1=.999999...?

You can say for most purposes 10*.999... = 9.999... However (10*0.999... - 0.999...) =/= 9, so this is a writing convention, not an identity.

J
You're still not grasping the concept of a truly infinite number. .999... is not in your series defined before (.9, .99, .999, ...) - if it was, it would have a definite index in it. As in, you can very clearly see that the first number has one digit, the second two, and so on. So if .999... was at any fixed spot in the list, it would also have a fixed number of digits in it, which by definition it does not.

Similarly for claiming that 10*.999... - .999... =/= 9, if you claim that's not the case, then what does it equal? 10*.999... cannot have a finite number of digits, since if it did, .999... would have a finite number of digits. Thus 10*.999... must have an infinite number of digits. And since we know it starts with 9, it essentially must be equal to 9 + .999... it's a little weird to think of, but if you actually allow yourself to believe that .999... = 1, then it actually makes more sense overall.
 
You're still not grasping the concept of a truly infinite number. .999... is not in your series defined before (.9, .99, .999, ...) - if it was, it would have a definite index in it. As in, you can very clearly see that the first number has one digit, the second two, and so on. So if .999... was at any fixed spot in the list, it would also have a fixed number of digits in it, which by definition it does not.

Similarly for claiming that 10*.999... - .999... =/= 9, if you claim that's not the case, then what does it equal? 10*.999... cannot have a finite number of digits, since if it did, .999... would have a finite number of digits. Thus 10*.999... must have an infinite number of digits. And since we know it starts with 9, it essentially must be equal to 9 + .999... it's a little weird to think of, but if you actually allow yourself to believe that .999... = 1, then it actually makes more sense overall.

To the contrary, you are not grasping continuous. Within the definition of a limit is an uncountable set of points, only one of which is tractable. This is because the algebraic numbers are dis-continuous, though dense. They are infinitely close, but still not continuous. 0.999... is infinitely close to 1, but still not equal.

J
 
You can say for most purposes 10*.999... = 9.999... However (10*0.999... - 0.999...) =/= 9, so this is a writing convention, not an identity.

J

So what does (10*.999...)-9 = ?

Seems to me under your system you can't say .999... = .999...
 
So what does (10*.999...)-9 = ?

Seems to me under your system you can't say .999... = .999...

If you could write the difference, it would not be the difference, sort of like Quantum Mechanics.

That is the reason for the simplifying transform. If (10*.999...)-9 = 0, things are much less messy. Further, it seems to always work when applied to physics. Maybe different colors in a Mandelbrot picture. Maybe not.

J
 
One thing which is certainly having a limit to infinity is how weird and contrived the justifications become just to try to refute a proved truth.
 
In maths? Not that I'd recall. I've made plenty of mistakes corrected by people with less education, but don't remember that a person with substantially less education would have made a relevant point that I would have dismissed.

If someone makes a comment on maths that I disagree with, I ask for his/her justification and present my own, at least that's the principle, sometimes I won't bother. Maths is self contained that way, if you know the definitions, you can get by with reasoning.

However, I've several times mistakenly thought that I know some things better than I do, due to superficial studying or mere arrogance. I could dig some cases out from this forum too, but won't bother right now.

So, you and Terx say that education may be hindrance on understanding the opposition's view here, but have you considered this: maybe your lack of education prevents you from understanding the case made against your argument? If you have thought about it, what convinced you that that's not what happening? How did you become sure that you really understood all the arguments presented here?

Some of the things said here aren't easily digested. For example the axioms of the real numbers that I linked to and the epsilon-delta method of proving things, people spend time understanding it. They ask questions, do exercises, get helped... Even after the first course it may take some time to digest the whole picture.

In this thread it looks like many posters just scanned through those posts and deemed them irrelevant. I wish I would have had students who understood not only the axioms of the real numbers, but also all the ramifications of them and the epsilon-delta method from a single reading of a forum post. Unfortunately, I've never come across with one.

What you're saying is very reasonable and here's where you got me:
You find your education to be hindrance all the time? Have you tried to study harder?

No, I know what you meant.

My entire discussion here has only pertained to the meta-argument because
a) that's what interests me
not unrelated b) I do lack the background to debate this equality, so why would I
following b.ii) have had no reason not to accept all the math explanations for the equality.

Your quote above, pointing out that I called my own discipline political economy while also referring to economics as my discipline while drawing a disciplinary difference between the two, demonstrated that you were following the meta argument just fine, too.1

Here's my invitation to you: in threads where the experts understand something others do not, meet the others where they actually are.2 Try to understand where they are coming from, instead of having disdain for their twin lacks of understanding and deference. The latter immediate sends of JerichoHill red flags who I have had the fortune of finally gaining the technical skills to find out many times when he was rolling his eyes and saying "because I'm an economist" his actual economics was wrong.3

When, for example, I had asked a question 3-4 times and never had it answered by the math people, I make a meta-argument calculation: these people don't know the answer, and might not understand the question. And since it was such an easy question, "yes, it seems that way, but it's still irrelevant" (as I surmised when akka indirectly answered much later) it demonstrated to me a high likelihood that the mathy people were stuck in a much smaller box than was useful for the discussion.

And if there's a lasting discrepancy between experts and others, the argument widens and the others build up more and more of a case, at which point there's more of a meta-argument to follow.

In other words: the longer people aren't met where they are, the less it seems like the experts are actually experts. Thus, the more important it is to be able to "win" the meta-argument to help everyone else get their bearings and stop the reasonable-sounding nonsense.

I think you will find this amenable since you just did it with me.

1. Indeed economics is a sub-discipline of political economy so the shorthand is something I can casually make to an in-crowd and they'll get it (and you got it too), but indeed what I stated was insufficiently consistent, and you charitably pointed it out.

2. like dutchfire does. He was the only person engaging brennan on brennan's terms and still bringing it back to 0.999... = 1.

3. Yes I understand the radical differences between a math proof and an applied economic assumption. It is the interchangeability of the meta-arguments—the credentialed appeal to one's personal authority with the frustrated disdain—that is similar. That meta-argument is all some of us have to go on until we are walked through another way.
 
Including but not limited to: what people are choosing to specifically discuss and in the way they are choosing to discuss it.
 
What else would you say this "process"ness applies to? I assume it would be to all repeating decimals (eg. 0.333, 0.111...) but how about irrational numbers such as pi?

How should we use your version of repeating decimals in a sum? I am sure someone has already presented you with something like this

34cc2f375a21d5f74b6bbaef6cfe35d2.png


how would you go about solving this?
I was truly startled that this worked. I tried to find the error, but all I could muster was the observation that this solution removes the infinite number when it subtracts x.
It allegedly only subtracts 1 x from the left side, leaving 9x, but since 10 times 0.999... only changes the first digit, the 0, to 9, but leaves the rest of it as it were, subtracting x effectively removes the infinite part out of the equation. My first thought was that this might be the key, however, do the same calculations with say 0.222... and it works just fine.
I would have to penetrate math more than I really care to to sufficiently explain this, but it appears to mwe like a weird quirk of the number 9 which allows for this solution to work. Something like because the number 9 is only one unit away from 10, modifying the equation with 10 allows you to end up with x=1 rather than as is the case with 0.222... with a coefficient.

Funny thing :)

Now you said you did not fully understand my position. See, using such academic maths, you will never understand it, I am afraid. My positon is not foremost about math as it is established as a discpline, but about quantities. Something closely related but apparently not quite the same as math. My position is that if 0.999... was not just a theoretical but actual quantity, then it would not be equal to one.
 
The fault in that 'proof' is the assumption that 10x0.999... = 9.999... any number written with a finite number of digits terminates with an (extra) 0 when multiplied by ten. Stating that 10x0.999... = 9.999... acts as though it is adding an extra 9 to the infinite number of nines, which opens the question of whether the operation [multiply by ten] is valid for 0.999...
 
The fault in that 'proof' is the assumption that 10x0.999... = 9.999... any number written with a finite number of digits terminates with an (extra) 0 when multiplied by ten. Stating that 10x0.999... = 9.999... acts as though it is adding an extra 9 to the infinite number of nines, which opens the question of whether the operation [multiply by ten] is valid for 0.999...
The operation is valid, and yes, you are correct that a finite number would end in a zero. But it's like asking what 10*pi is - it'd be 31.415926... but there simply is no last digit. As mentioned, infinities do weird things. It's a similar reason to why there are the same number (cardinality) of rational numbers as there are integers, even though every integer is a rational, and there are clearly rationals that aren't integers. Because there are (countably) infinite numbers of each, you can do funky operations with them that give results sometimes that are not always obvious.
 
I was truly startled that this worked. I tried to find the error, but all I could muster was the observation that this solution removes the infinite number when it subtracts x.
It allegedly only subtracts 1 x from the left side, leaving 9x, but since 10 times 0.999... only changes the first digit, the 0, to 9, but leaves the rest of it as it were, subtracting x effectively removes the infinite part out of the equation. My first thought was that this might be the key, however, do the same calculations with say 0.222... and it works just fine.
I would have to penetrate math more than I really care to to sufficiently explain this, but it appears to mwe like a weird quirk of the number 9 which allows for this solution to work. Something like because the number 9 is only one unit away from 10, modifying the equation with 10 allows you to end up with x=1 rather than as is the case with 0.222... with a coefficient.

Funny thing :)

Now you said you did not fully understand my position. See, using such academic maths, you will never understand it, I am afraid. My positon is not foremost about math as it is established as a discpline, but about quantities. Something closely related but apparently not quite the same as math. My position is that if 0.999... was not just a theoretical but actual quantity, then it would not be equal to one.

I don't understand your problem. At one point you say it does work with 0.222... and then you seem to say it doesn't.

I get:
x = 0.222...
10x = 2.222... = 2 + x
9x = 2
x = 2/9

Which is what I expect. How is this any different from 0.999... which yields 9/9 by the same process?

I don't know what to tell you. (as if I could "tell" you anything)

All we're doing is converting between decimal and rational representations of the same thing.
 
Infinities do weird things, yes. It seems to me that some of these proofs are being inconsistent with when they decide you can and cannot do things with infinities. Here you can just multiply all the digits by 10, there you can't stick something on the end of a string of infinite digits. All very arbitrary.
 
I don't understand your problem. At one point you say it does work with 0.222... and then you seem to say it doesn't.

I get:
x = 0.222...
10x = 2.222... = 2 + x
9x = 2
x = 2/9

Which is what I expect. How is this any different from 0.999... which yields 9/9 by the same process?

I don't know what to tell you. (as if I could "tell" you anything)

All we're doing is converting between decimal and rational representations of the same thing.

It should be
10x = 2.222... = 2 + x + r
9x = 2 + r
Where r is the residual.

J
 
It should be
10x = 2.222... = 2 + x + r
9x = 2 + r
Where r is the residual.

J

You're still not getting infinities. For example, in the following:
x = 0.222....
10x = 2.2222....
it's simple to prove that every digit in the 2.2222... number has to be a 2. Take the digit in the n-th spot. Well, from the 10x, he would have only been the digit in the n+1-th spot in the original number, which is a 2. And there are an infinite number of digits in the 2.222... number, since there were an infinite number of digits in the original number. Thus, the number is 2 + 0.222.... which has an infinite number of 2s, which by definition was our original x. Thus 10x = 2 + x, and x = 2/9, as we were all taught in grade 1.
 
It should be
10x = 2.222... = 2 + x + r
9x = 2 + r
Where r is the residual.

J

I'm reminded very strongly of Hilbert's hotel.

Hilbert's hotel has an infinite number of rooms all of which are occupied. The question is: how does the hotel accommodate a new arrival?

Spoiler :
The person in room 1 moves into room 2, the person in room 2 moves into room 3, and so on. So that room 1 is empty for the new guest.

https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel



The solution seemed absurd to me when I first encountered it. Though after thinking about it for a while, it now makes sense.

Do those saying that 0.999... =/= 1 also claim that Hilbert's hotel cannot accommodate another guest?
 
So what does (10*.999...)-9 = ?

Seems to me under your system you can't say .999... = .999...
If you could write the difference, it would not be the difference, sort of like Quantum Mechanics.

That is the reason for the simplifying transform. If (10*.999...)-9 = 0, things are much less messy. Further, it seems to always work when applied to physics. Maybe different colors in a Mandelbrot picture. Maybe not.

J

Dude what are you talking about with QM? "If you could write the difference, it would not be the difference" doesn't sound anything like QM.

Anyways please note that (10*.999...)-9 = 0 isn't even approximately true. Did you mean (10*.999...)-9 = 1 or (10*.999...)-9 = .999... what?
 
You're still not getting infinities. For example, in the following:
x = 0.222....
10x = 2.2222....
it's simple to prove that every digit in the 2.2222... number has to be a 2. Take the digit in the n-th spot. Well, from the 10x, he would have only been the digit in the n+1-th spot in the original number, which is a 2. And there are an infinite number of digits in the 2.222... number, since there were an infinite number of digits in the original number. Thus, the number is 2 + 0.222.... which has an infinite number of 2s, which by definition was our original x. Thus 10x = 2 + x, and x = 2/9, as we were all taught in grade 1.

Back at you. The residual is real, but not defined.

Just because you can get an arbitrarily large number, say n, of 2s (or 9s) after a decimal point does not mean that you can wave your hand a make residual disappear. Out there n+1 is a something non zero. You can (many have) use limits, and similar devices, to get around it, but that says the residual is negligible, not that it is zero.

J
 
Dude what are you talking about with QM? "If you could write the difference, it would not be the difference" doesn't sound anything like QM.

Anyways please note that (10*.999...)-9 = 0 isn't even approximately true. Did you mean (10*.999...)-9 = 1 or (10*.999...)-9 = .999... what?

In QM you can determine the position of a particle or it's velocity, but not both. In this exercise any number that that can be strictly expressed cannot be the difference.

Consider Thomae's function, AKA the Popcorn Function. It is infinitely discontinuous, at every rational number, but continuous everywhere else. In fact, it is integrable. The f(2/9) = 1/9. f(0.2222...) = 0.

I'm reminded very strongly of Hilbert's hotel.

Do those saying that 0.999... =/= 1 also claim that Hilbert's hotel cannot accommodate another guest?

No. They say that the first arrival can still make it to the restaurant before all the muffins are gone.

J
 
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