Is 1 = 0.999... ? Really?

Also Magnus,

I thought about your utility reply more.

You say that

A.) you want to get to the truth
B.) there’s more utility (if utility means truth) to 0.9… equaling 1

This can be resolved with second grade math:

Rounding.

Rounding is very useful, but we know it’s truth is only in its utility, not because it is actually an equality.

That’s why we see so many “magic numbers” appear, because people are rounding digits, not proving equalities.

Come to think of it, I’m not sure two scenarios are any different: two identical lines or one line with an infinite gap an infinite distance away. To say the gap is an infinite distance away is equivalent to saying it’s at the end of the line. But then what’s at the end of the other line? More line? For all intents and purposes, if we’re talking about “the end of the line”, I’d say it’s fair to say that’s where the lines end. So the gap has effectively been push out of the line and the two are once again identical.

(I realize the silliness of talking about the “end” of an infinite line, but I’m just following the logic that would come out of granting talk about a gap at the “end of the line”.)

Still debating the theoretical meaning of ‘1’?

Existence happens between the theoretical absolutes of 1/0.
There is no ‘one’ in reality - nor a nil. It is a mental abstraction that can refer to anything the mind detaches from space/time and places within vague space/time borders.
‘One’ is an idea, representing an arbitrary moment/place.
Like all ideas it can be defined by the mind - synthesized, manipulated, redefined and redefined, combined, in ways that go beyond the real.

It is a linguistic representation of a mental abstraction, created by the translation of sensual stimuli.

0.9999 is not one…it is a movement towards an absolute that does not exist, and therefore can never be attained.
It represents the fluidity of existence, in relation to the mind’s abstraction.

No 1 in reality eh?

I’m replying to you right now. You are a 1.

Nice try though!

Why does everyone think that 0.999… has to “build up”? Like it’s a process that needs time to complete?

If I say, “suppose you had a queue of 50 people,” do you say, “wait… okay, the 50th person has just been added. Now we can talk about it.” What if I say, “suppose you had a queue of 100 people”? Do we have to wait twice as long before continuing with the conversation?

0.999… is simply notation. It just stands for the idea of an infinite number of 9s. You’re supposed to imagine you already have an infinite number of 9s.

(And really, it doesn’t even represent that; it represents a quantity; the debate in this thread is: what is that quantity? Is it 1 or the next number before 1?)

Thanks for the masterclass on carrying and borrowing - you could have stuck to your kid’s website though - it’s that basic. No wonder you’re more qualified than professional mathematicians to determine these issues.
Are these processes how you’re distinguishing the standard algorithm from addition “in general”?

You realise that carrying and borrowing are only superficial accomodations for numerical systems that use positional notation? Without positional notation, let’s say we’re using a conceptual numerical system that had an infinite number of digits, you wouldn’t need to carry or borrow, because they don’t actually do anything to the quantity itself that is being denoted. The standard algorithm is addition with superficial decoration - your argument that they’re different is like saying a house is different to a house with graffiti on it.

This is probably indicative of a major difference between mathematicians and non-mathematicians: that mathematicians understand quantity at a deeper level since they traverse across multiple different representations depending on the job at hand and the best way to go about it. They don’t get hung up on the extraneous and incidental, they don’t confuse appearance for essence. This will be why non-mathematicians get so bewildered by (1=0.\dot9) because their intuitions lack adaptability to different representations of the same thing.

It’s not wrong and as mathematicians will tell you - explaining why is always necessary.
Simply saying it’s wrong backed by a vague objection and saying it’s unnecessary to prove it more is a far cry from mathematical proofs that can require obnoxious amounts of backing just to be complete.

Your main objection is that (\frac{9.\dot9}{10}\neq0.\dot9) right?
As above, you’re getting “carried” away by the superficial - thinking that (\frac{9.\dot9}{10}) literally shifts all the digits 1 space to the right (or the decimal point 1 space to the left) instead of that just being appearance, meaning there’s an extra “unmatched 9” at the “end” of the endless sequence of 9s compared to the “end” of the endless sequence of 9s in (0.\dot9)

The whole reason I explicitly used this notion for (\frac{s}{10}=\frac{9/{10^0}+9/{10^1}+9/{10^2}+…}{10}=\frac{9/1+9/{10}+9/{100}+…}{10}=9/{10^1}+9/{10^2}+9/{10^3}+…=\sum_{x=1}^\infty \frac9{10^x}) was to emphasise the one-to-one correspondence for “each decimal place” in both (\frac{s}{10}) and (s) but by representing each as a fraction instead of a decimal place - to try and help you from getting confused about superficial appearance of decimals.
This matches the (9/{10^1}) in (s) with the (\frac{9/{10^0}}{10}) in (\frac{s}{10}) (i.e. each 9 in the tenths column together for both s and s/10)
It does NOT match the (9/{10^1}) in (s) with the (\frac{9/{10^1}}{10}) in (\frac{s}{10}) (i.e. not the 0.9 in s with the 0.9/10=0.09 in s/10)

No doubt this still went over your head and you still got confused by the appearance of (9/{10^1}) looking the same in either, even though one was divided by 10 in (\frac{s}{10}) (obviously) and the other in (s) wasn’t (obviously).

This was exactly why it was a rephrasing of something you said to me, that “You have no clue what you’re talking about, so maybe you should just stop it and instead stick to the topic, what do you think?”

The whole point of me saying it was to demonstrate to you that what you just said was “one of the most pointless things you can say in a forum discussion.”

God, you miss literally everything…

You wonder why I’m getting so frustrated?
And you think I’m proud of it?!
I couldn’t be further from proud of the fact that even the most clear and obvious explanations mean nothing to you.

I love how you’ve just brought up Hilbert’s Hotel too, because it’s just another one of those thought experiments that demonstrates the indeterminate sizelessness of infinity (the hotel with infinite rooms in it) and therefore disproves all your nonsense about sizes of infinity. There’s still only 1 infinity (the hotel) whatever happens to it, and regardless of any finite operations/adjustments that you make to its rooms, it’s counterintuitively still full whilst also able to have spaces. Operating on infinity does nothing definite to the fact that it’s still infinite.

This is what indeterminate/undefinable/infinite/endless amounts to.

You can emphasise that, no matter how many 9s in (0.\dot9) fill the gap between it and (1), there’s always an infinite number of smaller infinitesimal gaps that could be filled, and I can emphasise that infinite recurring 9s will always fill these gaps and both/neither of us would be fully right, because infinity is undefinable, which is why things like (\frac1\infty) are invalid because it’s the same as saying (\frac1{undefinable}).
And since there’s nothing definite about how many 9s there are in (0.\dot9) (including any “spare 9s” compared to (9.\dot9)), this is the quality of endlessness that changes all your finite understandings of progressions like (0.9+0.1=1) and (0.09+0.01=1) and so on such that (1=0.\dot9) - there’s no gap because the indeterminacy of endlessness/infinity means there’s no way to determine any gap.

Yet another set of explanations that really really should put this topic to rest.
No doubt it won’t. Disappoint me further.

You are correct.
I was hasty in my expressions.
There is no movement towards the absolute, but away form it. Linear time is a movement towards increasing chaos, not increasing roder - that lies in the indeterminate past, as a hypothetical point in space/time - Big Bang.
It is a nihilistic inversion corrupting human language.
The more accurate image would be a movement towards absolute nil - away from the theoretical one.
The arrow of time points away from a singularity - the near absolute point we use to measure time - and represented by Yin/Yang.

Even in the metaphor of the line, the individuals in line are deteriorating - fragmenting, as they wait.

One is reference to order, since the mind cannot conceive chaos.
All language, including mathematics, measures order, so it tends to be positive, when, in fact, order is declining over space/time, not increasing.

Because that’s how any number is abstracted in the first place - a build up. (Step 1)

Then we use inferential logic for the value and say, “hey, this never ends” (step 2)

People (rightfully or wrongly) bounce back between the two ways.

If you’re going to articulate a disproof of a “proof” of an infinite sequence, you need to go back to step 1, there’s no other way to do it.

That’s exactly what I did with my argument.

It doesn’t mean that I don’t think step 2 is invalid, step 1 just needs to be revisited/redefined

There’s quantity, and there’s representation.

Representations are constructions that you have to build up before the construction “equals” a quantity (that was conceptually “already there” and didn’t need building).

Building the representation of (0.\dot9) needs endlessness, so writing out the representative construction never ends (or “ends” with indefinite continuation implied) with the quantity that it does equal, which is 1.
Any “gap” is at most hinted at by the representation, not the quantity that it’s building up to, but even then it is only hinted at to the uninitiated who impatiently second guess what the representation would look like if it were to be completed at any given finite point(s) along the way - and they thus conclude that it’ll never get there and there will always be a gap.

You don’t go (\frac{circumference}{diameter}\neq3.141592…\neq\frac4{1}-\frac4{3}+\frac4{5}-\frac4{7}+\frac4{9}-\frac4{11}+…\neq4\sum_{k=0}^\infty\frac{(-1)^k}{2k+1}) because there’s always got to be some “gap” at any conceptual “end” point to the representation implied by that contradiction in terms, that prevents it from ever equalling (\pi). No, they all equal the quantity “pi” because of the fact the construction contains some “incomplete” indetermination to it, not in spite of it.

A fraction with finite terms (C\div{d}) isn’t resolved, the explicit decimal places format isn’t finished, the infinite expansion of explicit fractions doesn’t terminate, the infinite series never gets to that “infinity” - the representation of (\pi) just pretends the irrational and transcendental quantity of pi is finite to make it easier and more compact to accept that all these incomplete constructions really do equal (\pi) in exactly the same way as (0.\dot9) really does equal (1).

Yet somehow, non-mathematicians will generally all accept any of these notations of (\pi) but not the notation of (1) as (0.\dot9). Forgive the pun, but this is entirely irrational.

I’m only able to make sense of some of what you say because the intelligence tests on which I score highly are mostly testing how well someone can extract patterns and sense from increasingly obscure sets of information. When Magnus says to me “It seems like you’re one of those rare people who can understand Ecmandu”, he seems to think that even though my ability to understand even extends as far as it does with you, I don’t understand his extremely simple explanations - the poor guy’s understanding is so consistently backwards.

I don’t have “man up” to address any argument, whether it’s one I support or not - I’ll address it fairly and rationally without partiality either way. I’d be quite happy to analyse any new arguments in this same way, but unfortunately I can’t quite get to the bottom of what you’re building here. Sometimes I feel like I’ve grabbed hold of something, but it goes away when I look into it a bit further.

In the seeming absence of some of your workings, and apparent jumps in your “notes”, I think maybe it boils down to some non-standard terminology that you’re using, like “convergence theory” (I don’t understand the exact process of getting the progressively different results that you’re presenting) or terminology that you’re using in a non-standard way such as the exact distinction “carrying left to right” and “carrying right to left”. Maybe an understanding of them will help me piece together these seemingly bizarre equations of yours.

Silhouette,

I like you, but damn are you arrogant sometimes! By saying this; I mean, man, can you shoot yourself in the foot sometimes!

“Convergence theory” is the idea that you can bound infinities to make them quantities …

For example:

0.999…

Is not a quantity.

1

Is a quantity.

Convergence theory states that an infinite sum (non convergent) equals a whole number (convergent)

What they lack is ability (or at the very least, incentive) to abandon logic when dealing with this subject.

The subject isn’t so complicated that it requires mathematical expertize. You don’t need mathematical expertize to understand that “2 + 2 = 4”. A bunch of mathematicians can’t declare that “2 + 2 = 5” and claim that the dissenters are wrong because they are merely amateurs.

The claim is that the two symbols ((0.\dot9) and (1)) do not represent the same quantity. A number of arguments have been put forward. You’re supposed to point out their flaws. The quoted doesn’t do that.

It’s important to note that it’s not necessary to show why Wikipedia proofs are wrong. There’s a much simpler proof that shows that they must be wrong – whatever their real flaws are.

I’m examining Wikipedia proofs for fun, not out of necessity. I might, in fact, be wrong about why Wikipedia proofs are flawed, but such a scenario wouldn’t render the main proof (which is a very simple proof) invalid.

This isn’t to say that I’m not sure about my claims regarding Wikipedia proofs. I’m pretty confident about them.

Thankfully, not everything is complicated in life, so not everything requires extensive proofs.

Again, the above isn’t pointing out flaws within an argument.

That’s correct.

((9 + \underline{0.\dot9}) \div 10 \neq \underline{0.\dot9})

The two underlined numbers don’t have the same number of 9’s.

In other words, the two infinite sums don’t have the same infinite number of non-zero terms.

Therefore, they aren’t the same number.

That’s my claim. It’s up to other people (which includes you) to point out flaws within my argument if they wish to do so.

That’s precisely the point of our disagreement. The (0.9) in (s) is paired with (0.09) in (s \div 10).

Let’s take (9.\dot9) and divide it by (10) and see what we get:

$$
9.0 \div 10 = 0.9\
0.9 \div 10 = 0.09\
0.09 \div 10 = 0.009\
\cdots
$$

I don’t wonder, I know. You suffer from low tolerance threshold. Not something to be proud of.

Considering that you talk about it publicly, yes.

If you can’t accept that Hilbert’s Hotel is nonsense, there is, I am afraid, no hope for you.

Notice that it’s not merely counter-intuitively full and not full. It does not merely go against intuition. It goes against logic.

Intuition is neither right nor wrong on its own. It’s that thing that is sometimes wrong and sometimes right. You can never know. So if a claim is counter-intuitive, it does not necessarily mean that it’s right or that it’s wrong. But if it goes against logic, then we have a problem, Sir.

A hotel that is both full and not full. Not a logical contradiction at all.

Now you’re conflating different concepts. Infinity/endless is not the same as indeterminate/undefinable.

That’s not true.

If you can’t determine whether there is a difference between two numbers, it does not follow that there is no difference between them.

Sorry to disappoint you.

Here’s a couple of super simple arguments for you to address. And when I mean “address”, I mean “point out their flaws”.

The argument that “Infinite quantity A - 1 < Infinite quantity A”.
viewtopic.php?p=2754790#p2754790

The argument that Hilbert’s Hotel is nonsense:
viewtopic.php?p=2754844#p2754844

Let me see you (or anyone else) point out their flaws.

Who ever said that the lines have to be added end to end? That was the point of having apples, oranges, and pears, each an infinite set.

And Ecmandu, there is a serious difference between “the concept orange” and “the concept of AN orange”.

Equally, there is a significant difference between “AN infinite line” (a finite quantity of a thing) and “the infinity of a line” (a quality of a single thing not being finite).

Explain this to me. What are your step 1 and step 2?

Step 1: build up a sequence of numbers in your mind (a decimal expansion).

Step 2: realize that the build up never ends (a repeating pattern, an irrational number).

This is how we come to realize that certain sequences are infinite, but we don’t have to do this every time. We remember. We say, “suppose you have (\pi),” which we know has an infinite decimal expansion because we’ve proven it already (or someone has).

Besides, we can also propose an infinitely expanding decimal, like 0.999… We say that 0.999… has an infinite number of 9s because we define it that way (and notate it with the …). We don’t need to build it up in our minds to confirm this, we propose that it just is infinite.

Even then, we get around that by inventing a representation of the representation–the dot in (0.\dot9)–which in turns just ends up being a representation of the quantity.

I’m not going to disagree with you here except to note:

What is an infinite orange? (which, if I didn’t say it that way, was a typo)

Gib, you’re showing holes in your knowledge here.

An infinite expansion doesn’t imply an irrational number.

What I mean by step 1 and step 2, is that even to find a finite number like the number 1, we have to explore.

Step number 2 is a completely different exploration and cognitive function, we abstract that it never ends!

Did someone say anything about an infinite orange? And why would he?

It’s merely a convenient way to describe certain properties of infinite sums.

(0.\dot9) approaches but never attains (1). This means two things: 1) not a single one of the partial sums of (0.9 + 0.09 + 0.009 + \cdots) (which is equivalent to (0.\dot9)) is equal to (1), and 2) the greater the number of terms that constitute a partial sum of (0.9 + 0.09 + 0.009 + \cdots), the closer it is to (1).

On the other hand, (1.\dot0) (which is an infinite sum equivalent to (1)) does attain (1). This merely means that it has a partial sum equal to or greater than (1). (In fact, if you’re evaluating the infinite sum from left to right, every single one of its partial sums is equal to (1).)

This alone proves that the two numbers aren’t equal.

I think that is the fundamental problem. I don’t think that it represents a “quantity”. It represents merely a quality. Quanta means finite and discrete.

Two infinite sets added maintains the quality of being infinite. But a quality does not affect quantity.

Here’s a simple visual representation of why (10 \times 0.\dot9 \neq 9 + 0.\dot9).

The image shows (10) instances of (0.\dot9).

The green line indicates the first (0.\dot9) among the ten instances. The number of terms of the equivalent infinite sum that is (0.9 + 0.09 + 0.009 + \cdots) will be represented using (\infty). So every occurence of that symbol from now on represents the number of terms of this sum (and no other infinite quantity.) That would be my standard infinity.

The blue line indicates (10) instances of (0.9) which is (9).

The red line indicates something interesting. It represents (0.\dot9) but is it the same quantity as the one indicated by the green line? Obviously not. You can see that the red rounded rectangle is narrower than the green rounded rectangle. In other words, the number of terms is not (\infty) but (\infty - 1).

Certainly, this proof can’t be accepted if one does not accept that (\infty + 1 > \infty) where every occurence of (\infty) represents the same infinite quantity. So maybe we should settle that question first?