The meaning of infinity

[Jokodo]

They don't. And that's exactly why it's the same. It's been demonstrated to you in at least three different ways in the last couple of weeks alone. If you want to call those proofs "tricks" or "slights of hand", it's up to you to show where they go wrong. If you can't, all you're doing is demonstrating that you have no clue what you're talking about.

You probably also believe that in a spherical coordinate system, 180° West and 180° East are 360° apart.

I showed you. You are claiming infinite operations complete.

9 * 1/9 immediately reduces to 1 by definition.

9 * 0.111... is an infinite operation that has no final value.

These are not the same thing.

Which is why you can't demonstrate anything without some transformation or by performing some operation chosen from left field..

We are talking about an infinite series.

It does not by magic change into something else.

The number whose decimal representation is 0.111... (let's call it Alex) is no different in kind than the number whose decimal representation is 0.125 (let's call this guy Tom). Neither is an infinite series. Both are reals. In fact, in base 9 it's Tom who has to be represented as 0.111... (while Alex is simply 0.1).

They don't "complete" or "become" something, and they're not series, infinite or otherwise -- and nine times Alex is 1.0 exactly whether you write it as decimal 0.111... * 9 or nonary 0.1 * 10 or octal 0.0707(07)* x 11

And since decimal 0.111... * 9 also happens to be 0.999... by simply multiplying every digit by nine, we have proven that 0.999... and 1.0 are equivalent descriptions of the same real number in the decimal system of notation, whether you like it or not.

If you don't understand that the decimal system is a convention that may look natural to beings who, by an accident of evolutionary history, have ended up with two hands of five fingers each, but nonetheless a fully arbitrary convention in the grand scheme of things, you really shouldn't be posting in a maths forum.

 

[untermensche]

I showed you. You are claiming infinite operations complete.

Which is why you can't demonstrate anything without some transformation or by performing some operation chosen from left field..

We are talking about an infinite series.

It does not by magic change into something else.

And since decimal 0.111... * 9 also happens to be 0.999... by simply multiplying every digit by nine

That's right.

That is all you have to do. Multiply every digit.

All of them.

Tell me when you are done.

You have defined a multiplication that never ends. I know this is very basic math.

You can assume it ends but that is not the same thing.

After a while these infinite leaps become second nature.

But they can't be done. It is not an operation that finishes.

They are merely defined to finish.

It is pretending to do something with infinity.

 

[Jokodo]

I showed you. You are claiming infinite operations complete.

Which is why you can't demonstrate anything without some transformation or by performing some operation chosen from left field..

We are talking about an infinite series.

It does not by magic change into something else.

And since decimal 0.111... * 9 also happens to be 0.999... by simply multiplying every digit by nine

That's right.

That is all you have to do. Multiply every digit.

All of them.

Tell me when you are done.

You have defined a multiplication that never ends. I know this is very basic math.

You can assume it ends but that is not the same thing.

I don't have to "assume it ends". I just have to know the result. All I have to know to know the result is that the result of "1" * nine in decimal notation is "9" independent of scale.

Also, are you or are you not claiming that the result of "0.111... * 9" in decimal is a different number from the result of "0.1 * 10" in base 9?

 

[untermensche]

That's right.

That is all you have to do. Multiply every digit.

All of them.

Tell me when you are done.

You have defined a multiplication that never ends. I know this is very basic math.

You can assume it ends but that is not the same thing.

I don't have to "assume it ends". I just have to know the result. All I have to know to know the result is that the result of "1" * nine in decimal notation is "9" independent of scale.

Also, are you or are you not claiming that the result of "0.111... * 9" in decimal is a different number from the result of "0.1 * 10" in base 9?

There is no final result.

It is an operation that goes on without finish.

I am claiming that mental tricks are played with infinity.

You perform infinite multiplications in an instant.

By knowing where you want to end up ahead of time.

Not by actually being able to perform them.

0.999.... approaches 1. If each 9 defined a value and a line connected the values it would be an infinite asymptote to a line representing the value of 1. It would never touch the line representing 1.

 

[Jokodo]

That's right.

That is all you have to do. Multiply every digit.

All of them.

Tell me when you are done.

You have defined a multiplication that never ends. I know this is very basic math.

You can assume it ends but that is not the same thing.

I don't have to "assume it ends". I just have to know the result. All I have to know to know the result is that the result of "1" * nine in decimal notation is "9" independent of scale.

Also, are you or are you not claiming that the result of "0.111... * 9" in decimal is a different number from the result of "0.1 * 10" in base 9?

There is no final result.

It is an operation that goes on without finish.

I am claiming that mental tricks are played with infinity.

You perform infinite multiplications in an instant.

By knowing where you want to end up ahead of time.

Not by actually being able to perform them.

0.999.... approaches 1. If each 9 defined a value and a line connected the values it would be an infinite asymptote to a line representing the value of 1. It would never touch the line representing 1.

I don't need to take two flights to Kiribati, one from Asia and one from the Americas, to tell that 180° East and 180° West are within spitting distance. The fact that they are follows from the what those descriptions mean within the system of spherical coordinates.

No more do perform any calculations to know what the result will be, if the result follows from the framework in which they're performed.

Are you, or are you not, denying that the statement in my first paragraph is true?

If you choose to reply to my post, don't forget to answer this question.

Also, you didn't answer my last question. For your convenience, I'll repeat and slightly reword it: Are you, or are you not claiming, that the result of 9 * (1/9) depends on which base you chose to represent 1/9 in?

 

[steve_bank]

So what is wrong with saying, at this point:

9x = 9/1

Is there an error there? If not, who cares what else you could say; Let's go with that perfectly true and correct statement, and conclude:

9x = 9
x = 1

This MUST be true, unless there is an error in the above.

NO! You missed the ...

10x = 9.999...

Which is true, but tells us nothing interesting, so why say it?

x = .9999

Again, NO.

x = .999...

Which is back where we started. That's true, but not useful; We know that x = .999... and what we want to know is what ELSE is x. The proof above shows us that
x = .999...
and also that
x = 1
so we must conclude that
.999... = 1

Simples.

Carry it out to as many digits as you like and you will get 0.9999...

Indeed. Or 1, which is the same thing, as we just proved.

let x = 5/10 + 5/100 + 5/1000 + ...
10x = 5/1 + 5/10 + 5/100 + 5/1000 + ....
10x = 5/1 + x
x = .5555
or
10x = 5 + x

Or more simply:
9x = 5

x = 5/9

Correct. So far so good...

If your technique is valid then 0.555... = 0.6

NO!!!

5/9 != 0.6

You need to concentrate, and think about what you are saying, or you will continue to make foolish errors.

There is a fundamental error.

So let x = 9/10 + 9/100 + 9/1000 + ... ok
10x = 9/1 + 9/10 + 9/100 + 9/1000 + .... ok
10x = 9/1 + x not ok

9/1 is the first term in the infinite series 9/1 + 9/10 + 9/100 + 9/1000 + ...., you can not alebraicaly separate the first term in the infinite series and add it to anothe infinite series. 9/1 + x or [9/1 ]+ [0.9 + 0.09 + 0 .009...] is not a valid operation.

And the counter aruments.

As infinite number of terms can not be reached when does 0.999.. become 1 You have to answer that mathemeatically, not conceptually.

If 0.999.. = 1 what does 1.9, 1.99,1.999... equal?

What does 0.6666.. equal?

What does 1. - 0.999.. equal taken term by term? 0.111.... not zero.

 

[bilby]

So what is wrong with saying, at this point:

9x = 9/1

Is there an error there? If not, who cares what else you could say; Let's go with that perfectly true and correct statement, and conclude:

9x = 9
x = 1

This MUST be true, unless there is an error in the above.

NO! You missed the ...

10x = 9.999...

Which is true, but tells us nothing interesting, so why say it?

Again, NO.

x = .999...

Which is back where we started. That's true, but not useful; We know that x = .999... and what we want to know is what ELSE is x. The proof above shows us that
x = .999...
and also that
x = 1
so we must conclude that
.999... = 1

Simples.

Indeed. Or 1, which is the same thing, as we just proved.

let x = 5/10 + 5/100 + 5/1000 + ...
10x = 5/1 + 5/10 + 5/100 + 5/1000 + ....
10x = 5/1 + x
x = .5555
or
10x = 5 + x

Or more simply:
9x = 5

x = 5/9

Correct. So far so good...

If your technique is valid then 0.555... = 0.6

NO!!!

5/9 != 0.6

You need to concentrate, and think about what you are saying, or you will continue to make foolish errors.

There is a fundamental error.

Not on my part there isn't.

So let x = 9/10 + 9/100 + 9/1000 + ... ok
10x = 9/1 + 9/10 + 9/100 + 9/1000 + ... ok
10x = 9/1 + x not ok

9/1 is the first term in the infinite series 9/1 + 9/10 + 9/100 + 9/1000 + ...., you can not alebraicaly separate the first term in the infinite series and add it to anothe infinite series. 9/1 + x or [9/1 ]+ [0.9 + 0.09 + 0 .009...] is not a valid operation.

Why not? I have bolded the series 'x' on the right hand side of the first two equations; what is different about the first instance of '9/10 + 9/100 + 9/1000 + ...' that renders it not equal to the second instance? They are the same; so why on Earth can't I refer to both as 'x', as defined by the first equation, and resulting in the third equation?

From which orifice have you pulled the bizarre and incorrect rule "you can not alebraicaly separate the first term in the infinite series and add it to anothe infinite series"? Both series are written in exactly the same way, using the same notation. How could they possibly NOT be equal to each other?

And the counter aruments.

As infinite number of terms can not be reached when does 0.999.. become 1 You have to answer that mathemeatically, not conceptually.

I have no idea what you are trying to say here. It is nonsense.

0.999... represents an infinite number of '9's after a decimal point; They don't need to be 'reached', they are defined to exist mathematically. That's what the "..." notation MEANS.

If 0.999.. = 1 what does 1.9, 1.99,1.999... equal?

1.9, 1.99, 2

What does 0.6666.. equal?

2/3

What does 1. - 0.999.. equal taken term by term? 0.111.... not zero.

Nope.

1 - 0.999... equals zero.

'taken term by term' is a meaningless description. The sum:

1 - 0.999... = x

doesn't represent a series of operations, or a set of tasks; It's a single subtraction operation, that has the answer:

x = 0

You need to concentrate, be rigorous, and think about what you are saying, or you will continue to make foolish errors.

 

[Juma]

9/1 is the first term in the infinite series 9/1 + 9/10 + 9/100 + 9/1000 + ...., you can not alebraicaly separate the first term in the infinite series and add it to anothe infinite series. 9/1 + x or [9/1 ]+ [0.9 + 0.09 + 0 .009...] is not a valid operation.

Yes you can. Why on earth wouldnt you be able to do that?
Maybe you confuse this with the fact that you cannot, generally, rearrange the order of the terms in an infinite sum?

And the counter aruments.

As infinite number of terms can not be reached when does 0.999.. become 1

That is no counter argument, it is just the ramblings of unter.

Please explain why you say that ”an infinite number of terms can not be reached”? What does it even mean?
Nobody needs to write out all the numbers to know what the sum is.

Let s =9 + 9/10 + 9/100 + 9/1000 + ...
we know it is convergent (the terms form a geometric series with sum a/(1-r) when r<1.)
Here r= 1/10 and a=9 so sum = 9/(1-1/10) = 9/(10-1)/10 = 10. Remove 9 from S and we get the requested 1.

 

[steve_bank]

bilby
Bombs proof fails for the staed reasons. Rigorously refute that.

Rigorously show exactly when 0.999.. =1 when infinity can never be reached. You do not understand?

"0.999... represents an infinite number of '9's after a decimal point; They don't need to be 'reached', they are defined to exist mathematically. That's what the "..." notation MEANS."

Exactly right, so if the end is never reached and decimal places are added infinitem by what mathematical process does it become 1? Answer with rigor.

If the method is solid show me how 1.9,1.99,.999... equals 2. You can not. 0.888.. = ?


x = .9999...
10x = [ 9 + .9 + .99...]

As you said pay attention to the ..., along with the brackets. 9 is the first term in an infinite series. You can not isolate the 9 as as a finite number from the infinite series [ 9 + .9 + .99 + .999...] If you want to argue that then do so by showing math theory that allows the operation. Algebra does not apply to infinite series like .9999...

In set theory there are ways to add infinite sets, but this is not it.

9 + [ .9 + .99 + .999...] does not algebraicaly equal [ 9 + .9 + .99 + .999...] That is bomb's mistake. You can't arbitrarily pull the 9 out of the first series.

If anything 9 + [ .9 + .99 + .999...] = [ 9.9 + 9.99 + 9.999...]

so

10x = 9 + x
10x = 9 + [ .9 + .99 + .999...]
10x = [ 9.9 + 9.99 + 9.999...]
x = [ .9 + .99 + .999...]

Take away the ... and it works. But then for any ka = kb + kc + kd

ka = k[b + c + d]
ka = k[b + c + d]
k/k = [b + c + d]/a
1 = [b + c + d]/a

Which is what bomb effectiveky did, the ... he added in the equations have no meaning.

 

[steve_bank]

9/1 is the first term in the infinite series 9/1 + 9/10 + 9/100 + 9/1000 + ...., you can not alebraicaly separate the first term in the infinite series and add it to anothe infinite series. 9/1 + x or [9/1 ]+ [0.9 + 0.09 + 0 .009...] is not a valid operation.

Yes you can. Why on earth wouldnt you be able to do that?
Maybe you confuse this with the fact that you cannot, generally, rearrange the order of the terms in an infinite sum?

And the counter aruments.

As infinite number of terms can not be reached when does 0.999.. become 1

That is no counter argument, it is just the ramblings of unter.

Please explain why you say that ”an infinite number of terms can not be reached”? What does it even mean?
Nobody needs to write out all the numbers to know what the sum is.

Let s =9 + 9/10 + 9/100 + 9/1000 + ...
we know it is convergent (the terms form a geometric series with sum a/(1-r) when r<1.)
Here r= 1/10 and a=9 so sum = 9/(1-1/10) = 9/(10-1)/10 = 10. Remove 9 from S and we get the requested 1.

I will have to look up geometric series. I will not comment until I read up on it.

Humor me. What does 1.9999... and 0.888.. conveger to? If 0.999.. goes to when I'd assume 1.999.. goes to 2, amd 0.888.. goes to 0.9.


It can not be reached because that is the definition of infinity, or am I missing something?

x = 0.999..
10x = 9 + [ .9 + .99 + .999...]
10x = [ 9.9 + 9.99 + 9.999...]
x = [ .9 + .99 + .999...]

Show my error. You can not isolate the 9 from [9 + .9 + .99 + .999...] and manipulaste it algebraically.

 

[bilby]

bilby
Bombs proof fails for the staed reasons. Rigorously refute that.

Rigorously show exactly when 0.999.. =1 when infinity can never be reached. You do not understand?

No, I do not.

WTF is 'reached' supposed to mean in this context? It's a number. There's nothing to 'reach'. It's not a series, a process, or something that's iterative. It's a number. It's another way of writing '1'. How do you 'reach' 1? it's just a number.

"0.999... represents an infinite number of '9's after a decimal point; They don't need to be 'reached', they are defined to exist mathematically. That's what the "..." notation MEANS."

Exactly right, so if the end is never reached and decimal places are added infinitem by what mathematical process does it become 1? Answer with rigor.

Gibberish cannot be answered, with or without rigor.

If the method is solid show me how 1.9,1.99,.999... equals 2. You can not. 0.888.. = ?


x = .9999...
10x = [ 9 + .9 + .99...]

As you said pay attention to the ..., along with the brackets. 9 is the first term in an infinite series. You can not isolate the 9 as as a finite number from the infinite series [ 9 + .9 + .99 + .999...] If you want to argue that then do so by showing math theory that allows the operation. Algebra does not apply to infinite series like .9999...

In set theory there are ways to add infinite sets, but this is not it.

9 + [ .9 + .99 + .999...] does not algebraicaly equal [ 9 + .9 + .99 + .999...] That is bomb's mistake. You can't arbitrarily pull the 9 out of the first series.

If anything 9 + [ .9 + .99 + .999...] = [ 9.9 + 9.99 + 9.999...]

so

10x = 9 + x
10x = 9 + [ .9 + .99 + .999...]
10x = [ 9.9 + 9.99 + 9.999...]
x = [ .9 + .99 + .999...]

Take away the ... and it works. But then for any ka = kb + kc + kd

ka = k[b + c + d]
ka = k[b + c + d]
k/k = [b + c + d]/a
1 = [b + c + d]/a

Which is what bomb effectiveky did, the ... he added in the equations have no meaning.

I am not isolating anything from any infinite series.

x is defined as 9/10 + 9/100 + 9/1000 + ...

At no step is this modified in any way.

10x is 9/1 + x

x is unchanged. It hasn't been manipulated, had anything separated from it, or had any other modifications, manipulations or changes.

It's consistently the same throughout the proof.

Explain to me how you concluded that I have done any operation at all on it - much less a disallowed operation.

 

[DBT]

Finite events occur in finite time.

If the past is some amount of finite events then the time they occurred in was finite as well.

Doesn't follow. It's not known if the BB was the beginning of time or not. It is not known whether our universe is a part of a greater system, a multiverse, infinite space, etc, but there is nothing to exclude the possibility.

It does follow.

Which is why all you can muster is an empty claim and no reason to think it does not follow.

Infinite time means infinite events which means infinite changes.

Infinite changes is like the positive integers. As defined there cannot be the possibility of reciting the all positive integers, you cannot reach the end, the final integer. Just as there can be no possibility of infinite changes ending as past changes are ended at the present.

If there was infinite time in the past that is like saying there were infinite changes that finished occurring at every present moment.

It is claiming an infinity completed.

A contradiction.

Only the finite completes.

The problem is that you ignore what I say and just repeat your claims. There is no claim of infinity being completed because events within an infinite system come and go.

Events are by their very nature finite. Events are finite even if space itself is infinite and eternal. All events, having a beginning, a middle and an ending, are complete events within the context of infinite space. Event come and go regardless of where and when they occur.

This does not mean that one has to 'wait an eternity for something to happen' because something is happening regardless of when or where. It is always now in relation to an observer/experiencer.

 

[untermensche]

It does follow.

Which is why all you can muster is an empty claim and no reason to think it does not follow.

Infinite time means infinite events which means infinite changes.

Infinite changes is like the positive integers. As defined there cannot be the possibility of reciting the all positive integers, you cannot reach the end, the final integer. Just as there can be no possibility of infinite changes ending as past changes are ended at the present.

If there was infinite time in the past that is like saying there were infinite changes that finished occurring at every present moment.

It is claiming an infinity completed.

A contradiction.

Only the finite completes.

The problem is that you ignore what I say and just repeat your claims. There is no claim of infinity being completed because events within an infinite system come and go.

AT ANY PRESENT MOMENT ALL the events in the past have completed.

NO more events in the past will take place AT THAT PRESENT MOMENT.

No more events can take place in the past at a present moment.

They are ALL completed.

They could not have been infinite.

Time in the past could not have been infinite.

 

[Jokodo]

Yes you can. Why on earth wouldnt you be able to do that?
Maybe you confuse this with the fact that you cannot, generally, rearrange the order of the terms in an infinite sum?


That is no counter argument, it is just the ramblings of unter.

Please explain why you say that ”an infinite number of terms can not be reached”? What does it even mean?
Nobody needs to write out all the numbers to know what the sum is.

Let s =9 + 9/10 + 9/100 + 9/1000 + ...
we know it is convergent (the terms form a geometric series with sum a/(1-r) when r<1.)
Here r= 1/10 and a=9 so sum = 9/(1-1/10) = 9/(10-1)/10 = 10. Remove 9 from S and we get the requested 1.

I will have to look up geometric series. I will not comment until I read up on it.

Humor me. What does 1.9999... and 0.888.. conveger to? If 0.999.. goes to when I'd assume 1.999.. goes to 2, amd 0.888.. goes to 0.9.

You'd assume wrong. 0.888.. is 8/9, just like 0.999... is 9/9. 8/9 is smaller than 9/10, but 9/9 is equal to 10/10

 

[untermensche]

There is no final result.

It is an operation that goes on without finish.

I am claiming that mental tricks are played with infinity.

You perform infinite multiplications in an instant.

By knowing where you want to end up ahead of time.

Not by actually being able to perform them.

0.999.... approaches 1. If each 9 defined a value and a line connected the values it would be an infinite asymptote to a line representing the value of 1. It would never touch the line representing 1.

I don't need to take two flights to Kiribati, one from Asia and one from the Americas, to tell that 180° East and 180° West are within spitting distance. The fact that they are follows from the what those descriptions mean within the system of spherical coordinates.

You need more than your assurances to prove 0.999... equal 1. One an undefined entity that has no final value. The other a final value.

You have proven nothing.

You have made manipulations which erase an infinitely small difference. You have found ways to round off an infinitely small amount through manipulation and assumption.

That is all you have done.

You merely assume they are the same thing because it does not matter.

But not mattering does not make an infinite asymptote ever touch the line.

Without changing or making manipulation prove 0.999 is the same thing as 1. With just words.

Why should I abandon the concept of asymptote and make your leap?

A decimal is a number.

A fraction is an operation.

There are problems when people see fractions as numbers.

 

[Jokodo]

There is no final result.

It is an operation that goes on without finish.

I am claiming that mental tricks are played with infinity.

You perform infinite multiplications in an instant.

By knowing where you want to end up ahead of time.

Not by actually being able to perform them.

0.999.... approaches 1. If each 9 defined a value and a line connected the values it would be an infinite asymptote to a line representing the value of 1. It would never touch the line representing 1.

I don't need to take two flights to Kiribati, one from Asia and one from the Americas, to tell that 180° East and 180° West are within spitting distance. The fact that they are follows from the what those descriptions mean within the system of spherical coordinates.

You need more than your assurances to prove 0.999... equal 1. One an undefined entity that has no final value. The other a final value.

You have proven nothing.

You have made manipulations which erase an infinitely small difference. You have found ways to round off an infinitely small amount through manipulation and assumption.

That is all you have done.

You merely assume they are the same thing because it does not matter.

But not mattering does not make an infinite asymptote ever touch the line.

Without changing or making manipulation prove 0.999 is the same thing as 1. With just words.

0.999 is not 1. They differ by 0.001 exactly.

0.999... is 9/9. Been there, done that. If you have specific questions about a specific step in my previous demonstration, ask them.

- - - Updated - - -

There are problems when people see fractions as numbers.

Really, coming from someone not distinguishing numbers and their representation?

 

[untermensche]

There are problems when people see fractions as numbers.

Really, coming from someone not distinguishing numbers and their representation?

Really.

To not understand a fraction is an operation, not a static entity, is a special kind of blindness.

1 divided by 9.

It can be shortened. But once shortened we shouldn't immediately forget what just happened.

 

[DBT]

It does follow.

Which is why all you can muster is an empty claim and no reason to think it does not follow.

Infinite time means infinite events which means infinite changes.

Infinite changes is like the positive integers. As defined there cannot be the possibility of reciting the all positive integers, you cannot reach the end, the final integer. Just as there can be no possibility of infinite changes ending as past changes are ended at the present.

If there was infinite time in the past that is like saying there were infinite changes that finished occurring at every present moment.

It is claiming an infinity completed.

A contradiction.

Only the finite completes.

The problem is that you ignore what I say and just repeat your claims. There is no claim of infinity being completed because events within an infinite system come and go.

AT ANY PRESENT MOMENT ALL the events in the past have completed.

NO more events in the past will take place AT THAT PRESENT MOMENT.

No more events can take place in the past at a present moment.

They are ALL completed.

They could not have been infinite.

Time in the past could not have been infinite.

You appear to be objecting to claims that I have made. Please read what I actually said.

 

[Artemus]

Really.

To not understand a fraction is an operation, not a static entity, is a special kind of blindness.

1 divided by 9.

It can be shortened. But once shortened we shouldn't immediately forget what just happened.

 Rational number:

In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.[1]

[1] Rosen, Kenneth (2007). Discrete Mathematics and its Applications (6th ed.). New York, NY: McGraw-Hill. pp. 105, 158–160. ISBN 978-0-07-288008-3.

1/9 is just one way of writing a particular rational number. Nothing "just happened". No one has to sit down with pen and paper and evaluate it out by long-hand. It simply is the number. Recognizing facts like this is what has allowed science and technology to advance. The religion is demanding that facts be rejected because they don't match your intuition, no different than rejecting evolutionary theory because it seems too complicated or the heliocentric solar system because you can see the sun circle overhead.

 

[Juma]

Yes you can. Why on earth wouldnt you be able to do that?
Maybe you confuse this with the fact that you cannot, generally, rearrange the order of the terms in an infinite sum?


That is no counter argument, it is just the ramblings of unter.

Please explain why you say that ”an infinite number of terms can not be reached”? What does it even mean?
Nobody needs to write out all the numbers to know what the sum is.

Let s =9 + 9/10 + 9/100 + 9/1000 + ...
we know it is convergent (the terms form a geometric series with sum a/(1-r) when r<1.)
Here r= 1/10 and a=9 so sum = 9/(1-1/10) = 9/(10-1)/10 = 10. Remove 9 from S and we get the requested 1.

I will have to look up geometric series. I will not comment until I read up on it.

Humor me. What does 1.9999... and 0.888.. conveger to? If 0.999.. goes to when I'd assume 1.999.. goes to 2, amd 0.888.. goes to 0.9.


It can not be reached because that is the definition of infinity, or am I missing something?

x = 0.999..
10x = 9 + [ .9 + .99 + .999...]
10x = [ 9.9 + 9.99 + 9.999...]
x = [ .9 + .99 + .999...]

Show my error. You can not isolate the 9 from [9 + .9 + .99 + .999...] and manipulaste it algebraically.

Tell me why you think that you cannot ”isolate 9 from the sum”?