Shodan: is at least part of the issue that representing 1/3 as .333333… does not mean “if you repeat the division long enough you eventually reach 0” - just that .333… is how mathematicians represent 1/3
Nailed it :).
There is a theorem (might have been a lemma, not sure who’s, you can look it up if you care) that every real number has a a decimal expansion. The proof for it is a proof by construction. Take the real number x and build up a digit sequence to represent it. When you follow the proof for a particular number one of two things happens. If there is an actual decimal expansion for the number, the process gives it to you after a finite number of steps. E.g, 9/8 => (1, 1.1, 1.12, 1.125), so 9/8 = 1.125. However, if there is no actual decimal expansion then the process cannot stop and you end up with an infinite expansion. But the limit of the sequence you create is the number x so you define it as being the representation for x. So 1/3 => (0.3, 0.33, 0.333, …), so it’s defined as 0.3~. But of course what that means is that no infinite decimal representation is ever equal to the number it represents; it is always infinitesimally smaller.
Chronos: watchwolf, with all due respect, perhaps you might want to sit this one out?
Mate, you hang in there. Keep at it and you’ll eventually get it.
Exapno Mapcase: You keep using the word infinitesimal without any definition.
How big is infinity? It’s not a number, it’s a concept; it has no size. However big you think it is, it’s bigger than that. How big is an infinitesimal? Again, it’s a concept, not a number. However small you think it is, it’s smaller than that. Except that it’s always greater than 0.
Exactly so. What’s the problem? Look at the definition of a “neighborhood” in real numbers. No matter how small it is, I can still always find a number, inside the neighborhood. I can always find a real number between any two real numbers.
The problem is that you’re imagining you can find a number between 0.999~ and 1, and that you simply cannot do. Any candidate you nominate for that number can be exploded, trivially, by looking out further along the decimal expansion.
They are the same value, the same image of, say, 7/7 or π/π. You see? That’s two (let me count them for you…, ONE…, then…, TWOOOOOO) ways to express the same fraction in decimal notation.
If I say infinity is bigger than any number that you can name, you could respond by saying “name the number that you think infinity is and I can always name a number bigger than that by adding one to the number you’ve named.” This is not a valid argument, because infinity is a concept, not a number. You can’t name a number that’s bigger than the concept of infinity. I hope you understand that.
When you say “find a number between 0.999~ and 1” and “looking out further along the decimal expansion” you’re making the same sort of argument, and it fails for the same reason. For every term in the infinite series there is a finite difference between that term and 1. An infinitesimal does not have a finite size. It’s infinitely small but still bigger than 0.
I once got into an argument with a guy who believed that 1 / 1 = 0, and that this fact was being suppressed by the international conspiracy of mathematicians for their own nefarious purposes. His main proof involved cutting up a pie. He had some auxiliary proofs, each of which consisted of finding a real world situation where he could show that three series, {a}, {b}, and {a/b}, each tended to 0. Of course he didn’t understand series or limits. He didn’t understand fractions and he “didn’t trust them.” Any argument I put to him provoked a response in which he restated his proof. Eventually I found a way to demonstrate that his proof didn’t work, without using fractions. Never heard from him again. But when I checked back a year later his website had disappeared. I like to think that my persistence paid off, but who knows?
Watchwolf, I think your problem is that you are trying to filter a problem in analysis through a set of tools you gained from a course in linear algebra. That is why you are finding it hard work.
Actually, 9/8 -> (1, 1.1, 1.12, 1.125, 1.1250, 1.12500, 1.125000…). It doesn’t end any more than 1/3 does. Both end up with a repeating sequence; it’s just that for 1/3 that sequence is 3, while for 9/8 it’s 0.
And watchwolf, your biggest mistake was in claiming that there was a homeomorphism between any two vector spaces. That’s only true if they’re both the same dimension, and also both the same cardinality. But the cardinality of the set of decimal expansions is greater than the cardinality of the set of fractions, so you can’t use that at all. You could do something with the set of repeating decimal expansions, but that’s ultimately going to just fall back onto the fact that they’re both representations of the rational numbers. And ultimately, we’re just dealing with a one-dimensional space here anyway, so the fact that they’re vector spaces doesn’t really add anything (in fact, the general results about vector spaces are all ultimately derived from the corresponding results about 1-D spaces).
Nope: here, you fail. You’re claiming that there is a number between 0.999~ and 1.0, but you cannot define it. You cannot tell us what it is, nor construct it. You’re in the same position as claiming “10^540 is infinity,” which I can disprove by observing “10^541.”
Your very own argument works against you; you cannot perform the task you define.
I once got a very nice phone call from a guy who disagreed with the concept of “nine.” He didn’t think that the digit 9 belonged in our number system. I listened to him for a while, and asked a few obvious questions – “What’s eight plus one?” – and, in due course, told him that I couldn’t agree with him, and rang off.
I suspected as much, I did kinda feel myself derailing there. Again thank you for these corrections, I’m fighting my own ignorance here and it’s not really taking longer than I thought. I sometimes jump down old rabbit holes without checking for water first, so I do appreciate you grabbing my tail and pulling me out before I drown.
What the hell? It’s your word. You put it in your OP. It’s up to you to define it.
Cantor answered these questions 150 years ago. You seem to refuse to accept the answers, but I can’t figure out why or what you’re putting in their place. It’s not infinitesimals, because those already have a place that’s not useful in infinities.
How about you stop screwing around and give a nice straightforward post telling us why the last 150 years of universal acceptance of how to handle infinity should be thrown out.
What I said to engjs goes for you too. You’ve been singularly unhelpful - hell, willingly obfuscating - in this whole thread. It was engjs who said that 0.333~ is not equal to 1/3 and that pi isn’t the same as the number it represents. (Even watchwolf said represents and not notation, which I see as a different argument than the one you are talking about.) I’m asking questions to gain information and clarify the nonsense that’s been posted, much of it by you.
watchwolf49: Also, the infinitesimal is a concept, why are you comparing it to zero?
Let me try it another way. Let us suppose you plot the function y = 1/3 - 1/x, then inspect the behaviour of the coordinates (x,y). The curve is asymptotic to the line y= 1/3. Clearly, as x tends to infinity, y tends to 1/3. But at no point does the curve ever touch the line, so y is never actually equal to 1/3. It is always smaller, and it becomes infinitesimally smaller as x becomes infinitely larger. That’s the definition of infinitesimal. It’s not an actual number, it’s the concept “keeps getting smaller.”
Chronos: Actually, 9/8 → (1, 1.1, 1.12, 1.125, 1.1250, 1.12500, 1.125000…). It doesn’t end any more than 1/3 does. Both end up with a repeating sequence; it’s just that for 1/3 that sequence is 3, while for 9/8 it’s 0.
I didn’t want to get into complexities, but since you want to make the point… In the first sequence, every term after 1.125 is equal to 1.125; adding zeros doesn’t change that. So the series has reached its limit and then never varies from it. On the other hand, no term in the second series is ever equal to its limit. So instead of saying stops/never stops I could have said reaches the limit/never reaches the limit. Does this make any difference to the argument? I can’t see that it does. More interestingly, the version of the proof I found with a quick google search used < rather than <=, which means that it would give the sequence (1, 1.1, 1.12, 1.124, 1.1249, 1.12499, 1.124999, …) which really does confuse things. Because every finite number except zero has two decimal representations, one finite and one infinite.
How convenient, then, that the number 1/3 is not defined by the touching of an asymptote to a line.
I can construct an asymptotic line between my car and the next lamp-post, but that doesn’t keep me from hitting it if I steer wrongly. Zeno’s Paradox? Not convincing in a world that rejoices in calculus.
No … let’s use the function we’ve been discussing all along … y = 1/3 … the infinitesimal of x here is all but meaningless. There’s no asymptote, y never approaches infinity, and there’s nothing in the function that’s getting smaller … it’s always 1/3.
I’m done with you …
I skimmed the article on Wikipedia, and what Exapno Mapcase says here makes a hell of a lot more sense than anything you’ve posted. Be reminded it makes a hell of a lot more sense than anything I’ve posted as well … so let’s try to be cognitive here and quit rinky-dinky with the amatuers … deal with the folks who know what they’re talking about …
engjs, maybe I missed it in reading the thread, but did you ever answer what you mean by the symbol “=”?
If I understand your argument, your assertion is that the arrow never reaches the tree, so long as we represent the arrow’s flight by the infinitely long decimal expansion. Which, it seems to me, is another way of saying that a decimal expansion and its limit are not equal; there’s always that infinitely small distance between the expansion and the limit. Have I got that correct?
In that case, it seems to me that you are simply saying that the meaning of “=” is not the same as the meaning ascribed to it by the others arguing with you in this thread. So perhaps we need to agree on definitions before continuing to talk past each other?
Chronos: Not true. 1/3 is a finite number, and yet it has only one decimal representation.
Ya got me. I should have said: Every real number that has a finite decimal representation also has an infinite one, except 0.
Trinopus: I can construct an asymptotic line between my car and the next lamp-post, but that doesn’t keep me from hitting it if I steer wrongly. Zeno’s Paradox? Not convincing in a world that rejoices in calculus.
I think most school children understand that the curve can never touch the line, because that can never happen except when x = infinity, and x can never equal infinity because there is no such number as infinity. Type “asymptote” into Google and see what it tells you.
What’s this got to do with Zeno’s paradox?
watchwolf49: No … let’s use the function we’ve been discussing all along … y = 1/3
You’ve missed the point. y = 1/3 - 1/x in a asymptotic to y = 1/3, which in this case means that the distance between the two is strictly monotonically decreasing but never 0. It could only ever be 0 if there were an actual number called infinity that x could be equal to, and that is nonsense. If you think that the two actually can touch then you’re in Trinopus’s position; you haven’t understood high school maths and I’m not sure that it’s possible to help you further.
0.3~ does not mean 0. followed by all the threes, because that is nonsense. It does not mean 0. followed by a number of threes whose count is a number called infinity, because that is also nonsense. It mean 0. followed by a non-terminating sequence of threes. Because the sequence of numbers formed by stopping the process after each three forms a strictly monotonically increasing series, and because the sequence has no final three, the sequence can never be equal to 1/3. It’s impossible for the same reason that it is impossible for the two function to ever touch. That’s why in order to use 0.3~ to represent 1/3 you have to define it as being equal to 1/3.
