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[Korey]

And since .333... is a static number, that is how I understand .999... to be a static number. If it is static how can it exist as another number!

It doesn't exist as another number.

They are the same number. That's the point.

This will blow your mind:

1/9 = .111...
2/9 = .222...
3/9 = .333...
4/9 = .444...
5/9 = .555...
6/9 = .666...
7/9 = .777...
8/9 = .888...
9/9 = .999...

Also: 1.999... = 2

 

[Tntnnbltn]

I just don't see how or why. That implies there is no such number/distinction as .999999

0.999999 is a different number to 1. Infact, any number like that with a finite number of 9s is different to 1. It's only when you have an infinite number of 9s that it equals 1.


Same applies for 2.999... = 3..., 1.428999... = 1.429..., 5.642858396843723999... = 5.642858396843724

 

[DownLikeBCPowder]

Why? .999999 is a number, and it's different from .999....

Yes, yes, I meant .999... and you know very well that is what I meant, as it has been the subject the entire time thanks.

My point being is that I do not understand what gives .999... the ability to morph to 1. Where does that stop? If it is 1, why is there an infinite string of 9s? Why ever publish that number if it is not literally what is implied.

 

[DownLikeBCPowder]

0.999999 is a different number to 1. Infact, any number like that with a finite number of 9s is different to 1. It's only when you have an infinite number of 9s that it equals 1.


Same applies for 2.999... = 3..., 1.428999... = 1.429..., 5.642858396843723999... = 5.642858396843724

Isn't this rounding? Sure, for all intents and purposes, 1.345399999 is 1.3454, but technically it is not the same. Which is why I do not understand why .999... is 1. Sure, for all intents and purposes it may as well equal 1, but it does not appear to be technically 1, where I find problem

I just don't see how it doesn't bleed. If .999... infinite is essentially 1, why even bother with 9s? Why even bother designating it as 9 rather than 1!

 

[Korey]

Yes, yes, I meant .999... and you know very well that is what I meant, as it has been the subject the entire time thanks.

My point being is that I do not understand what gives .999... the ability to morph to 1. Where does that stop? If it is 1, why is there an infinite string of 9s? Why ever publish that number if it is not literally what is implied.

Huh? Why not?

.333... * 3 = .999...
1/3 * 3 = 1

These are the same equation. Except in one of them "1" is expressed as .999...

 

[zoku88]

Yes, yes, I meant .999... and you know very well that is what I meant, as it has been the subject the entire time thanks.

My point being is that I do not understand what gives .999... the ability to morph to 1. Where does that stop? If it is 1, why is there an infinite string of 9s? Why ever publish that number if it is not literally what is implied.

It doesn't morph. It's just a feature of base 10 that that happens. It happens with other bases too (just not with repeated 9's).

Not sure about your stopping question.

Just a different representation.

Why publish other equivalent rationals, like 3/3 4/4?\

Isn't this rounding? Sure, for all intents and purposes, 1.345399999 is 1.3454, but technically it is not the same. Which is why I do not understand why .999... is 1. Sure, for all intents and purposes it may as well equal 1, but it does not appear to be technically 1, where I find problem

I just don't see how it doesn't bleed. If .999... infinite is essentially 1, why even bother with 9s? Why even bother designating it as 9 rather than 1!

No, it's not rounding. It would be rounding if any of the original numbers were terminating.

I'm not sure what fact you aren't accepting. Do you think convergence is an approximation or something?

 

[KHarvey16]

Yes, yes, I meant .999... and you know very well that is what I meant, as it has been the subject the entire time thanks.

My point being is that I do not understand what gives .999... the ability to morph to 1. Where does that stop? If it is 1, why is there an infinite string of 9s? Why ever publish that number if it is not literally what is implied.

No I actually didn't know you meant that. A lot of things are being thrown around.

I don't know if you saw this before but the answer might help. What is the numerical difference between .999... and 1 in your mind? If you say it's just infinitely small then you understand why it's 1, since infinitely small can only mean 0.

 

[Tntnnbltn]

Why ever publish that number if it is not literally what is implied.

We usually write it as "1", because more people understand it that way, it's easier to do calculations when it's written that way, it's shorter to write, etc.

I mean, we could go writing "1" as "582/582". They are the same number. It's just easier when we write it as "1".

Isn't this rounding? Sure, for all intents and purposes, 1.345399999 is 1.3454, but technically it is not the same.

Technically it is the same.

 

[Korey]

Isn't this rounding? Sure, for all intents and purposes, 1.345399999 is 1.3454, but technically it is not the same. Which is why I do not understand why .999... is 1. Sure, for all intents and purposes it may as well equal 1, but it does not appear to be technically 1, where I find problem

I just don't see how it doesn't bleed. If .999... infinite is essentially 1, why even bother with 9s? Why even bother designating it as 9 rather than 1!

Please read this wikipedia page: http://en.wikipedia.org/wiki/0.999...#Skepticism_in_education

You fall under their very first bullet point in "Skepticism in Education" section:

Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of infinitesimals. There are many common contributing factors to the confusion:

* Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a paradox, which is amplified by the appearance of the seemingly well-understood number 1.

Your confusion simply originates from your assumptions and misguided preconceptions of how numbers "should be".

 

[DownLikeBCPowder]

No I actually didn't know you meant that. A lot of things are being thrown around.

I don't know if you saw this before but the answer might help. What is the numerical difference between .999... and 1 in your mind? If you say it's just infinitely small then you understand why it's 1, since infinitely small can only mean 0.

How does infinitely small mean zero to you? It is not zero, it is just infinitely small. Considering that, the difference between .999... and 1 is difficult to express, but is clearly visible in the visual representation of .999... and 1. Clearly there is a difference, though it is obviously minute. ESSENTIALLY, it may be close to equaling 1. If I have slept %.999... of the day, I have slept nearly the entire day, almost an immeasurably close amount to being the full day, however it is not the full day. I just can't see how it is exactly the same number when there is a difference, even if so "small" of one.

We usually write it as "1", because more people understand it that way, it's easier to do calculations when it's written that way, it's shorter to write, etc.

I mean, we could go writing "1" as "582/582". They are the same number. It's just easier when we write it as "1".

Well, in the case of 582/582, it is literally 1.

Technically it is the same.

I just don't see how. They are completely different answers.

 

[zoku88]

How does infinitely small mean zero to you? It is not zero, it is just infinitely small.

This is true. They are different. Except, when talking about real numbers (which we are right now,) the only infinitely small number is 0.

Archimedean property

 

[DownLikeBCPowder]

Please read this wikipedia page: http://en.wikipedia.org/wiki/0.999...#Skepticism_in_education

You fall under their very first bullet point in "Skepticism in Education" section:


Your confusion simply originates from your assumptions and misguided preconceptions of how numbers "should be".

I actually read the entire thing before I even posted here (the entire segment). I still find myself in complete disagreement.

 

[KHarvey16]

How does infinitely small mean zero to you? It is not zero, it is just infinitely small. Considering that, the difference between .999... and 1 is difficult to express, but is clearly visible in the visual representation of .999... and 1. Clearly there is a difference, though it is obviously minute. ESSENTIALLY, it may be close to equaling 1. If I have slept %.999... of the day, I have slept nearly the entire day, almost an immeasurably close amount to being the full day, however it is not the full day. I just can't see how it is exactly the same number when there is a difference, even if so "small" of one.

If there is a small difference how can .999...% really be infinity? A difference means it mist stop, but infinity means it doesn't so there is no difference.

 

[DownLikeBCPowder]

If there is a small difference how can .999...% really be infinity? A difference means it mist stop, but infinity means it doesn't so there is no difference.

So what is .888..???

 

[Tntnnbltn]

If I have slept %.999... of the day, I have slept nearly the entire day, almost an immeasurably close amount to being the full day, however it is not the full day.

(Actually you've slept 1% of the day, not the full day)

Well, in the case of 582/582, it is literally 1.

And "0.999..." is literally 1. Which was the point I was making about how we don't write it like that. Because "1" is simpler.

 

[zoku88]

So what is .888..???

.8888..... or 8/9, I guess

If you're talking about base 10. There is no finite number that has an infinitely small difference (0) from it.

 

[DownLikeBCPowder]

(Actually you've slept 1% of the day, not the full day)

I see my typo at this point, but you understand what I'm implying. I have not quite slept 1% of the day, even if it is so close as to essentially BE 1%, it is still not 1%, as there is an infinitely small fraction of that 1% which I have not slept! Ugh.

And "0.999..." is literally 1. Which was the point I was making about how we don't write it like that. Because "1" is simpler.

Again, unless .999... grows at some point, there is no reason to believe it will ever become 1, as it is .999...

I hate mathematics and I am going to bed.

.8888..... or 8/9, I guess

If you're talking about base 10. There is no finite number that has an infinitely small difference (0) from it.

I don't understand how .8888 doesn't just experience this magical growth at some point and become .888...9... which somehow grows again until it's 1.

I just don't follow where the additional growth comes in .999..

 

[Tntnnbltn]

So what is .888..???

0.888... is just 0.888... .

Unless you were using Base 9, I think. Then 0.888... = 1?

Again, unless .999... grows at some point, there is no reason to believe it will ever become 1, as it is .999...

Grows?

 

[zoku88]

Again, unless .999... grows at some point, there is no reason to believe it will ever become 1, as it is .999...

I hate mathematics and I am going to bed.

You admitted that the difference was infinitely small (which is 0 in real numbers.) Why do you need more convincing?

There is no growing.

 

[DownLikeBCPowder]

0.888... is just 0.888... .

Unless you were using Base 9, I think. Then 0.888... = 1?


Grows?

Yes, to me this just looks like .999.. is experiencing a "growth" an "addition" an "extra" at some point to be come 1. I just don't understand (because I am retarded) how this number increases to become another number.

You admitted that the difference was infinitely small (which is 0 in real numbers.) Why do you need more convincing?

There is no growing.

So now "infinitely small" is zero? At this rate, why even have numbers. It's clear I don't understand math.

 

[zoku88]

Yes, to me this just looks like .999.. is experiencing a "growth" an "addition" an "extra" at some point to be come 1. I just don't understand (because I am retarded) how this number increases to become another number.

I'm confused. Did you reject the difference between .9999* and 1 is infinitely small?

So now "infinitely small" is zero? At this rate, why even have numbers. It's clear I don't understand math.

Property of real numbers.

http://en.wikipedia.org/wiki/Archimedean_property

Real numbers if the first example.

 

[itschris]

So now "infinitely small" is zero? At this rate, why even have numbers. It's clear I don't understand math.

Correct. The real number system does not contain infinitesimals. It's called the Archimedean property.

The field of the real numbers is Archimedean both as an ordered field and as a normed field. It is chiefly because the real numbers are obtained as the completion of the rational numbers, which themselves satisfy the axiom in both sense, with respect to an absolute value structure compatible with the ordering.

In the axiomatic theory of real numbers, the non-existence of nonzero infinitesimal real numbers is implied by the least upper bound property as follows. Denote by Z the set consisting of all positive infinitesimals. This set is bounded above by 1. Now assume by contradiction that Z is nonempty. Then it has a least upper bound c, which is also positive, so c/2 < c < 2c. Since c is an upper bound of Z and 2c is strictly larger than c, 2c must be strictly larger than every positive infinitesimal. In particular, 2c cannot itself be an infinitesimal, for then 2c would have to be greater than itself. Moreover since c is the least upper bound of Z, c/2 must be infinitesimal. But 2c and c/2 cannot have different types by the above result, so there is a contradiction. The conclusion follows that Z is empty after all: there are no positive, infinitesimal real numbers.

One should note that the Archimedean property of real numbers holds also in constructive analysis, even though the least upper bound property may fail in that context.

 

[DownLikeBCPowder]

Correct. The real number system does not contain infinitesimals. It's called the Archimedean property.

Let me just clarify something.

If I write on a sheet of paper .9999 with 9s all the way to pluto, but only to pluto, it is still not 1. However, if it is infinite in nature, that is when it "becomes" or "exists as" 1?

If that's the case, why doesn't .88888... increase to .888888888888888888888888..9 at some point?

or .111... ? etc

 

[zoku88]

Let me just clarify something.

If I write on a sheet of paper .9999 with 9s all the way to pluto, but only to pluto, it is still not 1. However, if it is infinite in nature, that is when it "becomes" or "exists as" 1?

Yea. I suppose you can say it that way. ("exists" as would be better than "becomes")

 

[KHarvey16]

Let me just clarify something.

If I write on a sheet of paper .9999 with 9s all the way to pluto, but only to pluto, it is still not 1. However, if it is infinite in nature, that is when it "becomes" or "exists as" 1?

Yes. .9999 with 9's all the way to Pluto isn't infinite, so it isn't 1.

 

[Korey]

I actually read the entire thing before I even posted here (the entire segment). I still find myself in complete disagreement.

Ok I'm trying to help you wrap your head around it.

First of all, accept the fact that .999... = 1. It is a widely accepted fact among all mathematicians and people who teach math. In other words, everybody...except most people who encounter this for the first time because it goes against what they think is intuitive since .999... is such a special case. This is a good first step so that you can begin trying to prove to yourself that this is indeed the case, which may make it easier to accept.

---

Think of it this way: between two distinct real numbers, there are an infinite amount of other real numbers.

Between 1.492 and 1.493 is 1.4925, etc.

There are no numbers inbetween .999... and 1, not even one. Therefore they are not distinct numbers. Therefore they are the same number.

And there is nothing "special" about .999... = 1. It happens all over the place, 2.42999... = 2.43, 4.999... = 5, etc.

---

And again, stop thinking of .999... as a tourist who's going somewhere. It's not trying to reach anything. It's a completely static number. It's not "essentially" trying to be 1. It's not trying to be anything. It's hard for the human mind to understand the concept of infinity.

 

[SmokeMaxX]

I don't understand how .8888 doesn't just experience this magical growth at some point and become .888...9... which somehow grows again until it's 1.

I just don't follow where the additional growth comes in .999..

Intelligent design.

Spoiler

TBH though, you're clinging to the notion that .999 repeating needs something added to it to reach 1. It doesn't. You cannot add anything to it to ever make it one. If there is no difference between two numbers, those numbers are the same.

 

[DownLikeBCPowder]

And again, stop thinking of .999... as a tourist who's going somewhere. It's not trying to reach anything. It's a completely static number. It's not "essentially" trying to be 1. It's not trying to be anything. It's hard for the human mind to understand the concept of infinity.

That's really the reason I don't get it, because it's not "going anywhere".

I just don't see how infinite 9 equal auto-up, how infinite 8 doesn't equal auto-up and so forth.

 

[zoku88]

If that's the case, why doesn't .88888... increase to .888888888888888888888888..9 at some point?

or .111... ? etc

Well, one problem is that .888....9 doesn't really make sense. (only you're saying that there are not an infinite number of 8's or something.)

For your question.

Make some number that is what you describe. .88....9 (wtv that is supposed to me.)

Then make the sequence that builds .88888* and subtact that sequence from .8...9.

You'll notice that the difference has a nonzero lower bound. (it is decreasing, just not to zero.)

 

[Korey]

Let me just clarify something.

If I write on a sheet of paper .9999 with 9s all the way to pluto, but only to pluto, it is still not 1. However, if it is infinite in nature, that is when it "becomes" or "exists as" 1?

If that's the case, why doesn't .88888... increase to .888888888888888888888888..9 at some point?

or .111... ? etc

There is no such thing as ".xxx...y"

 

[DownLikeBCPowder]

Well, one problem is that .888....9 doesn't really make sense. (only you're saying that there are not an infinite number of 8's or something.)

For your question.

Make some number that is what you describe. .88....9 (wtv that is supposed to me.)

Then make the sequence that builds .88888* and subtact that sequence from .8...9.

You'll notice that the difference has a nonzero lower bound. (it is decreasing, just not to zero.)

I am saying that, unless I am misunderstanding, I do not see a point "of measure" as you fellows would put it, for .8888... to .9. (or for a lesser example, .88....9). If infinite 9s equals 1 because there is no viable point of measure due to the nature of infinity, I do not see how that same rule does not apply for the other numerals when given in terms of infinity..

 

[Gruco]

Thread needs more Aristotle and Tortoise.

 

[zoku88]

I am saying that, unless I am misunderstanding, I do not see a point "of measure" as you fellows would put it, for .8888... to .9. (or for a lesser example, .88....9). If infinite 9s equals 1 because there is no viable point of measure due to the nature of infinity, I do not see how that same rule does not apply for the other numerals when given in terms of infinity..

I have no idea what you're trying to say. Are you saying that there's no point in expressing .1 as .999*?

If you don't see a point in that, that's ok. You just have to understand that it isn't wrong to express .999* as 1, since they are the number.

If you're confused about the .888*. convince yourself with this.


Take a calculator.

Type in .889 or something.

Subtact from it: .888, .8888, .88888, . 888888 And keep on doing that until you can tell me what the difference between .889 and .888* is.

Then do the same for .8889 and .88889.

Now, do this for 1 and approximations of .999*

1-.999*, 1-.9999* and so on and so on. You'll notice that at no point does the difference grow.


There is no "rule" being applied here. It's just that there don't exist numbers between the two, so they must be equal.

 

[DownLikeBCPowder]

Alas, as much as I want to delve into this mystery and understand why the other numbers do not seem to have the same properties as 9, it must wait until tomorrow.

Thanks for the communication and I'll be back to further attempt to align my misunderstanding with the way of the 9.

 

[peakish]

If that's the case, why doesn't .88888... increase to .888888888888888888888888..9 at some point?

or .111... ? etc

0.999... doesn't increase to 1 at some point (which is finite), just as 0.888... will never increase to 0.888...9 at "some point," it'll continue infinitely.

0.888... -> 8/9 just as 0.999... -> 9/9 = 1


If you can agree on the point that two numbers are the same if there is no difference between them, 0.999... = 1 follows from the fact that you can't create a difference between them. Never, ever. As soon (at every point) as you stop the series to create your 0.000...1 difference the 0.999...9 series continues.

Or, as another poster put it in the visualization of an apple divided into infinitely many pieces, every piece will be volume-less so if you remove one (which would be the difference) you haven't removed any volume at all. If every piece had some volume you could always tell exactly how many pieces there are, which contradicts the fact that you started with an infinite number of pieces.

 

[Tntnnbltn]

I just don't see how infinite 9 equal auto-up, how infinite 8 doesn't equal auto-up and so forth.

It's a special property of 'Base (whatever)'.

Base 10 (what we use) contains the following numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. After "9" you move into the tens column and start again. In the Base 10 number system, any infinite recurring trail of 9s is equal to the number which looks 'above it' (i.e. 0.99... = 1, 2.499... = 2.5)


Base 4 contains the following numbers: 0, 1, 2, 3. After "3" you move into what we would call the tens column. So the first ten numbers in Base 4 are: 1, 2, 3, 10 [4], 11 [5], 12 [6], 13 [7], 20 [8], 21 [9], 22 [10] (base 10 equivalent in square brackets). For the Base 4 system, any recurring trail of 9s is equal to the number which looks 'above it'. (i.e. 0.33... = 1, 13.333... = 20)


Hexadecimal is a Base 16 system used in computing containing the following numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. I won't go through the details of how hex works (it's the same principle as the others), but in hex a number of 0.FFF... = 1, 2.5FFF... = 2.6, 9.FFF = A.


Basically, this rule applies whenever you have a recurring trail of the highest possible unit in any base system.

 

[Ashes]

There is no debate here. No mathematician in the world disagrees. Maths is not about opinions. This is mathemetically proven.
1. 1 and 2 are different numbers.
2. .9 and 1 are different numbers.
3..99999 recurring is the same as 1.
4. you imply that there is 0.0000.....1 difference.
5. But that series of zero is infinite long, so that *1* at the end never comes.
6. and logic dictates that it can never come.
6.And since that difference is always zero: there is no difference between the two numbers. It is simple.

Which of these sentences do you disagree with? please be specific.

 

[Son of Godzilla]

There is no debate here. No mathematician in the world disagrees. Maths is not about opinions. This is mathemetically proven.
1. 1 and 2 are different numbers.
2. .9 and 1 are different numbers.
3..99999 recurring is the same as 1.
4. you imply that there is 0.0000.....1 difference.
5. But that series of zero is infinite long, so that *1* at the end never comes.
6. and logic dictates that it can never come.
6.And since that difference is always zero: there is no difference between the two numbers. It is simple.

Which of these sentences do you disagree with? please be specific.

0.000...1 is the infinitely small number directly above 0. It is not equal to zero and it is not measurable, but it is the difference between 1 and .999_

 

[Tntnnbltn]

0.000...1 is the infinitely small number directly above 0. It is not equal to zero and it is not measurable, but it is the difference between 1 and .999_

You are talking about infinitesimals, and they don't exist in our number system.

 

[peakish]

0.000...1 is the infinitely small number directly above 0. It is not equal to zero and it is not measurable, but it is the difference between 1 and .999_

Hey, my 0.000...01 is smaller than your number there.

 

[Korey]

0.000...1 is the infinitely small number directly above 0. It is not equal to zero and it is not measurable, but it is the difference between 1 and .999_

Again, there's no such thing as .xyz...y

 

[Ashes]

at godzilla: what do you make of point 5?

 

[zoku88]

0.000...1 is the infinitely small number directly above 0. It is not equal to zero and it is not measurable, but it is the difference between 1 and .999_

Not it isn't. That's a finite small number. You just terminated it right there. Unless you're saying that there's an infinite number of 0's between the the 0's you wrote and 1 (which doesn't make sense.) And of course, since that number doesn't make sense, all like numbers between .999* and 1 don't exist, and thus, the difference must be nothing: 0.

 

[Xapati]

0.000...1 is the infinitely small number directly above 0. It is not equal to zero and it is not measurable, but it is the difference between 1 and .999_

If something is infinitely small, it's zero, which is what you're struggling to accept. Otherwise it's a finite number and I can always find one smaller. There is no infinity-1 which is somehow infinity yet not.

I just don't see how infinite 9 equal auto-up, how infinite 8 doesn't equal auto-up and so forth.

What would 0.8... "auto-up" to? Doesn't matter anyway, because whatever number you decide on, I can always find a number that's in between. 0.8... and whatever you pick. That's the thing, there is no number between 0.99... and 1 because the difference is infinitely small.

Edit: And if you don't understand any of this infinitely small stuff then argue the proof I posted a while back, which uses a neat trick to get rid of the repeating 9's:

Once you accept that 0.99... = 1, it becomes a lot easier to understand why.

 

[Ashes]

I'll try metaphor now. ^ He (those who are finding this difficult) has made what he thinks is a logical jump. Not realizing that lthat logical jump requires him never to land *and* land at the same time. And somewhere there he is satisfied that yes, somehow, this reality makes logical sense and is real.


4. you imply that there is 0.0000.....1 difference.
5. But that series of zeros is infinite long, so that *1* at the end *never* comes.
6. and logic dictates that it can *never* come. (It really can't)

(Maybe you are mixing tenses up or something. Think of it this way: you are talking about a future that you think is possible but can never come. This is illogical. )

and how about a visual explanation:

Time: 11 am gmt>>>>time:billion years from 11gmt>>>>>>>>>>>>>>>>> trilllion years>>>>>>>>>>>same rule>>>>
0.99999999>>>>>>>>This never stops>>>>>This never stopped>>>>>>>>nope still 999999999>>>>>>>>>>>>>>>
0.000000000>>>>>>>This never stops>>>>>Where and when do I put the 1?>> here?>>>>> Where in this space/time continium do I put the 1?

Spoiler

did this help?

 

[Son of Godzilla]

Edit: And if you don't understand any of this infinitely small stuff then argue the proof I posted a while back, which uses a neat trick to get rid of the repeating 9's:

Once you accept that 0.99... = 1, it becomes a lot easier to understand why.

That doesn't get rid of the repeating 9's because the operation on the infinite repetition never ends and you can't continue.

Again, there's no such thing as .xyz...y

Then why is it represented so easily and understood by anyone?

Not realizing that lthat logical jump requires him never to land *and* land at the same time. And somewhere there he is satisfied that yes, somehow, this reality makes logical sense and is real.

You can still jump without having anywhere to land.

 

[Evlar]

That doesn't get rid of the repeating 9's because the operation on the infinite repetition never ends and you can't continue.

You appear to be claiming that 0.9999... has the strange quality of being unusable in subtraction. I wonder what leads you to this remarkable conclusion.

If 0.9999... is a real number (and it is) then it MUST be possible to perform subtraction on it.

 

[Inflammable Slinky]

That doesn't get rid of the repeating 9's because the operation on the infinite repetition never ends and you can't continue.


Then why is it represented so easily and understood by anyone?


You can still jump without having anywhere to land.

There is no operation of infinite repetition. what you described does not exist. with the real number .999... (with the ... representing the infinite amount of 9s to follow), nothing is happening because the number is a static.

Do you understand what i mean by static? 2 is static. .5 is static. 1/3 (otherwise written as .333...) is static. Pi is static (although it is not a member of the rational set but that's something different). You have to realize, there is no operation, no addition of 9s happening to .999...

Again, if you have difficulty understanding this, what number that is an element of the Real Set, exists between .999... and 1?

Keep in mind that .000...1, which in this case means an infinite series of zeros terminating in 1 does not make logical sense in the Real Set, and therefore is not an element of the Real Set.

(In fact, you can see the logical contradiction of .000...1 just by thinking about it, since an infinite series of numbers, by definition, cannot terminate, yet you say that it does terminate in 1. Its like saying there exists a square triangle in Euclidean geometry)

 

[Son of Godzilla]

(In fact, you can see the logical contradiction of .000...1 just by thinking about it, since an infinite series of numbers, by definition, cannot terminate, yet you say that it does terminate in 1. Its like saying there exists a square triangle in Euclidean geometry)

It doesn't terminate, but if it did it would do so in a 1 not a 0. It's more like drawing a triangle that has infinite long sides. It's still a triangle.

Again, if you have difficulty understanding this, what number that is an element of the Real Set, exists between .999... and 1?

What, you expect me to type it out?

 

[DownLikeBCPowder]

(In fact, you can see the logical contradiction of .000...1 just by thinking about it, since an infinite series of numbers, by definition, cannot terminate, yet you say that it does terminate in 1. Its like saying there exists a square triangle in Euclidean geometry)

And this is what I don't get, I don't see the logical contradiction. In fact, thinking about it makes it worse.

Since an infinite series of numbers cannot terminate, as you imply, .000... never manages to tag that .00...1 on there, but on the same notion, wouldn't .999.. never get the digit, either? Thus it is not 1?