1=.999999...?

[Souron]

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.

Only if one infinity is equal to another. If you reject the notion that one more than infinity is equal to the same infinity, then this proof falls apart. This is the Hilbert's Hotel problem.

Specifically if x is defined as above it would have a number of twos after 0 with a quantity we can label x_digits. This quantity is infinite. Then 10 x would have x_digits -1 digits after zero. This quantity is also infinite, but the argument is that it is nevertheless not equal to x_digits. There is one less digit.

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?

It helps if you say that Hilbert's hotel cannot accommodate another guest. I'm not sure if it's necessarily or not.

 

[Souron]

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.

But is the so called carnality of infinite sets measuring the same thing as the carnality of finite sets? Sure, if you take a countable infinity, and add or subtract or add one element, you get a countable infinity. This is a useful thing to say, because it allows you to compare it to uncountable infinite sets. But it does not have the other properties of finite cardinality, so is it really right to say it's the same thing? Are there not in some sense more rational numbers than integers? More integers than primes?

 

[brennan]

Can Hilbert's hotel accomodate another guest if all its rooms are occupied? Of course it can! All occupants simply move into the next room.

Does this really work?

The 'solution' is reliant upon the impossibility of conceiving of an occupant that has no room to move to, since any occupant of the next room will be moving into another room whose occupant will be moving... and so on, therefore you could say that from one point of view it appears to work, but isn't this fooling us just because infinity is not a clearly defined amount?

Let's reframe the problem a little:

Simply add the requirement that none of the occupants of the hotel can move unless there is an empy room for them to move into, otherwise one occupant will obviously end up without a room. The problem is now insoluble:

Can Hilbert's hotel accomodate another guest if all its rooms are occupied? Of course not, all of its rooms are occupied.

The solution appears to be just a word game.

These things arise because infinity is not clearly definable. Take the notion that the cardinality of the set of Integers is the same as that of the even numbers, this relies upon the bijection of the two sets: for every integer (Z) you can conceive of you can also conceive of a corresponding even number (2Z). But simply look at the problem from another angle and it becomes clear that this is the same sort of word game and a little reframing proves the opposite result: For every even number that can be conceived there are obviously twice as many integers as even numbers and therefore it is obvious that the sets cannot biject.

 

[Atticus]

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.

This thread is full of arguments that meet the anti-equation folks where they are. For example the one that starts with 0.999... * 10. I haven't seen need to repeat them, but mostly to present argument against Onejayhwak, and to meet him where he is, imo, is to present the definitions.

Then, there's also the problem you have with conspiracy theorists: they can reinterpret the facts, change the methods of verification of truth etc. whenever they are backed into a corner, and don't ever have to admit they're wrong. That's one reason why this thread is so stupid: some people's motivation isn't to learn or understand, but to "win".

To modify the your metaphor, when I try to meet them where they are they move somewhere else.

This thread is full of people moving the goal lines or just outright ignoring counterarguments when they don't suit them.

For this reason I put on the table the definitions of the real numbers, limits, series etc., so that if someone disagrees with them, he can tell where they go wrong. I've also asked for clarification for some people's definitions that conflict them.

The authority thing isn't meant to be a surrogate for argument, I've mentioned it mostly to inform that I'm not pulling from the hat the things I say, and also to warn possible bystanders who might have difficulty to discern what is nonsense and what is not. If you read back, the only appeal to authority I've made as an argument against the claim that 0.999... != 1 is that I don't lie when I tell the usual definitions of things. That's a thing anyone can check. For example, if you don't believe that a sum of a series is defined as the limit of the partial sums, you can go to library and check it. (For example Rudin's Real and Complex analaysis is a one of the standard references when it comes to the basic analysis). The argument I've presented doesn't rely on the authority.

therefore it is obvious that the sets cannot biject.

f:Z -> 2Z: f(n) = 2n
is a bijection.

You should check how bijections, infinity and cardinality of sets are defined.

 

[onejayhawk]

Talking about definitions, it appears 0.999... is not well defined. If it is, I have not been able to find a source. There are two possibilities.

0.999... = sum from 1 to infinity (9*10^-n) or
0.999... = lim[sum from 1 to infinity (9*10^-n)] = 1

Common usage favors the first rendering. That they are not equal is easily shown. The first is on a segment open at 1. The second is not.

J

 

[Lohrenswald]

0,999... doesn't have a special defenition. It's just a number, like any other (and equal to 1)

 

[azzaman333]

Talking about definitions, it appears 0.999... is not well defined. If it is, I have not been able to find a source. There are two possibilities.

1 = 1 or
1 = lim[1] = 1

Common usage favors the first rendering. That they are not equal is easily shown. The first is on a segment open at 1. The second is not.

J

FTFY.

 

[Bootstoots]

How did this thread get so long? I mean I could go back and look but I'm not sure I want to.

But really, it's as true as 2+3=5. Or more to the point, 1/2+1/4+1/8+... = 1. It's just a geometric series, folks. They converge.

 

[Mouthwash]

I have a forty page long thread? Almost feel like a real forumer now!

 

[AdrienIer]

How did I miss this thread...

Jay the sum from 1 to infinity is a concept that is literally defined as the limit when n goes to infinity of the sum from 1 to n. So your two examples are the same thing (actually the second one is not well written : you're not taking the limit of anything)

Edit : Oh the amount of ignorance in this thread is depressing. What is worse is the refusal by some to accept the many correct proofs of the equation that were proposed.

 

[Mouthwash]

And now I'm really curious as to why it's suddenly twenty-seven pages long. Was a discussion moved or something?

 

[Mise]

Hygro's point about meta-discussion has gone unanswered, which is partly my fault for being blind to it at the time. The error is in thinking that those who argue 0.999...=/=1 are trying to prove that 0.999...=/=1, when in fact they are trying to prove that they are very smart.

 

[Hygro]

ahaha I should announce that I surrender to the gentleman who pointed out that maybe I not only was wrong but had my stated position in the meta-game backward.

 

[Mise]

ahaha I should announce that I surrender to the gentleman who pointed out that maybe I not only was wrong but had my stated position in the meta-game backward.

Sorry, I don't know if I've misunderstood either this response here or your "meta-discussion" posts earlier in the thread. The way I understand it, you're saying that you think =1ers should meet =/=1ers where they are, rather than making maths arguments exclusively. In other words, they should change the way they respond, if they want to convince =/=1ers. I'm saying that there are basically two types of =/=1ers: (A) ones who don't understand the maths yet and think it counter-intuitive, but who are otherwise willing to listen and are interested in being converted, and (B) ones who are primarily interested in making counter-arguments. =1ers are appealing to group A when they make their maths-based arguments; they might convert 90% of A if they do this, because, well, the maths is well-established, and makes a lot of sense. (They might convert a far lower percentage in economics or politics, which are far more debatable subjects.)

However, there are a bunch of group B people, who dress themselves in group A clothes, and who =1ers engage as if they were in group A. And when they do that, they get bogged down, because really, group B isn't interested in being convinced or in understanding the maths or whatever. They are engaging in the debate solely to express an opinion of their own; solely to be contrary; solely to prove their intellectual chops. It's like a footballer (soccer player) who won't play as a team because they are more interested in performing tricks and showing off their skill, than in actually winning the football match. When you say that we should meet them where they are, this works and makes a lot of sense for people in group A. In politics especially, meeting people where they are is a huge failing of the left in this election cycle, and your posts on this are pretty prescient on that. On page 1 of this thread, group A was in the majority of =/=1ers, and they were largely convinced by the maths-based arguments put forth by the clever =1ers. But this thread, by the end of it, after 20-odd pages, no longer contained a majority of group A. Instead, it was mainly a handful of group B arguing against the =1ers. At this point, "meeting them where they are" isn't even possible, because group B are not interested in being convinced. At this point, it's best to just stop engaging with group B: identify their number, and just ignore them.

 

[Ziggy Stardust]

I remember this thread!

 

[Lohrenswald]

@Mise
On one hand I feel like you're right.
On the other hand I see that these group B people are obviously wrong, and look very silly saying the same thing over and over again.
But then again since I know 0,999...=1 I guess my perspective gets skewed

 

[Hygro]

Sorry, I don't know if I've misunderstood either this response here or your "meta-discussion" posts earlier in the thread. The way I understand it, you're saying that you think =1ers should meet =/=1ers where they are, rather than making maths arguments exclusively. In other words, they should change the way they respond, if they want to convince =/=1ers. I'm saying that there are basically two types of =/=1ers: (A) ones who don't understand the maths yet and think it counter-intuitive, but who are otherwise willing to listen and are interested in being converted, and (B) ones who are primarily interested in making counter-arguments. =1ers are appealing to group A when they make their maths-based arguments; they might convert 90% of A if they do this, because, well, the maths is well-established, and makes a lot of sense. (They might convert a far lower percentage in economics or politics, which are far more debatable subjects.)

However, there are a bunch of group B people, who dress themselves in group A clothes, and who =1ers engage as if they were in group A. And when they do that, they get bogged down, because really, group B isn't interested in being convinced or in understanding the maths or whatever. They are engaging in the debate solely to express an opinion of their own; solely to be contrary; solely to prove their intellectual chops. It's like a footballer (soccer player) who won't play as a team because they are more interested in performing tricks and showing off their skill, than in actually winning the football match. When you say that we should meet them where they are, this works and makes a lot of sense for people in group A. In politics especially, meeting people where they are is a huge failing of the left in this election cycle, and your posts on this are pretty prescient on that. On page 1 of this thread, group A was in the majority of =/=1ers, and they were largely convinced by the maths-based arguments put forth by the clever =1ers. But this thread, by the end of it, after 20-odd pages, no longer contained a majority of group A. Instead, it was mainly a handful of group B arguing against the =1ers. At this point, "meeting them where they are" isn't even possible, because group B are not interested in being convinced. At this point, it's best to just stop engaging with group B: identify their number, and just ignore them.

I think this is practical. I was in group A not insofar as the math could convince me, because I never got that far in math to "know" this, but insofar as to understand, after this thread, that the semantic differences lead to tortured irrelevance because math equation notation is not meant to imply a process like an executable code.

I think there's value to having B people. Beyond that I'm like

 

[warpus]

Imagine what Trump would say about this thread if he had to say something about it to the American people. The fact that 0.999... can equal 1 would probably fly right over his head. Sad.

 

[onejayhawk]

How did this thread get so long? I mean I could go back and look but I'm not sure I want to.

But really, it's as true as 2+3=5. Or more to the point, 1/2+1/4+1/8+... = 1. It's just a geometric series, folks. They converge.

Which brings you to the point of the discussion. It's a thought experiment like the original tortoise and Achilles race. Convergence is not identity. Modern math says it's close enough. Look back on the discussion and see how often limits are used. The definition of limit is just rigorous mathspeak for close enough.

J

 

[warpus]

Modern math does not say it's "close enough". Modern math says that they are equivalent.