An infinite question: Why doesn't .999~ = 1?

[netzweltler]

@watchwolf49

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 …

“…” is clearly defined:

n = 1, t = 0: I move my pen from point 0 to point 0.9 of the number line
n = 2, t = 0.9: I move my pen from point 0.9 to point 0.99 of the number line
n = 3, t = 0.99: I move my pen from point 0.99 to point 0.999 of the number line
…

For every n ∈ ℕ there is a line on the list.
For every n ∈ ℕ is valid that we don’t reach 1.

ℕ is clearly defined as the infinite set of the natural numbers { 1, 2, 3, … }.

[netzweltler]

@Budget_Player_Cadet

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!

Nothing happens after an infinite number of steps. There is no action specified after an infinite number of steps. Not on this list:

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
…

Please present your list which specifies what happens after an infinite number of steps.

@Budget_Player_Cadet

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?

The distance is not defined, because the position of 0.999… is not defined.

[Budget_Player_Cadet]

@netzweltler

“…” is clearly defined:

n = 1, t = 0: I move my pen from point 0 to point 0.9 of the number line
n = 2, t = 0.9: I move my pen from point 0.9 to point 0.99 of the number line
n = 3, t = 0.99: I move my pen from point 0.99 to point 0.999 of the number line
…

For every n ∈ ℕ there is a line on the list.
For every n ∈ ℕ is valid that we don’t reach 1.

And this doesn’t matter, because we’re looking at what happens once we’ve done all infinity steps. The result we’re looking for is not present in ℕ. There is no step n, n ∈ ℕ that encapsulates what we’re doing. If we don’t reach 1, then after an infinite number of steps, where are we? What is our distance from 1?

Again, I refer you to the example with the balls in the barrel. It seems like, if we add a net of 9 balls every step, we should have an infinite number of numbered balls. But no matter which number you choose, I can point to the step where we necessarily removed that ball. Infinity is weird and counterintuitive. This isn’t even a particularly weird or unintuitive case; it’s perfectly well-understood what happens at a convergent limit. And you still don’t get it.

Yes, at no discrete step during the “process” do we reach 1. It doesn’t matter - the process can be accurately described as “Getting infinitely close to 1”, and as a result, after an infinite number of steps, we are infinitely close to 1 - meaning that there is 0 distance between 0.99… and 1, and therefore they are the same number.

[Budget_Player_Cadet]

@netzweltler

Nothing happens after an infinite number of steps. There is no action specified after an infinite number of steps. Not on this list:

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
…

Please present your list which specifies what happens after an infinite number of steps.
The distance is not defined, because the position of 0.999… is not defined.

(Emphasis mine)

:smack:

The entire point of applying a limit in this case is to see what happens after an infinite number of steps. And we do it because looking at the individual steps on the list does not give us the full picture. If we haven’t already finished all of the infinite number of steps, we’re not looking at 0.99…; we’re looking at 0.9999999 or 0.999999999999999999 or even

0.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999

Which is still a far cry from 0.99… because it doesn’t go on literally forever. And contrary to your assertions, the position 0.99… is very well-defined: it is 3 x (1/3). 1/3 is 0.33… and 0.99… is simply three times that. Or we can use your definition; 0.99… is the sum of the convergent infinite row i(n) = 9*10^(-n). That works too, as long as you don’t blunder right into Zeno’s Paradox and refuse to accept any attempt to resolve it.

[netzweltler]

@Budget_Player_Cadet

The entire point of applying a limit in this case is to see what happens after an infinite number of steps. And we do it because looking at the individual steps on the list does not give us the full picture. If we haven’t already finished all of the infinite number of steps, we’re not looking at 0.99…; […]

The point is: We have finished an infinite number of steps by t = 1. At t = 1 we can state that the only actions are the actions on the list. And according to these actions the state of the pen is pretty undefined at t = 1. It is definitely not at position 1. Unless you present another list of actions.

[The_Great_Unwashed]

@netzweltler

No. A number has a well-defined position on the number line.

So, 0.999… is not a number?

And by extension 0.333… is also not a number? Lots of eight-year-olds are going to be saddened by this news.

BTW, I think you should have come right out and said all this in your first post
"*what are you talking about 0.999... = 1 is nonsense because 0.999... isn't even a number*".

I, for one, would have saved quite a bit of time.

[naita]

@netzweltler

The point is: We have finished an infinite number of steps by t = 1. At t = 1 we can state that the only actions are the actions on the list. And according to these actions the state of the pen is pretty undefined at t = 1. It is definitely not at position 1. Unless you present another list of actions.

Ignore actions. They clearly give a nonsense answer, even if you can’t admit it. Continuous motion.
The pen moves from 0 to 10 (just for fun). As it approaches and passes 1 it either passes by a point that can be described as 0.999… or it passes by one with a finite number of 9’s.

If it does pass by 0.999… how large is the distance between that and 1? I’m not waiting for your answer, as I’m sure it’s that the distance is not defined and that you’re going to reiterate your step by step process which everyone else understands traps you in non-mathematics.

The actual answer is that it is 0.

[netzweltler]

@The_Great_Unwashed

So, 0.999… is not a number?

And by extension 0.333… is also not a number? Lots of eight-year-olds are going to be saddened by this news.

BTW, I think you should have come right out and said all this in your first post
"*what are you talking about 0.999... = 1 is nonsense because 0.999... isn't even a number*".

I, for one, would have saved quite a bit of time.

My position was clear from the very beginning:
The distance |1 - 0.999…| is undefined, because the position of 0.999… on the number line is not defined.

[Budget_Player_Cadet]

@netzweltler

My position was clear from the very beginning:
The distance |1 - 0.999…| is undefined, because the position of 0.999… on the number line is not defined.

You’re wrong. 0.99… is well-defined, and it clearly corresponds to a point on the number line. Interestingly enough, you can reach this point by performing the supertask you continue to define: just keep getting infinitely closer to 1, until after infinite steps, you have 0.99… and you also have your position on the number line: 1.

[DSYoungEsq]

Why, why, why are y’all bothering? He’s not going to admit he’s wrong, because he’s just not going to.

[The_Great_Unwashed]

netzweltler, I am sure I am not the only one who isn’t convinced that your arguments are made in good faith.

You have side-stepped issue after issue and have showed no preparedness to play the game and argue directly to the points made. I feel obliged to do some mathematics for you, this isn’t how it should work, you should be doing this yourself.

We’ll suppose that you are right, that 0.999… < 1

If p ∈ (0, 1) then p < √p < 1

So 0.9 < √0.9 < 1 so the first digit after the decimal point of √0.9 must be 0.9
and 0.99 < √0.99 < 1 so the first digits after the decimal point of √0.99 must be 0.99
and 0.999 < √0.999 < 1 so the first digits after the decimal point of √0.999 must be 0.999

It should be obvious that as 0.999… is an infinite string of 9s, that its square root must also be an infinite string of 9s.

i.e. 0.999… = √0.999…

What number(s) are there for which x = √x ?
There, no deltas, no epsilons, no limits as n approaches infinity. High School maths at most.

[septimus]

@The_Great_Unwashed

What number(s) are there for which x = √x ?

So it seems that netzweltler has overturned the venerable D’Alembert-Gauss Theorem, since that quadratic has THREE roots: 0, 1, and 0.9

At your behest, I’ll bezt Mr. Nezt: … x[SUP]2[/SUP] - 3x + 2 has FOUR roots. [/sarcasm]

[Riemann]

There have been many interesting posts from BPC and others, but the didactic exercise is obviously a waste of time at this point. It’s not as though this is a student who wants to learn something. Just like relativity cranks and creationist cranks, he’s a mathematics crank who’s just going to keep repeating his misconceptions and ignorance of mathematics ad nauseum without regard for any explanation that’s offered.

[watchwolf49]

The nice thing about plunging down rabbit holes is there’s a hookah smoking caterpillar to be found.

103rd …

@netzweltler

“…” is clearly defined:

Then please state the definition you are using in the following, because (…) can be defined as a long comma … requiring the reader to pause a bit longer than a comma but not so long as would be the case with a period.

@netzweltler

n = 1, t = 0: I move my pen from point 0 to point 0.9 of the number line
n = 2, t = 0.9: I move my pen from point 0.9 to point 0.99 of the number line
n = 3, t = 0.99: I move my pen from point 0.99 to point 0.999 of the number line
…

For every n ∈ ℕ there is a line on the list.
For every n ∈ ℕ is valid that we don’t reach 1.

ℕ is clearly defined as the infinite set of the natural numbers { 1, 2, 3, … }.

Again, infinity is not a natural number … so asking us to pick a line n (∈ ℕ) is nonsense … it is your process that is flawed … as well as your definition of a number … and of course your logical assumption that infinity is just a very large finite number is wrong … your citations are absent … you seem to have difficulties with basic addition … long division is just too far “out there” … and that you should say that anything about mathematics should be “ill-defined” show a lack of even the simplest of understanding on the subject.

Take the pill and go through the little door, take the next pill and jump over the fence, second door on the left … have a few puffs if it is lawful for you to do so in your jurisdiction.

My final advice to you is to bring more humility with you when you log on to the message board. Here, there are professionals in most all the fields of human endeavor. So when someone with an advanced college degree in mathematics weighs in on a question about mathematics, you’d be wise to respect that information. Try to be more appreciative of their time explaining things.

[74westy]

netzweltler, I understand that you don’t consider 0.999… to be a number. I’m curious what you do consider to be a number.

Is -1 a number?
Is 1/2 a number?
Is the square root of 2 a number?
Is the square root of -1 a number?

Is something like 0.999… objectively not a number or could we create a self-consistent definition that maps strings like 0.999… to numbers?

[septimus]

@septimus

So it seems that netzweltler has overturned the venerable D’Alembert-Gauss Theorem, since that quadratic has THREE roots: 0, 1, and 0.9

At your behest, I’ll bezt Mr. Nezt: … x[SUP]2[/SUP] - 3x + 2 has FOUR roots. [/sarcasm]

Is my face ever red … again. :eek:

x[sup]2[/sup] - 2x + 2 = 0 is a quadratic with no less than eight distinct solutions!

1 + i
1 + 0.9 i
1 - i
1 - 0.9 i
0.9 + i
0.9 + 0.9 i
0.9 - i
0.9 - 0.9 i

[watchwolf49]

You’re scaring me, septimus …

[The_Great_Unwashed]

@septimus

Is my face ever red … again. :eek:

x[sup]2[/sup] - 2x + 2 = 0 is a quadratic with no less than eight distinct solutions!

1 + i
1 + 0.9 i
1 - i
1 - 0.9 i
0.9 + i
0.9 + 0.9 i
0.9 - i
0.9 - 0.9 i

I know, I was so embarrassed on your behalf I couldn’t bring myself to say anything.

[Trinopus]

@netzweltler

The point is: We have finished an infinite number of steps by t = 1. At t = 1 we can state that the only actions are the actions on the list. And according to these actions the state of the pen is pretty undefined at t = 1. It is definitely not at position 1. Unless you present another list of actions.

Actually, here’s where you’re wrong. In your model, at t=1, then pen has “jumped” an infinite number of times, and has come to rest on 1.0

So far, you’ve declared that it doesn’t…but you have never actually demonstrated this. If you’d actually, y’know, do the math, you’d realize where you’ve blundered.

By the way, what is the square root of 0.999~? And are you able to construct a proof by contradiction, beginning with the assumption that 0.999~ does equal 1.0? You seem to enjoy cutting and pasting your same old dreary goop over and over, but you don’t seem to have any willingness to engage in actual constructive mathematics.

[Chronos]

The exercise about putting balls into and out of a box is an unnecessary distraction from this thread, since it’s discontinuous, and so the limit isn’t meaningful. The limit for 0.9, however, is perfectly well-defined.