## Is 1 = 0.999... ? Really?

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## Is it true that 1 = 0.999...? And Exactly Why or Why Not?

Yes, 1 = 0.999...
13
38%
No, 1 ≠ 0.999...
18
53%
Other
3
9%

### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:The link you just sent me implies that I’m not allowed to make ANY argument that shows FOR A FACT that infinite and finite behave differently (supposedly (according to you) by my own reasoning).

I am not sure I understand what you're saying here.

Your argument about me contradicting myself by having every boy step forward and still all be holding hands is a fantasy of yours! It violates YOUR reasoning! Not what I’ve presented in this thread.

How? What's wrong with it? Which part do you disagree with?

That's precisely what it is.
"Let's keep the debate about poor people in the US specifically. It's the land of opportunity. So everyone has an opportunity. That means everyone can get money. So some people who don't have it just aren't using thier opportunities, and then out of those who are using them, then most squander what they gain through poor choices, which keeps them poor. It's no one else's fault. The end."

Mr. Reasonable
Magnus Anderson
Philosopher

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### Re: Is 1 = 0.999... ? Really?

I made a claim earlier that infinite sequences are infinite sequences.

(You can find it here.)

You responded to it by saying:

Ecmandu wrote:[T]rue, but also definitely false.

Magnus wrote:How can a statement be both true and false? Isn't that a logical contradiction?

You ignored these questions.

So I'm going to ask you once again:

How can a belief be both true and false?

"Let's keep the debate about poor people in the US specifically. It's the land of opportunity. So everyone has an opportunity. That means everyone can get money. So some people who don't have it just aren't using thier opportunities, and then out of those who are using them, then most squander what they gain through poor choices, which keeps them poor. It's no one else's fault. The end."

Mr. Reasonable
Magnus Anderson
Philosopher

Posts: 4825
Joined: Mon Mar 17, 2014 7:26 pm

### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
Ecmandu wrote:The link you just sent me implies that I’m not allowed to make ANY argument that shows FOR A FACT that infinite and finite behave differently (supposedly (according to you) by my own reasoning).

I am not sure I understand what you're saying here.

Your argument about me contradicting myself by having every boy step forward and still all be holding hands is a fantasy of yours! It violates YOUR reasoning! Not what I’ve presented in this thread.

How? What's wrong with it? Which part do you disagree with?

That's precisely what it is.

Magnus man, you are so incredibly naive. It makes sense to me though that if you can’t understand the implication that:

0.111... = 1/9

And

1/9 = 0.111...

They imply EACH OTHER and because they imply each other, they are equalities!

That you’d have the inability to understand that this is an argument about god!?

You’re amazing at inequalities ! (That’s an insult)

You are not good at equalities! (That’s an insult)
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

What does it mean to say that $$0.111\dotso$$ implies $$\frac{1}{9}$$?
"Let's keep the debate about poor people in the US specifically. It's the land of opportunity. So everyone has an opportunity. That means everyone can get money. So some people who don't have it just aren't using thier opportunities, and then out of those who are using them, then most squander what they gain through poor choices, which keeps them poor. It's no one else's fault. The end."

Mr. Reasonable
Magnus Anderson
Philosopher

Posts: 4825
Joined: Mon Mar 17, 2014 7:26 pm

### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:I made a claim earlier that infinite sequences are infinite sequences.

(You can find it here.)

You responded to it by saying:

Ecmandu wrote:[T]rue, but also definitely false.

Magnus wrote:How can a statement be both true and false? Isn't that a logical contradiction?

You ignored these questions.

So I'm going to ask you once again:

How can a belief be both true and false?

What I’m saying is that it’s true on the surface, but totally false!

Let me give you the example of why I was sent to hell, and then hell beyond hell:

My argument was simple:

If you make suicide and homicide as easy as you could possibly make it (set suicidal and homicidal tension to zero), that whatever survived, would have inherent purpose to live! That’s the solution to ethics!

The argument was flawless!

I was wrong!

The problem on a higher plane of existence with this argument is that you can’t destroy existence, this “flawless” argument only sends people to hell.

Your argument from your mind seems flawless to you, but it is false!
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:What I’m saying is that it’s true on the surface, but totally false!

So you're saying that the statement ("Infinite sequences are infinite sequences") is false and that it only appears to be true?

And you're also saying that some infinite sequences are finite and some are infinite?

Is that correct?
"Let's keep the debate about poor people in the US specifically. It's the land of opportunity. So everyone has an opportunity. That means everyone can get money. So some people who don't have it just aren't using thier opportunities, and then out of those who are using them, then most squander what they gain through poor choices, which keeps them poor. It's no one else's fault. The end."

Mr. Reasonable
Magnus Anderson
Philosopher

Posts: 4825
Joined: Mon Mar 17, 2014 7:26 pm

### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
Ecmandu wrote:What I’m saying is that it’s true on the surface, but totally false!

So you're saying that the statement ("Infinite sequences are infinite sequences") is false and that it only appears to be true?

And you're also saying that some infinite sequences are finite and some are infinite?

Is that correct?

Magnus! I like you because you stick with it!

Obviously tautologies aren’t false: “infinite sequences are infinite sequences”

With math, as with MANY aspects of life, it isn’t that simple!

Here’s an infinite sequence:

0.333...

ALSO a NUMBER!!!

ALSO 1/3rd!!!

Multiple equalities are true for EVERY possible number!! That’s the way it works.

No! I’m saying EVERY possible number is represented by a finite and/or infinite sequence and the reverse!

It’s really not hard to prove that magnus!
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:Here’s an infinite sequence:

0.333...

That's not an infinite sequence.

No! I’m saying EVERY possible number is represented by a finite and/or infinite sequence and the reverse!

If infinite sequences are not infinite sequences then what kind of sequences are they?

Are they sequences that are both finite and infinite?
"Let's keep the debate about poor people in the US specifically. It's the land of opportunity. So everyone has an opportunity. That means everyone can get money. So some people who don't have it just aren't using thier opportunities, and then out of those who are using them, then most squander what they gain through poor choices, which keeps them poor. It's no one else's fault. The end."

Mr. Reasonable
Magnus Anderson
Philosopher

Posts: 4825
Joined: Mon Mar 17, 2014 7:26 pm

### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
Ecmandu wrote:Here’s an infinite sequence:

0.333...

That's not an infinite sequence.

No! I’m saying EVERY possible number is represented by a finite and/or infinite sequence and the reverse!

If infinite sequences are not infinite sequences then what kind of sequences are they?

Are they sequences that are both finite and infinite?

0.333... right?

Well it’s not an infinite SUM! 0.333... is a discernible pattern that goes on forever (thus an infinite sequence!)

This is the problem we’ve been having. Your second quote encapsulates this problem! “If infinite sequences are not infinite sequences, then what kind of sequences are they?”

Well... tautologically, an infinite sequence is an infinite sequence! No argument from me there!

They are ALSO finite! In the form of a simple step procedure (a finite algorithm). Every infinite sequence is equal to a finite algorithm; they are equal.

That’s all my point was.
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Let me try to explain why $$0.333\dotso$$ is not an infinite sequence.

When we ask "Is $$0.333\dotso$$ an infinite sequence?" what we're asking is "Does the symbol $$0.333\dotso$$ represent an infinite sequence?"

This means that we're asking what the symbol $$0.333\dotso$$ represents and NOT what the symbol $$0.333\dotso$$ is in itself.

When we ask "Are numbers sequences?" what we're asking is "Does the word <number> represent a sequence?"

Since the word "number" does not represent itself, we do not care about the fact that the word "number" is a finite sequence of letters. In other words, the fact that the word "number" is a finite sequence of letters does not mean that what the word "number" represents (= symbolizes = signifies) is a finite sequence of letters (or a sequence at all.)

In the same exact way, the fact that the symbol $$0.333\dotso$$ is a finite sequence of characters does not mean that it represents a finite sequence of characters (or a sequence at all.)

That symbol, on its own, is a sequence consisting of $$8$$ characters. But the symbol itself does not represent a sequence of $$8$$ characters.

$$0.333\dotso$$ is a symbol representing certain number. And it's not the only symbol that represents that number. There are many other symbols representing the same number. There are finite sequences of characters such as $$0.3 + 0.03 + 0.003 + \cdots$$ as well as infinite sequences of characters that we cannot write down (since they are infinite) but that we can represent using other symbols (e.g. what is represented by the statement "a sequence that starts with $$0$$ followed by $$.$$ and an infinite sequence of $$3$$'s" is an example of such a symbol.)

Anything can be represented using any kind of symbol. The fact that you can represent a number using an infinite sequence DOES NOT MEAN that that number is an infinite sequence.

You can represent numbers with horses. That does not mean that numbers are horses.

The symbol is not the symbolized.
"Let's keep the debate about poor people in the US specifically. It's the land of opportunity. So everyone has an opportunity. That means everyone can get money. So some people who don't have it just aren't using thier opportunities, and then out of those who are using them, then most squander what they gain through poor choices, which keeps them poor. It's no one else's fault. The end."

Mr. Reasonable
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Sure... the map is not the territory.

We also have a property called “inferential logic”. We know that “x” implies “y”.

We also have inferential proofs.

The most basic inferential proof (and this always happens with any infinity). Is:

1,2,3,4,5,6,7,8,9,10,11,12 etc...

We know inferentially that’s no number repeats and it always grows in size.

Can any being hold all those numbers in its head to prove it ? NO! That’s why it’s called an inferential proof!

We infer from the sequence!

This type of proof is used in all infinity math!

What makes it? A sequence, a set of instructions, a discernible pattern!

Yes! 0.333... is a discernible pattern. That’s all a sequence is or need be
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:Can any being hold all those numbers in its head to prove it?

What does that mean?
(I asked this question before, you ignored it.)
"Let's keep the debate about poor people in the US specifically. It's the land of opportunity. So everyone has an opportunity. That means everyone can get money. So some people who don't have it just aren't using thier opportunities, and then out of those who are using them, then most squander what they gain through poor choices, which keeps them poor. It's no one else's fault. The end."

Mr. Reasonable
Magnus Anderson
Philosopher

Posts: 4825
Joined: Mon Mar 17, 2014 7:26 pm

### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
Ecmandu wrote:Can any being hold all those numbers in its head to prove it?

What does that mean?
(I asked this question before, you ignored it.)

I probably didn’t answer it because I was so incredulous that you don’t know what it means!!

It means what it says!!!

For example:

Can you hold every (implied) digit of 0.999... in your head at once?

Of course you can’t! Nobody can!
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

What does it mean to say that someone can hold something in their head?

Does it mean that someone's head is big enough for that thing to fit inside?

In other words, does it mean that the volume of one's brain is equal to or greater than the volume of that thing?

If this is the case, what's the volume of the infinite sequence that is $$(1, 2, 3, \dotso)$$?

If the volume of every number in that sequence is exactly one infinitesimal $$cm^3$$, then the total volume of that sequence (assuming that only its elements have volume) is exactly $$1cm^3$$ (since infinity times infinitesimal is $$1$$) which makes the volume of that infinite sequence smaller than every human brain.
"Let's keep the debate about poor people in the US specifically. It's the land of opportunity. So everyone has an opportunity. That means everyone can get money. So some people who don't have it just aren't using thier opportunities, and then out of those who are using them, then most squander what they gain through poor choices, which keeps them poor. It's no one else's fault. The end."

Mr. Reasonable
Magnus Anderson
Philosopher

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Joined: Mon Mar 17, 2014 7:26 pm

### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:It means what it says!!!

You were asked to clarify, not to repeat what you said.

"Let's keep the debate about poor people in the US specifically. It's the land of opportunity. So everyone has an opportunity. That means everyone can get money. So some people who don't have it just aren't using thier opportunities, and then out of those who are using them, then most squander what they gain through poor choices, which keeps them poor. It's no one else's fault. The end."

Mr. Reasonable
Magnus Anderson
Philosopher

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### Re: Is 1 = 0.999... ? Really?

On the other hand, the fact that something is larger than one's head does not mean that it is a process. There are many things that are larger than our heads e.g. airplanes. Does not mean that airplanes are processes.
"Let's keep the debate about poor people in the US specifically. It's the land of opportunity. So everyone has an opportunity. That means everyone can get money. So some people who don't have it just aren't using thier opportunities, and then out of those who are using them, then most squander what they gain through poor choices, which keeps them poor. It's no one else's fault. The end."

Mr. Reasonable
Magnus Anderson
Philosopher

Posts: 4825
Joined: Mon Mar 17, 2014 7:26 pm

### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:On the other hand, the fact that something is larger than one's head does not mean that it is a process. There are many things that are larger than our heads e.g. airplanes. Does not mean that airplanes are processes.

I like that analogy!

You can hold a plane in your “minds eye”, you cannot hold 0.999... in your “minds eye”

Does that help?

These are very abstract metaphors here!
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

In order to be able to see or visualize anything, one needs time. If you're sitting in front of a TV and I'm flashing a picture on it (let's say the picture is shown for no longer than $$1$$ millisecond), you wouldn't be able to tell me anything about it. The more you want to see or visualize something, the more time you need. If it takes you one second to visualize a single apple, it will take you an infinite number of seconds to visualize an infinite number of apples. If it takes you one infinitesimal second to visualize a single apple, it will take you a second to visualize an infinite number of apples.

Most importantly, the fact that we can't see or visualize something within a finite period of time does not mean that that thing is a process.
"Let's keep the debate about poor people in the US specifically. It's the land of opportunity. So everyone has an opportunity. That means everyone can get money. So some people who don't have it just aren't using thier opportunities, and then out of those who are using them, then most squander what they gain through poor choices, which keeps them poor. It's no one else's fault. The end."

Mr. Reasonable
Magnus Anderson
Philosopher

Posts: 4825
Joined: Mon Mar 17, 2014 7:26 pm

### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:In order to be able to see or visualize anything, one needs time. If I made you sit in front of a TV and flashed a picture on it (let's say the picture is shown for no longer than $$1$$ millisecond), you won't be able to tell me anything about it. The more you want to see or visualize something, the more time you need. If it takes you one second to visualize a single apple, it will take you an infinite number of seconds to visualize an infinite number of apples. If it takes you one infinitesimal second to visualize a single apple, it will take you a second to visualize an infinite number of apples.

Ahh... that’s an interesting argument. You gave me something new to work with.

If infinity is not a process(as I say it is), then the odds of an infinitesimal (say us in an infinite universe) are exactly zero percent! Actually, and I say this to the best mathematicians on earth (and hopefully you can understand it magnus!) if there are always infinitely larger infinities, and our cosmos is finite (or a lesser order of the highest infinity), than at the highest cardinality, there is a zero percent chance for our entire universe to exist!

If however, hyperreals can’t exist, then our odds of existing are not zero percent, but rather 100%.
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:
Magnus Anderson wrote:In order to be able to see or visualize anything, one needs time. If I made you sit in front of a TV and flashed a picture on it (let's say the picture is shown for no longer than $$1$$ millisecond), you won't be able to tell me anything about it. The more you want to see or visualize something, the more time you need. If it takes you one second to visualize a single apple, it will take you an infinite number of seconds to visualize an infinite number of apples. If it takes you one infinitesimal second to visualize a single apple, it will take you a second to visualize an infinite number of apples.

Ahh... that’s an interesting argument. You gave me something new to work with.

If infinity is not a process(as I say it is), then the odds of an infinitesimal (say us in an infinite universe) are exactly zero percent! Actually, and I say this to the best mathematicians on earth (and hopefully you can understand it magnus!) if there are always infinitely larger infinities, and our cosmos is finite (or a lesser order of the highest infinity), than at the highest cardinality, there is a zero percent chance for our entire universe to exist!

If however, hyperreals can’t exist, then our odds of existing are not zero percent, but rather 100%.

I figured out a way to put this in layman’s terms...

Let’s say that only one penny exists in all of existence, but you have an infinite number of dimes (they never stop) - your odds of finding the penny are zero percent! As if it didn’t and never did exist.
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:Let’s say that only one penny exists in all of existence, but you have an infinite number of dimes (they never stop) - your odds of finding the penny are zero percent! As if it didn’t and never did exist.

I'd say the probability of finding a penny is $$\frac{1}{\infty + 1}$$. That's close to $$0$$ but it's not $$0$$.

The question I want to ask is:

How does this prove that the word "infinity" refers to a process?
"Let's keep the debate about poor people in the US specifically. It's the land of opportunity. So everyone has an opportunity. That means everyone can get money. So some people who don't have it just aren't using thier opportunities, and then out of those who are using them, then most squander what they gain through poor choices, which keeps them poor. It's no one else's fault. The end."

Mr. Reasonable
Magnus Anderson
Philosopher

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Joined: Mon Mar 17, 2014 7:26 pm

### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
Ecmandu wrote:Let’s say that only one penny exists in all of existence, but you have an infinite number of dimes (they never stop) - your odds of finding the penny are zero percent! As if it didn’t and never did exist.

I'd say the probability of finding a penny is $$\frac{1}{\infty + 1}$$. That's close to $$0$$ but it's not $$0$$.

The question I want to ask is:

How does this prove that the word "infinity" refers to a process?

That’s a good question!

We have inferential proof.

We can inferentially prove the sequence 1,2,3,4,5,6,7. Etc... not only goes on forever (without repeat), but always increases in size.

We can’t actually “see” in our “minds eye” ALL of it!!

ALL is not a “word that infinity understands”. These are metaphors to be sure!

All assumes completion, such as the number 1, it’s abstracted as an “object” (again, a type of metaphor)...

But! Is infinity (or an infinite sequence) an object like the number 1 is?

The answer is a resounding “no”. Because it’s not an object, it’s a verb, it’s action itself.
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:We have inferential proof.

What is inferential proof?

We can inferentially prove the sequence 1,2,3,4,5,6,7. Etc... not only goes on forever (without repeat), but always increases in size.

I can agree that the sequence $$(1, 2, 3, \dotso)$$ has no end and that it contains no repetitions. However, I cannot agree that it increases in size.

Can you provide a syllogism?

If you cannot expose your reasoning, all we can do is tackle your claims by providing counter-arguments i.e. arguments that prove the opposite of what you say.

This means that, unless you expose your reasoning, all I can do is say that the term "infinite sequence" does not refer to something that exists in time which means that it cannot increase in size and that it's not a process in the first place. I can back this up by quoting Wikipedia. But I've already done most of the work, so the only thing that is left right now is to ask you to expose your reasoning. Such a feat requires a degree of self-consciousness and I'm not really sure you can pull it off but one can always hope.
"Let's keep the debate about poor people in the US specifically. It's the land of opportunity. So everyone has an opportunity. That means everyone can get money. So some people who don't have it just aren't using thier opportunities, and then out of those who are using them, then most squander what they gain through poor choices, which keeps them poor. It's no one else's fault. The end."

Mr. Reasonable
Magnus Anderson
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Joined: Mon Mar 17, 2014 7:26 pm

### Re: Is 1 = 0.999... ? Really?

Magnus,

No possible being can hold “all” of an infinite sequence in their “minds eye” at once, all they can do is to infer it.

An object has an end. Say, a couch. Thus we call it a couch. Very simple. An infinite number of couches?

No way! We can infer it, but we cannot count it.
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Here's a flowchart demonstrating a structured approach to discussing ideas, one that is conducive to resolving disagreements. (This is no quick fix, so no promises of the form "Make everyone agree within a record period of time!" are made. My sole claim is that it's an approach that is better than other approaches. If it takes more than $$1,000$$ pages for people to come to agreement using this approach, it merely means that other approaches would have taken much longer than that.)

Ecmandu loves algorithms, so he should be able to appreciate this flowchart:
How to Discuss Ideas

It's a pretty simple flowchart, actually.

So when I say that someone did not address a claim I made, I am simply saying that they did not follow the steps outlined in this algorithm.

The key part is the idea that the right way to respond to mere assertions is different from the right way to respond to arguments.

For example, if I make a statement that $$0.9 \neq 1$$, without explaining how I arrived at such a conclusion, the adequate way to respond to it, in case you disagree with it, is by offering a counter-argument i.e. by providing an argument that proves that $$0.9 = 1$$. For example:

$$3 \times \frac{1}{3} = 1$$
$$\frac{1}{3} = 0.333\dots$$
$$3 \times \frac{1}{3} = 3 \times 0.333\dotso = 0.999\dotso$$
Therefore, $$1 = 0.999\dotso$$.

Whether or not the argument is sound, whether or not its conclusion is true, whether or not the person understood the original claim and whether or not the person presenting such an argument is stupid or smart are completely irrelevant. The response is an adequate one for the simple reason that it provides a counter-argument to a previously made claim.

However, that wouldn't be a proper response in the case I offered the following argument . . .

1) $$1$$ is a number
2) $$0.999\dotso$$ is not a number because it does not end
3) A number cannot be equal to something that is not a number
4) Therefore, $$1$$ is not equal to $$0.999\dotso$$

Here, you have to show what's WRONG with the presented argument and not merely argue against the conclusion. You have to explain why you think the argument is UNSOUND. Is it logically invalid? If so, why? Are there any premises that are false? If so, which ones and why?

But if all you do is COMPLAIN about how other people are not up to your expectations, then you will get nowhere. You will waste your time and even achieve the opposite of what you wanted to achieve in the first place.
"Let's keep the debate about poor people in the US specifically. It's the land of opportunity. So everyone has an opportunity. That means everyone can get money. So some people who don't have it just aren't using thier opportunities, and then out of those who are using them, then most squander what they gain through poor choices, which keeps them poor. It's no one else's fault. The end."

Mr. Reasonable
Magnus Anderson
Philosopher

Posts: 4825
Joined: Mon Mar 17, 2014 7:26 pm

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