And (I suppose you know) I have complete faith in delta-epsilon proofs, I just don’t believe that they are as intuitive as **Trinopus **wants them to be.
Non-mathematicians already bent on the idea that 0.999… is different to 1 will fairly quickly home in on the construct 0.000…1 – that “looks” like a very reasonable thing to propose – in fact it’s the apparent compliment of the 0.999… in question. It’s the bit that the deniers suppose is missing.
Getting all lawlerly and demanding they play the game by “our” rules isn’t as persuasive as one might hope.
IOW, a bullshit number, kinda like a zillion to the jillionth power.
Your lone 1 is in a specific place in the decimal number you’ve postulated. And that in itself points out the impossibility of an infinite number of zeroes preceding it.
0.000…1 is the “compliment” of 0.999…9. This generally means that it doesn’t matter how many 0’s or 9’s you put in, as long as it is a finite number of 0’s or 9’s. So in the case of the OP, we have 0.333…3 ≠ 1/3, which is true.
There are laws that govern math at this level, and they can never be broken UNLESS we specifically say so. I think this is the problem we’re dealing with here, we have certain symbols we use that have established definitions. We can use these symbols in other ways but when we do it is our responsibility to specify the differences.
For example, F=ma has a generally accepted meaning. If we are using these symbols for something else, we have to say so; F = energy, m = mass, a = speed of light squared. The Master is using 0.333… to mean an infinite number of threes, the OP is using 0.333… to mean a finite number of threes. Therefore 0.333…[sub](The Master)[/sub] ≠ 0.333…[sub](The OP)[/sub].
If you’re saying it’s impossible to have an infinite number of ZEROS followed by a ONE, how can you expect “me” to believe that we can have a ZERO-POINT followed by an infinite number of NINES.
Besides, I think I can have an infinite number of zeros followed by a one, it’s the limit of this suite (0.1, 0.01, 0.001, …). It’s already been stipulated that suites can have such weird limits (i.e. (0.9, 0.99, 0.999, …))
You should note you are arguing with the already persuaded. The obstinate and yet-to-be-persauded others are not going to be impressed with the mere assertion that 0.000…1 is a “bullshit number”.
Certainly that’s the ending of the game you’re playing.
I am indeed saying that, and I’ve explained why.
I don’t expect to be able to get “you” to believe anything.
Oh, you can think anything you want. There’s no force under heaven that can stop you.
The limit to that sequence is zero. ‘Limit’ is a mathematical term of art, and has a precise definition, which I’ve given for the case of sequences . Each term in the sequence is a finite number of zeroes, followed by a 1, but the sequence converges to, IOW has a limit of, zero, and not some made-up number that exists nowhere on the number line - and if it did, it would be indistinguishable from zero anyway.
I have not only asserted it, I’ve provided reasons. Feel free to ignore them.
But here’s one more. If your bullshit number is distinct from zero, then there’s a number in between it and zero. (An infinitude of them, actually.) Give one for-instance.
Perhaps you should re-read The Great Unwashed more carefully, and acknowledge that he’s playing devil’s advocate here? Perhaps you realize that, but your confrontational tone suggests that you don’t.
At this point I think every active participant in this conversation understands that. I think that The Great Unwashed was challenging the intuitive “game” proposed by Trinopus, and showing that it doesn’t necessary provide a definitive intuitive resolution for (other) people who may have questioned the mathematics on intuitive grounds. I think he has a point.
Oh noes, isn’t that one of them-there Delta-Epsilon proofs again?
Okay, I’ll try…
What was my number again? Oh yeah…
0.000…1
And you want a number smaller than that but bigger than zero? How about…
0.000…01
This game is not as hard as I was led to believe.
I said before that the game ain’t over until the gong sounds. I meant to say the game ain’t over until the gong stops sounding.
Ah, then we have a paradox … either “it keeps doing that” or it stops and we find a 1.
There’s no intuitive explanation for this, 0.999… doesn’t look like 1, it doesn’t feel like 1, it doesn’t smell like 1 and I’m pretty sure it doesn’t tastes like 1. 99.999…% of the world’s population will live happy productive lives never acknowledging that 0.999… equals 1.
Well, in introductory calculus, at least, they certainly are. “Given any delta…” That’s my point: you do have to posit an initial point of departure. It can be in variable form – “delta” – rather than a concrete number. But the point is, given that delta I can find the epsilon that fills the conditions.
It’s valid enough to serve as a mathematical proof. What do you propose to counter it with? Intuition?
. . . What was my number again? Oh yeah…
0.000…1
And you want a number smaller than that but bigger than zero? How about…
0.000…01
As noted above: these constructions are not defined. You’re making stuff up.
Also, you’re proving my point, not yours. I can always construct a number, still greater than zero, smaller than yours. You’re doing my work for me, but you are not making your own point.
Are you actually arguing, or, as has been claimed, just playing devil’s advocate? Because you’re not doing a very good job of either.
Nitpick that doesn’t invalidate your general point: At least the way it’s commonly formulated in introductory calculus, you;re given an epsilon and have to construct a delta.
If one is going to argue “My delta is 0.000…1”, then one might as well instead argue “My delta is quazxt”. Neither 0.000…1 nor quazxt is a valid way to specify any number.
I’ve always been shaky on Green’s Theorem and the associated Functions. I took classes out of order relative to the rest of my grad school cohort, and so when I took Mathematical Methods II, it was “Well, you’ll all be learning about these in E&M II, so we’ll skip over them”, and when I took E&M II, it was “Well, you all learned all about these in MM II, so we’ll skip over them”.
Okay let’s begin with me. I have a first class honours degree in Mathematics. I teach Maths in a lycée (French high-school). I have never sold Crystal Meth.
I am utterly convinced of the soundness of delta-epsilon proofs.
At the same time, I am not persuaded that such proofs are in the least bit intuitive. For the record, I find it hard to believe that many find them intuitive. And furthermore, so what if they are intuitive? Maths is one long lesson in why not to trust our intuition but instead to trust, well, um..., to maths.
Let me say more: the "game" is proposed that the limit denier give a number representing the difference between the proposed limit and what they see to be the real value.
They say, I dunno, how about 0.00000000000000000000000001?
And then the smug mathematician says, Aha, what about the term 0.99999999999999999999999999 ? That's closer to 1 than your specified difference.
So they reply, fine 0.0000000000000000000000000000000000000001 then!
And the self-assured reply is 0.99999999999999999999999999999999999999999
Here the clever mathematician is laughing to themselves saying, idiot n00b can't see that I can always propose a value closer than their delta.
Meanwhile the idiot n00b is saying wft is wrong with these people? I never said I knew what the "real value" was, I'm just guessing here, and so what if they can propose an epsilon smaller than my delta, well I can always propose a delta that is smaller than their epsilon!
There's not a great deal of intuition going on on either side.
I see nothing wrong with the number 0.000…1 if we accept it to be the limit of the sequence (0.1, 0.01, 0.001 …). Likewise for 0.000…01 which is the limit of the sequence (0.01, 0.001, 0.0001 …).
I believe both these numbers are equal to exactly 0. This is not a matter of intuition. Indeed no-one who believes that 0.999… <> 1 is going to be persuaded that 0.000…1 = 0. They’re two sides of the same coin.
What is intuitive to me?
1/3 = 0.333... I believe that pure and simple. Likewise :
1/3 + 1/3 = 0.666... and furthemore :
1/3 + 1/3 + 1/3 = 0.999...
If I were aiming to persuade someone of this fact, I think I'd start with some "less controversial" cases :
X = 0.131313...
100X = 13.131313...
so 99X = 13
and X = 13/99
Then I'd have them do the long division.
So they get to see that 13/99 = 0.131313... and that there's a nice little trick to find the fraction given the infinite decimal.
It might take a few cycles, say 1/11 = 0.090909... and 1/7 = 0.142857142857...
I find this method considerably more persuasive than calling their numbers bullshit (when they aren't), or proclaiming my rightness by mere dictat of being better looking than them.
It’s been my experience in life that those that feel the need to crow that they have won an argument have often missed the point.