For the nth time, yes we can, it falls where 1 is. Because it is equal to 1.
The notation 0.999… does not represent a process, there’s nothing “to do”. If it were a process you would have a point – it would be the same point that Zeno made, but that “paradox” has long since been resolved, so still not much of a point.
Wait, let me get this straight: You can recognize 3.14159 as pi, even though it only matches for six digits, but you can’t recognize 0.9 as 1, even though it matches for an infinite number of digits?
Achilles is at the one end of the football field; the tortoise is at the 90-yard line but moves at 1/10 the speed of Achilles.
t = 0; Achilles (with pen in pocket) moves from the 0 to the 90-yard line.
t = 0.9; Achilles moves from the 90-yard line to the 99-yard line.
t = 0.99; Achilles moves from the 99-yard line to the 99.9 yard line
…
We don’t reach the tortoise (or the 100-yard line) on this list of actions.
For the 92nd time …
Ahhhh … you’ve committed the logical fallacy of “moving the goalposts” here … your first 90 posts didn’t include the specification of “but not reach the limit”. In effect now, you’ve admitted you were wrong and are quickly changing your argument so it doesn’t look like you’ve been wrong all this time.
Unfortunately, you’ve not defined what “but not reach the limit” means exactly. We can start by defining what the distance is between the limit and the actual value we reach after infinite iterations. Please spare no details, either tell us which branch of mathematics you’re using or give us the Reader’s Digest version of this new branch of mathematics you’ve developed of this claim you’re making.
The rafters have broken loose.
Infinitesimal is a well defined word, look it up in any calculus textbook. I get the feeling you’ve never taken calculus and indeed it’s a two year course in college just because understanding this well established concept of the infinitesimal is kinda sorta that fucking hard. That’s why we’re all being very patient with you, you seem very bright and intuitive but just missing this little detail …
The kingstuds have snapped in the middle.
Here we have the logical fallacy of “the strawman” … you’ve changed what I actually posted “you cannot define π on the number line using your methods” into some thing else and then argued against that something else. Once again this has the effect of admitting you are in error, it’s impossible to argue against what I said, therefore you argue against something else.
The shearwall is beginning to twist.
It is coincident with 1 using the method you described in your first 90 posts … using the method in your 91st post we can only say this cannot be determined until you edify us on the definition of “not reaching the limit” of an infinite convergent series.
… and now your house has come crashing down.
I can, but apparently you can’t. Nor can you seem to give me a number between the two, even though there would be an infinite number of examples if they were indeed different numbers.
0.999… is NOT a point in (0, 1). It’s not a point on the number line at all. Why should I respond to a question which presupposes that 0.999… is a point on the number line? You will get strange results if you presuppose that. So, you better give a counter-example that shows how we reach the limit although infinitely many commands tell us NOT to do so.
Of course, the addition of 0.9 + 0.09 + 0.009 + … can be treated as an infinite process.
3.14159 is a good approach to π for most technical purposes. 0.99999 is a good approach to 1 for most technical purposes. So what?
Correct. This list of actions is not a complete description of what really happens. Achilles moves continuously from 0 to the 100-yard line. He arrives there by t = 1.
For obvious reasons every line on the list has the same property:
I move my pen to get closer to the limit but not reach the limit
t = 0: I move my pen from point 0 to point 0.9 of the number line
t = 0.9: I move my pen from point 0.9 to point 0.99 of the number line
t = 0.99: I move my pen from point 0.99 to point 0.999 of the number line
…
Tell me if you can find a line on the list for which this does not hold true.
Yes, if they were different numbers. Two different well-defined numbers. 1 is a well-defined point on the number line, 0.999… is not.
I don’t care if it is on the number line or not. I asked what its squareroot was (or rather, just what the first few digits of its square root are).
Elsewhere you have accepted that 0.999… is greater than all of {0.9, 0.99, 0.999, … } :
So even if you are right and we cannot place it on the number line, you do agree that it is still a number.
WHAT ARE THE FIRST FEW DIGITS OF ITS SQUARE ROOT?
98th …
For your claims 1 to 90 and 92 to 98, then you’ll reach the limit on the last line, the one with the ellipsis (…). That’s what the ellipsis typically means in this context, “continue this convergent series until it converges on the limit”.
For your 91st claim, this is still unclear until you define the ellipsis there. Does it mean the same as above, in which case you have a serious contradiction (the theorems of calculus prohibit both converging on a limit and not reaching the limit at the same time) … or does it mean “after a very very large but finite number of steps”?
I was going to suggest a new numbering system, S, where we limit the digits to the right of the decimal place to 10[sup]100[/sup]. With this set then we are allowed to have two different elements of S for which there exists no element of S in between them. This gives us the property of being able to converge on the limit yet not reach the limit. This might be useful in Quantum Mechanics, where we have an actual physical property that has this feature, there is no quantum valve between 0.000···00135 and 0.000···00136 (where ··· = 10[sup]92[/sup] zeroes and 1 = 10[sup]100[/sup] quantum units (about a half a teaspoon)).
Of course there will be the technological problem of building a pen with a tip one quantum in diameter … failing this then your pen will cross over 1 on your second step …
I don’t know why I persist but …
It’s indistinguishable from yours, except mine is more sensical.
Why do you continue to cut-and-paste incomprehensible gibberish over and over? What pen? What is t? Even if your first nine requires 90 minutes to draw, the third will take just 0.9 minutes, the fifth a half-second, the twelfth only several nanoseconds — impossible.
Yes, yes, we get that your endlessly repeated gibberish is intended to depict some analogy, but an analogy for what? It makes no sense.
Spend some time, if you can, formulating your thesis into simple English with no metaphiers. Drop the pen. Drop the incorrect concept of time. Drop the number line if you don’t understand it. Explain in simple English (without the meaningless “t = 0.99” etc.) what your point is.
Except that this is a nonsensical and meaningless way of looking at 0.99… for all the reasons previously examined. Why the hell should we defined 0.99… as “Constantly getting closer to 1 but not reaching it”? That’s not a useful or meaningful definition for the number. That’s got no basis in calculus whatsoever. You’re talking about summing a row, and even then you have no idea what you’re talking about, because it’s trivial to define a row with exactly that characteristic that behaves wildly different in the limit than in any of the individual parts!
And even then, it completely ignores the fact that the limit is what happens after an infinite number of steps! Not during the steps, after an infinite number of them. And the behavior of a system after infinite steps does not have such a trivially basic relationship to the behavior of the system within any given step! As I previously demonstrated with the balls in the barrel; if you look at any given step, there are countless balls in the barrel, each numbered, and with each step, we only add more… But after an infinite number of steps, it is impossible for any ball to be in the barrel, because for any ball we name, we can point to a step where we removed that ball.
This is like baby’s first infinity; you don’t seem to get that infinity is not a number and that infinite sets of tasks do not behave the way finite sets of tasks do. This is stuff I learned in middle school, or which can be learned by watching Vsauce. Please, learn the basics.
Or, failing that, here’s even more basic logic for you that you seem to not grasp or to be unwilling to address. After infinite steps of moving your pen closer and closer to the limit of 1, what is the distance between your pen and 1? If there is no distance, then your pen is on one, and the paradox is resolved. If there is a distance, why can I then point to a step where we clearly passed the number your distance describes?
Credit where credit is due: this is actually accurate. Because 0.99… is not included in (0,1). Because it’s equal to 1.
:smack:
It clearly corresponds to a point on the number line. That point is 1.
Of course it can. But then you run into Zeno’s paradox, and the entire set of issues you can’t seem to get past here. There are far better, far more reasonable ways to define 0.99…; yours simply seems to be the only one that allows for your particular brand of confusion. Meanwhile, you don’t understand infinity, you don’t understand limits, and you’ve got like 3 or 4 people here explaining to you, all in different ways, where you’ve gone wrong. I strongly recommend you heed their words.
We are like the links of an anchor chain, Grasshopper, sometimes we have to lay in fish poop until the wind changes.
That’s a very pretty example of the weirdness of infinities.
I’m not sure your conclusion is right though – surely there’s an infinite number balls left, just none of them are numbered with whatever number you care to name (can name)?!
An aside: does "your" series, **10 - 1 + 20 - 1 + 30 - 1 + ...**, converge to 0 with Ramanujan summation?Grin! I love that’n. Of course, it can be addressed by a change in notation…
As Martin Gardner set it up, each day ten balls are put in, and one is taken out. Say on day 1 balls 1-10 go in, and ball 1 comes out. On day 2, balls 2-20 go in, and ball 2 comes out. By this notation, the barrel is empty “after infinity,” because any specific numbered ball has been taken out.
So… Change the notation. On day 1, ball 1a through 1j are put in, and ball 1a is taken out. On day 2, balls 2a through 2j are put in, and ball 2a is taken out.
In this notational system, ball 1b is never taken out, and so the barrel is not empty.
(ETA: This is just another way of observing that “infinity minus infinity” is not defined.)
If netzweltler could come up with a notational system that backs up his claims, he might have something worth rebutting. He tried with the moving pen deal, but it failed. Maybe he can come up with something new.
Well, each ball we put in is numbered with a natural number, and we cannot find a ball afterwards that has a natural number on it…
It’s weird. 
Also, I don’t know what Ramanujan summation is, I’m afraid.
No. A number has a well-defined position on the number line.