x is the summation of the infinite series 3 * 10^-1 + 3 * 10^-2 + …
That’s how I’m using the notation 0.333…
Does that clarify matters?
I saw this post when I was on the john looking to see responses to one of my posts, and thought I just have to join in, because I understand the equivalence well and I’m good at and like explaining things
but at this point is there anybody who doesn’t understand? Does the OP get it yet?
it has to do with the whole convergence thing - that is, when you’re imagining “It gets closer and closer to ____, but what if you did it forever, like you had already finished doing it to infinity?” So yeah, .9999999… gets closer and closer to 1, so if you did it forever, you’d be at 1. Which makes sense, since the digits repeating as a decimal each refer to a ninth. .1111… is 1/9, .22222… is 2/9, etc. All those can be shown with the equation proving as such, but ultimately you have to understand it conceptually.
On a related topic, I believe that for any stuff like this to truly make sense, your series has to converge, and do it quick enough that your sum reaches one finite number.
Which is why I think videos like this:
http://www.youtube.com/watch?v=w-I6XTVZXww
Are B.S.
Conceptually it makes no sense.
With something like .999999, or the area under a hyperbole (I think that’s what it is) or whatever, it makes sense to say “what if you had done it to infinity?”, or “What number does it closer and closer to?”. But all the numbers put together do not converge. The sum just gets greater and greater. That they can do some silly transformations with algebraic-ish math doesn’t change the reality.
I was even talking to a physics teacher recently and he said their claim that they use this in string theory isn’t true.
(my underlining)
I don’t get why you are taking it to mean anything other than the infinite decimal 0.333… that is intended (I see there is an element of Devil’s Advocate in your post). That this value falls in the infinite series of intervals (3, 4), (3.3, 3.4), (3.33, 3.34) etc. we can agree on.
Anyway, “there may be more than one number of this sort” is the point where I shall in turn disagree. If there are two different numbers with that property then they have a difference, call that difference epsilon. Now each of those intervals has an amplitude, namely 0.1, 0.01, 0.001… and we can navigate through that list to find an amplitude, delta, which is smaller than epsilon. At this point the game is up, the assumption that there are two distinct values with the given property leads us to a contradiction.
I understand how you’re using the notation. But the person you’re arguing with has a different interpretation of the notation, and I’m illustrating how they might react with that in mind.
Speaking of reality, that’s probably why I never seem to be able to cross the street. First, I need to get half way across, then I need to cross the first half of what’s remaining, and then the first half of that, . . . Indeed, “the sum just gets greater and greater”. Reality, man!
I believe that EdwinAmi was referring to the “proofs” that 1+2+3+4+5+… = -1/12
I’m proposing a general interpretation of decimal notation motivated by the idea that notation A denotes a value strictly less than notation B just in case the same holds of the most significant digit at which they differ, as many seem to assume when discussing how 0.999… must be less than 1, while still agreeing with the traditional notation on terminating decimals. This is accomplished by the use of such “half-closed, half-open” intervals (it makes no difference for 0.333…, but consider the difference it makes for 0.999…!).
I’ve noted before that the standard interpretation of decimal notation uses closed intervals, but I’m exploring what sense we can make of the desires of those who insist on doing things differently.
That is exactly what I am doing; I am playing Devil’s Advocate. The people who argue that 0.999… < 1 aren’t generally very good at formalizing their ideas; I’m taking the role of providing such formalizations.
[quote]
That this value falls in the infinite series of intervals (3, 4), (3.3, 3.4), (3.33, 3.34) etc. we can agree on.
[quote]
Sure. I don’t think you’ll find anyone objecting to that.
Given the presumption that any two distinct values necessarily differ by an amount greater than one of 0.1, 0.01, 0.001, … . But the people at whom the arguments in this thread are directed generally do not grant that presumption. They have a different framework in mind. Hence “there may be more than one number of this sort”.
[Of course, these folks should be made to understand the framework everyone else is standardly discussing. But as I said, I am playing Devil’s Advocate; I often find it the best way to sharpen my own understanding]
We had a thread discussing this, which you may or may not find of interest.
But it doesn’t. The distance of the width of the street is finite. That’s what I’m saying. It DOES “converge” if you will.
I’m open to hearing how 1+2+3+4… could ever reach some number, but so far I haven’t heard anything convincing.
I understand what you’re doing as an intellectual exercise, but I think it’s likely to confuse 7777777 more than clarify things for him. After all, it was a similar intellectual exercise that convinced him that …999 = -1. If someone has a shaky grasp of normal high school algebra, it does them no good to confuse them with a welter of different algebras with different axioms.
It’s used in string theory alright, and also in vanilla quantum field theory—it can be used to calculate the so-called Casimir attraction between two conductive plates in vacuum, for example. In string theory, it can be used to determine the spectrum (i.e. the excitations) of the bosonic string. To be sure, both quantities can be calculated in ways that don’t depend on the sum 1 + 2 + 3 + 4 + … = -1/12, but it’s intriguing that it can be used in such a way.
There’s absolutely no problem with this way of summing mathematically, by the way. It’s simply a different sort of summation from that usually used; the ‘everyday’ summation doesn’t attach a value to 1 + 2 + 3 + 4 + … and similar series, but there are notions of summation that do, and these notions seem to be useful, as shown by the examples where they can be used to calculate the correct magnitude of a physical quantity.
I think 7777777 is irredeemably confused. The exercise is for other people’s benefits (e.g., my own).
Indeed. That seems to be the final say on these weird maths. On the other thread, somebody said “if it’s defined a certain way…”, which is kind of what you’re saying here.
Yes, if you have very specific definitions and conceputalizations of what you’re doing, you can use these maths. What laymen like us are saying is that it doesn’t directly correlate to anything you can do in the real world that forms intuitive math. Everything you can do in rational-number arithmetic can be “done” in real life. Any child can pick up multiple rocks, thereby “adding” them. You can easily draw standardized lengths and “add” or “multiply” fractions of them, making larger lengths.
But whatever the hell these supposed sums of non-convergent series are, it doesn’t directly correlate to anything in real life, though the calculations may be ultimately useful. This is similar to imaginary numbers. The square root of a negative number is intuitively impossible, as well as by definition (of what a square root is) impossible by the “normal” math that people are used to using. Nonetheless, said calculations are useful, because they create a two-dimensional number line that follows certain rules. “Traditional” or “intuitive” math would say it’s impossible, but you’re just ignoring that and creating a new math.
Regarding what I said above:
I know many mathematicians and scientists would argue whether there is an “intuitive” or “real” math that exists, and claim it’s all equally conceptual.
But I would completely disagree and propose what perhaps we might call a “Chomsky” theory of math and mathematical understanding
Anybody familiar with Chomsky’s proven theories on linguistics? Basically, people DO indeed have some innate, pre-programmed language skills. This should be obvious when viewed through a biological lense, but for whatever reason it was a subject of debate.
What I’m saying is math is the same way. People have some innate, biologically pre-programmed sense of numbers, or the concept of scaling and counting, and tend to think of such things in terms of what they can directly do in reality; that is, our understanding evolved based on what we could actively do in the real world.
At some point, being able to understand how many spears you made (integral counting), or how much meat you’ve divvied up (rational numbers/continuous units) for the tribe, or how far away the next tribe is (again, rational number lines) became a huge evolutionary advantage.
Being able to do or understand highly conceptual and abstract math such as negative square roots, or sort of understand how different versions of “infinity” can be added up, were not evolutionary advantages.
Like all other things that ultimately derive down to biology, this is a pretty easy pill to swallow. Indeed, there have been experiments that show that all humans have an innate sense of geometry (they tested people with different home languages and levels of mathematical training, right down to nomadic tribespeople who still live paleolithic style). Well duh, guys, people have been building pyramids since 2,000 BC, and frankly the construction process was much the same as nowadays and the understood all the important issues building such a thing. Clearly a lot of this stuff is biologically pre-programmed (not like the complete math, but the ability to learn and understand it, eh, you know what I mean)
*None of this is to say I agree with Noam Chomsky’s politics, just his linguistic theory was on the money
The problem with the video is not that it’s wrong. It isn’t. The problem is that it isn’t clear about the fact that it’s using a non-standard definition of summation. It’s expanding the standard definitions of symbols like “+” and “Σ”. The new definitions give the same results for convergent series but also assigns values to non-convergent series in a self consistent way.
It’s like taking the definition of addition of integers and coming up with an analogous operation for finding the “sum” of two real numbers. The new real-addition operation should give the same results as integer addition when its operands have zero to the right of the decimal point but extends the definition to assign values to the “sums” of other pairs of real numbers. The new operation has proven to be useful in mathematics and other areas like advanced physics.
The problem the video and 7[sup]7[/sup] both have is they don’t make it clear when they are using standard definitions and when they are not. The video does it because it’s cheeky and 7[sup]7[/sup] does because he/she doesn’t know any better.
Not true. There are societies and languages where it’s not possible to count numbers higher than ten or three or whatever. There are even some where it’s not possible to count. Chomsky was wrong about innate linguistic abilities, and many linguists agree about how wrong his ideas are. I will explain at further length later, but I have to leave for something immediately.
Codswallop. I clearly said not all math is pre-programmed, but the ability to understand it following certain tendencies. Are you going to tell me it is impossible for people to learn to count and learn math? Are they permanently disjoined from all math just because of their original learning?
Well, then, they’d be ignoring a huge body of evidence, wouldn’t they?
Animals can’t talk. People can. Are we really going to argue that there’s a biological element to that?
Colibri for best mod?
Really just a nitpick, but I’m not sure the statement below is correct:
It may still be the case that the underlying ability may have had other advantages.
So, although I’ll grant you that the ability to work with imaginary numbers per se has no evolutionary advantages, the underlying ability to say ‘what if’ or to devise rules that apply only to subsets of things but not to every thing, may well confer advantages.
Re-responding:
Perhaps. But I am not convinced that the best way to deal with someone who is struggling with the conflict between their intuition and what they’ve heard about some piece of mathematics is to hide from them the existence of other pieces of mathematics capturing their intuition.
Indeed, I think this often leads to doubling down: “I have an understanding in my head of the way things work. I keep hearing people tell me I’m mistaken; that the One True Way of things is different. But the picture in my head is so clear, even if I have trouble describing it to others; it must be they that are mistaken. If only I could get them to see the picture I can, they would recognize their errors and agree with my understanding”. This is the kind of thinking fostered by the One True Way illusion.
It seems to me better to bring it all out into the open: acknowledge the existence of multiple frameworks in which the same concepts might be explored (not necessarily limited by prior design, as we can, ourselves, make up any rules for any game we might care to look into at any time), discuss how we might capture the intuitions one cares about, and also discuss the merits of other frameworks which do things differently. And, yes, if one is to be equipped to communicate with others, it behooves them to recognize and understand the standard ways in which others talk.
But I’m not very fond of slapping people’s hands for wanting to explore on their own (the very thing to be encouraged in mathematics!) in a direction other than the standard, and particularly not fond of lying to them about the alleged nonsensicality of thoughts which can perfectly well be made sense of, however unorthodox. I hate the reduction of math to received wisdom from unquestioned authority.
[Also, I don’t think anyone’s algebra skills, in the sense of manipulating polynomial equations and so on, have been shown lacking. Problems have just been with understanding how infinitary decimal notation is customarily interpreted. Which plays such a negligible role in ordinary life that it hardly matters if some people have their own ideas how to use it.]