1) + 10^(-2) + 10^(-3) + ... 10^(-n))) EQUALS 1

    and it is what you are thinking of with your X and it is what I've been trying to explain all along! Basically, you have given an intuitive, verbal definition of "limit." This what math teachers try to get math students to do on their own prior to all the fancy notation and symbols.
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  • HalationEffect
    replied
    I just knew it was going to be a problem of semantics in the end!

    I was expressing it as a layman, you were expressing it as a mathematician... and it took a while for our differing explanatory methods to asymptotically approach each other (rimshot)
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  • Qqmike
    replied
    kubicle, yes, kind of cutesy regarding the little "9 times" proof, it begs the question.
    But also, the argument 1 - X = 0.999 ... normally makes sense, but not this time as we are abusing notation when we write the symbol 0.999.... . Familiar example to many people is e, which we know to be a certain well-known limit, this limit:

    e = limit (n --> infinity) (1 + 1/n)^n

    But the compact notation, 0.999.... gives no clue to most people that it is actually a limit, the limit of a series, the value of that limit is 1, or the limit equals 1, or the series (represented here) converges to a limit and that limit is 1, or the limit of the series (represented here) equals 1, and so on.
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  • kubicle
    replied
    I won't touch the mathematics (much), but philosophically (and logically) for two numbers (or the representations of two numbers) to be different, one should be able to put something "between" them, in other words:

    There should be a non-zero X that would make 1 - X = 0.999... true.

    If there is, the numbers are different.
    if there isn't, the numbers are the same.

    Side note, I'm sensing a logical problem with the first "simple proof" on the wiki:
    1/9 = 0.111...
    9 x 1/9 = 0.999...
    1 = 0.999...
    However, 1/9 = 0.111... is true only if 1 = 0.999... which is the thing we're trying to prove, so we're using the conclusion as a premise in circular logic (it's true if it's true).
    Last edited by kubicle; Jul 30, 2013, 02:48 AM.
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  • Qqmike
    replied
    Sorry, but this might be worth it, a post to keep peace here :-)

    jlittle, your post kind of triggered this for me in a helpful way ...
    Snowhog's post, #19, the first paragraph, is good, and I had to read it carefully.
    Key, from that paragraph: the symbols "0.999..." and "1" represent the same number.
    That is good!

    I think maybe we are all basically on the same page here, HallationEffect, but it has been a matter of terminology, and maybe even more so, perspective. The culprits inciting riot have been these words: equal, represent, same as, and importantly (for me) "is defined to be." By 0.999... is meant an infinite string of symbols, which is meant to be that infinite summation we have hammered to death, which is actually a limit (as the number of summands approaches infinity of a series of finite sums), which by modern methods of limit theory can be shown to be EQUAL to the real number 1 (KEY: it does not define the element 1, but the limit is defined to be "1"). As a consequence, in particular, we may use the symbols 0.999.... and 1 interchangeably, at least in certain contexts. In fact, we may write 0.999.... = 1, for example. My focus here has been on viewing the reals as an algebraic field where we have agreed to represent the unique multiplicative identity by the symbol "1." This is a modern view. But there is the important historical view, too, which has its roots in computation; where early mathematicians worked to understand, define, and compute infinite summations (of which 0.999.... is an example).

    So, I do now see what you guys may have been trying to say, and from what perspective. You are stuck on the (perhaps partly historical) definition and numerical value (the limit or convergence) of that infinite summation; I am stuck on viewing the reals as an algebraic field (where the symbol 1 is clearly--though arbitrarily--set aside, defined, for a special purpose, the multiplicative identity). Sorry for missing what I think may have been driving your perspective on this. I think we'd all agree on this stuff ... once we knew what it is we are each talking about! Don't tell me now that this is about apples and oranges; you might be right.
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