## 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
41%
No, 1 ≠ 0.999...
16
50%
Other
3
9%

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

Well, I'm still waiting for you to begin addressing my points.

Ecmandu wrote:You wrote an entire message about hyperreals

I wrote a message, yes, but I am not really sure it was a message about hyperreals.

Are you sure? Can you check one more time?

and I responded to it!

You didn't directly respond to any of the points I raised.

1) I said that mathematical sets have no notion of beginning and no notion of end which means that your definition of infinity cannot be applied to them.

You said NOTHING in response to this.

2) I said that it's sequences, and not sets, that have the ability to possess a beginning and an end.

You said NOTHING in response to this.

3) I said that sequences that have an end but no beginning are infinite (that they are infinite even though they have no end because the number of elements they consist of is larger than every integer.)

You said NOTHING in response to this.

4) I said that sequences that have BOTH a beginning and an end are infinite (that they are infinite even though they are bounded from both sides because the number of elements they consist of is larger than every integer.)

You said NOTHING in response to this.

5) I said that mathematical sets and sequences do not exist in time which implies they aren't processes (since processes exist in time.) (And since an infinite set, being a set, is not a process, it also follows that it is not a never-ending process of increase.)

You said NOTHING in response to this.

6) I said that non-mathematical sets and sequences that do exist in time are of no relevance to this topic. (Because this topic is entirely about mathematics and because mathematical entities do not exist in time.)

You said NOTHING in response to this.

And how about things I didn't say but should have said? For example, the fact that your definition of infinity as that which is endless does not imply that infinity is a never-ending process (something you insist the word "infinity" implies.)

Perhaps you need to learn how to hold a conversation first?

So instead of directly responding to what I said, you responded to something you imagined . . . infinitesimals, hyperreals, numbers such as 0.333pi333... which I have no idea what they represent and so on.

Is

0.333pi333...

A number or not a number?

It's not a number because Pi is not a digit. I don't see how this thing you came up with out of nowhere is of any relevance whatsoever. Care to explain?
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:Well, I'm still waiting for you to begin addressing my points.

Ecmandu wrote:You wrote an entire message about hyperreals

I wrote a message, yes, but I am not really sure it was a message about hyperreals.

Are you sure? Can you check one more time?

and I responded to it!

You didn't directly respond to any of the points I raised.

1) I said that mathematical sets have no notion of beginning and no notion of end which means that your definition of infinity cannot be applied to them.

You said NOTHING in response to this.

2) I said that it's sequences, and not sets, that have the ability to possess a beginning and an end.

You said NOTHING in response to this.

3) I said that sequences that have an end but no beginning are infinite (that they are infinite even though they have no end because the number of elements they consist of is larger than every integer.)

You said NOTHING in response to this.

4) I said that sequences that have BOTH a beginning and an end are infinite (that they are infinite even though they are bounded from both sides because the number of elements they consist of is larger than every integer.)

You said NOTHING in response to this.

5) I said that mathematical sets and sequences do not exist in time which implies they aren't processes (since processes exist in time.) (And since an infinite set, being a set, is not a process, it also follows that it is not a never-ending process of increase.)

You said NOTHING in response to this.

6) I said that non-mathematical sets and sequences that do exist in time are of no relevance to this topic. (Because this topic is entirely about mathematics and because mathematical entities do not exist in time.)

You said NOTHING in response to this.

And how about things I didn't say but should have said? For example, the fact that your definition of infinity as that which is endless does not imply that infinity is a never-ending process (something you insist the word "infinity" implies.)

Perhaps you need to learn how to hold a conversation first?

So instead of directly responding to what I said, you responded to something you imagined . . . infinitesimals, hyperreals, numbers such as 0.333pi333... which I have no idea what they represent and so on.

Is

0.333pi333...

A number or not a number?

It's not a number because Pi is not a digit. I don't see how this thing you came up with out of nowhere is of any relevance whatsoever. Care to explain?

Of course pi is not a digit. Pi is an algorithm. It’s a process that never ends!

Like I stated before:

In infinity, if you have a series that is every other number and use this as proof that Infinite series have different sizes!

All I have to do is write an algorithm that they all take a step back or forward like this:

If the first number doesn’t correspond then you either send all the other numbers up one step or you send all the numbers back one step.

With infinities, either way you choose will still
Have everyone holding everyone’s hand.

With finiteness, this is not possible ...

Magnus,

You are treating infinite as finite!
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

I am waiting for you to do two things:

1) address at least one of my points

2) stop abusing "Quote" button

Here's a song to that end:

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

Magnus,

Your last two post haven’t been discussions, they’ve been you saying, “I lost the debate and I don’t know what else to do”
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

You have to respond to the points that I raised.
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:You have to respond to the points that I raised.

I’ll grant you ignorance here.

In discussions on Internet forums, it’s on you to bring up AGAIN!!!! that which you think I’m ignoring!!!

If you’re really lazy, just make a link to the post you think I ignored! And tell me how I ignored it
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:Of course pi is not a digit. Pi is an algorithm. It’s a process that never ends!

Pi is neither an algorithm nor a process. (The word "algorithm" and the word "process" mean two different things. Process is what you get when you execute an algorithm. Algorithms themselves aren't processes. They are merely descriptions of processes.)

Most importantly, even if Pi is a terminating decimal, it still wouldn't be a digit.
Last edited by Magnus Anderson on Tue Jun 16, 2020 2:04 am, edited 1 time in total.
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:In discussions on Internet forums, it’s on you to bring up AGAIN!!!! that which you think I’m ignoring!!!

If you’re really lazy, just make a link to the post you think I ignored! And tell me how I ignored it

Well, I already did. Just a couple of posts ago. You even quoted it. Here it is. I made $$6$$ different claims and you have yet to respond to a single one.
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
Ecmandu wrote:In discussions on Internet forums, it’s on you to bring up AGAIN!!!! that which you think I’m ignoring!!!

If you’re really lazy, just make a link to the post you think I ignored! And tell me how I ignored it

Well, I already did. Just a couple of posts ago. You even quoted it. Here it is. I made $$6$$ different claims and you have yet to respond to a single one.

I read the first sentence, and holy shit! I missed that post! Sorry Magnus ! I’ll read it now! Our last four posts have been a misunderstanding
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

How does one miss a post that they quoted and responded to? (:
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

In short, to all of it, if you believe infinities have beginnings and ends (both), then we have no discussion or conversation.

It’s implied that the endlessness of infinity means that infinity is not an object. It can only be described as a process.

You also completely ignored my point that in infinite sets, if you choose every other number as “non-correspondent” that you can use the simple algorithm of taking 1,2,3 steps forwards or backwards and EVERYONE will still be holding hands again, which is impossible in finite sets.

How about you discuss THAT! Instead of ignoring it for 3 pages!!
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Care to show exactly where?

I am curious to see if what you're saying here (that you responded to each one of my points before I raised them) is true.

In short, to all of it, if you believe infinities have beginnings and ends (both), then we have no discussion or conversation.

I didn't say that "infinities" can have beginnings and ends. I said very clearly that it is infinite sequences that can have beginnings and ends. And I further explained that this is because the word "infinite" simply means "a number larger than every integer". So when we say that a sequence is infinite, we're not actually saying that it's without an end (where by "end" I mean "the last element in the sequence"), but rather, that the number of its elements is larger than every integer. This is exactly the same thing we do with sets. When we say that a set is infinite, we are not saying that it is without an end (because the word "end" is not defined with respect to sets) but that the number of its elements is larger than every integer.

I am pretty sure you did not directly respond to the following point (point #1) anywhere in this thread:

Magnus wrote:1) I said that mathematical sets have no notion of beginning and no notion of end which means that your definition of infinity cannot be applied to them.

Take any finite set of your choice. You can take the set of ternary digits, for example.

$$\{1, 2, 3\}$$

What does it mean to say that this set has an end?
(You are NOT allowed to answer with "It means the number of its elements is an integer".)

What's the last element in this set?
There is no such thing because sets have no order.

It’s implied that the endlessness of infinity means that infinity is not an object. It can only be described as a process.

How does the word "endless", which means no more than "without an end", imply a never-ending process of increase?

You also completely ignored my point that in infinite sets, if you choose every other number as “non-correspondent” that you can use the simple algorithm of taking 1,2,3 steps forwards and backwards and EVERYONE will still be holding hands again, which is impossible in finite sets.

How about you discuss THAT! Instead of ignoring it for 3 pages!!

I responded to that several times in the past.

The problem with that claim of yours is that it contradicts your earlier statements.

You earlier statements imply there's no one-to-one correspondence between the two sets.

Take a look here.
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus,

We have two separate debates in this thread.

0.999=1 (or not)
Orders of infinity exist (or not)

I’m going to resolve both in this post.

9/10+9/100+9/1000... does equal 1 if you add

1/10+(0/10) + 1/100+ (0/100) + 1/1000 etc...

To make the argument that 0.999...=1, you have to argue that addition and non operators are EQUAL!!!

A contradiction! So either addition means something, or all numbers are equal to each other without even using operators.

I’m using (to make this proof) an expanding algorithm, as long as it keeps moving (and carrying), 0.9 + 0.1 always equals 1 and 0.99 + 0.01 always equals 1!

Have you personally EVER seen an infinite sequence that’s not an algorithm? Have you EVER seen a bound infinity all at once?

Nobody ever has nor ever will. Because infinity just keeps going and going and going. By definition!

That means that no matter how far you look, it just keeps going, because the algorithm doesn’t terminate! in the mathematically ideal sense of what infinite expansion or infinitesimals are defined as.

Thus: it’s a process.
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:I’m going to resolve both in this post.

In other words, you are going to ignore everything I said and simply present a new argument of yours.

That's not a good way of making friends.

Have you personally EVER seen an infinite sequence that’s not an algorithm? Have you EVER seen a bound infinity all at once?

It has nothing to do with seeing and everything to do with defining.

The term "infinite sequence" does not refer to an algorithm. (And it also does not refer to a process.)
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
Ecmandu wrote:I’m going to resolve both in this post.

In other words, you are going to ignore everything I said and simply present a new argument of yours.

That's not a good way of making friends.

Have you personally EVER seen an infinite sequence that’s not an algorithm? Have you EVER seen a bound infinity all at once?

It has nothing to do with seeing and everything to do with defining.

The term "infinite sequence" does not refer to an algorithm. (And it also does not refer to a process.)

The only POSSIBLE way that your sentence is true is if you’ve seen ALL the members of an infinite sequence in your head!! If you did or do that, then yes, infinity is not an algorithm or process.

That’s quite a claim Magnus ... that you hold every number in your head!!!!

Actually, it’s laughable!
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

An algorithm is a finite sequence of instructions on how to perform a task.

A process is what you get when you run, execute or follow an algorithm.
(Note that the two words aren't synonymous. Not only that but algorithms aren't even processes.)

An infinite sequence is not an algorithm for two reasons:

1) because algorithms are FINITE sequences (whereas infinite sequences are INFINITE sequences)

2) because algorithms are sequences of SPECIFIC THINGS (namely, instructions on how to perform a certain task) (whereas infinite sequences can be sequences of pretty much anything e.g. animals)

It has ABSOLUTELY NOTHING to do with whether one can hold all of the members of an infinite sequence in one's head or not.
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:An algorithm is a finite sequence of instructions on how to perform a task.

A process is what you get when you run, execute or follow an algorithm.
(Note that the two words aren't synonymous. Not only that but algorithms aren't even processes.)

An infinite sequence is not an algorithm for two reasons:

1) because algorithms are FINITE sequences (whereas infinite sequences are INFINITE sequences)

2) because algorithms are sequences of SPECIFIC THINGS (namely, instructions on how to perform a certain task) (whereas infinite sequences can be sequences of pretty much anything e.g. animals)

It has ABSOLUTELY NOTHING to do with whether one can hold all of the members of an infinite sequence in one's head or not.

Magnus,

Honestly man! This is absurd!

Algorithms can execute finite or infinite. That’s like the most basic “duh” thing in the world!
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### Re: Is 1 = 0.999... ? Really?

Even if algorithms are infinite (and not finite) sequences of instructions, they are STILL infinite sequences of instructions and not merely infinite sequences (since infinite sequences can be sequences of pretty much anything whereas algorithms are specifically sequences of instructions.)
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:Even if algorithms are infinite (and not finite) sequences of instructions, they are STILL infinite sequences of instructions and not merely infinite sequences (since infinite sequences can be sequences of pretty much anything whereas algorithms are specifically sequences of instructions.)

There is NO algorithm that is an infinite sequence of instructions!!! ALL algorithms are a finite sequence of instructions that CAN imply an infinite sequence!
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

That's why you need to learn how to speak properly.

This sentence right here . . .

Ecmandu wrote:Algorithms can execute finite or infinite.

. . . is probably not a proper English sentence.

Two things:

1) Algorithms cannot (and so they do not) execute anything. (It is computers that can execute things, things such as algorithms.)

2) Finite or infinite what?

ALL algorithms are a finite sequence of instructions that CAN imply an infinite sequence!

These algorithms you speak of do not "imply" an infinite sequence. Rather, they produce an infinite sequence as their output. In other words, their output is an infinite sequence. The problem is that the output of an algorithm and the algorithm itself are TWO DIFFERENT THINGS. They aren't one and the same thing.
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:That's why you need to learn how to speak properly.

This sentence right here . . .

Ecmandu wrote:Algorithms can execute finite or infinite.

. . . is probably not a proper English sentence.

Two things:

1) Algorithms cannot (and so they do not) execute anything. (It is computers that can execute things, things such as algorithms.)

2) Finite or infinite what?

ALL algorithms are a finite sequence of instructions that CAN imply an infinite sequence!

These algorithms you speak of do not "imply" an infinite sequence. Rather, they produce an infinite sequence as their output. In other words, their output is an infinite sequence. The problem is that the output of an algorithm and the algorithm itself are TWO DIFFERENT THINGS. They aren't one and the same thing.

Why are you disagreeing with me and then repeating what I stated?

Algorithms IMPLY either a finite or infinite sequence!

That’s what I said!
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Algorithms IMPLY either a finite or infinite sequence!

You are misusing the word "imply".

The algorithms you speak of do not "imply" infinite sequences. Properly speaking, they produce them as their output.

And while I agree that there are algorithms that produce infinite sequences as their output, I disagree that that is a proof that infinite sequences are algorithms.
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Let us consider the following statements:

1) The output of an algorithm is the algorithm itself

2) Algorithms are processes

Of course, both statements are false, but let us accept them as true and see where they lead us.

I believe that the two statements represent the premises in your argument that infinity is process. So what I'm going to try to show now is that accepting them as true leads to conclusions that contradict some of your previous claims.

There are algorithms that output a binary digit. They output a bit: either $$1$$ or $$0$$. This means that bits are the output of some algorithms.

Given that 1) bits are the output of some algorithms, and 2) the output of an algorithm is the algorithm itself, it follows that bits are the algorithms that produced them. And given that algorithms are processes, it follows that bits are processes just as well.

The conclusion is that BITS ARE PROCESSES.

This isn't true but that's not the main thing here.

The main thing here is that it contradicts your earlier claim that bits (being integers) are objects.
Last edited by Magnus Anderson on Tue Jun 16, 2020 7:20 pm, edited 2 times in total.
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
Algorithms IMPLY either a finite or infinite sequence!

You are misusing the word "imply".

The algorithms you speak of do not "imply" infinite sequences. Properly speaking, they produce them as their output.

And while I agree that there are algorithms that produce infinite sequences as their output, I disagree that that is a proof that infinite sequences are algorithms.

You’re strictly talking about non-sentient computers.

To humans!!! Algorithms IMPLY!!!
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Not really.

Magnus wrote:You are misusing the word "imply".

The algorithms you speak of do not "imply" infinite sequences. Properly speaking, they produce them as their output.
Magnus Anderson
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