Guys 0.9999999999999999999999999 is not 1, because there is no number inbetween. That is the point of infinity, so we can put the numbers next to each other without having to think that there is an infinite amount of numbers in between. 0.999999999999 is not 1 because there IS a decimal showing the difference. But it is infinitely long. That is the point of infinity. There is a difference, but it is extremely small. Because 1/3 is NOT 0.33333333333333333333... You CAN NOT DEFINE 1/3 OF 1! Then do not multiply a non-existant number by 3 to get 3/3 of 1, because 1/3 of 1 times 3 = 3/3 of 1 = 1. But 0.3333333333333... is not and will never be 1/3.

OK thanks. Please let us know when you've finished having every math book out there rewritten to accomodate your proclamation.

The problem you're having is that you actually believe there to be "space" between 1 and 0.999~.
There is not. This is mathematically proven and has been demonstrated to you numerous times.

C'mon Cosmicpsore, you gotta read more carefully than that.

I did read.... That's why I pointed out that you are wrong.
....Read more carefully for yourself next time.

As I said, you actually believe there is a space between 0.999~ and 1. This space, even if you call it "infinitely small", does not actually exist. There is no space. It is nonexistent.
This is why you are confused, and why you're wrong that it is an inconsistency. You're the one making the inconsistency. It does not exist otherwise.

Math is internally consistent in every way. You really have absolutely no argument.
You could argue that math is inconsistent with reality if you'd like, but that is not what you are doing.

This is implying the existence of "infinitely small" values. These 'values', however, are typically ignored because they have no true value in mathematics, and no relationship to reality.
As you can see from above, we still reach the same conclusion as we would without it... Simply because it literally has no value.
This is why there is no "space" between 0.999~ and 1. There is literally no value between them. Therefore it is logically and internally consistent.

And unless you can prove the existence of an actual value for "infinitely small" numbers, then they will never have meaning within mathematics.

Why do you think repeating the same thing I've already read will make me change my perspective? I've already stated that I understand.

You said it was a "logical inconsistency".
If you understood what I've stated to you, then you would also understand that it is not and that you were wrong.

Since you have neither admitted to this error, or explained that you understand in a way that would be acceptable for me to see you have accepted this error.... then I can only assume you needed more explanation.

As well, I repeated nothing. Apparently you are still refusing to read people's posts.... again.

Just answer me two things.....
Do you still believe '0.999~ = 1' is a "logical inconsistency", yes or no?
If 'No' then: Did you actually read the explanation in my previous post, yes or no?

Since you have neither admitted to this error, or explained that you understand in a way that would be acceptable for me to see you have accepted this error.... then I can only assume you needed more explanation.

What error was there to admit to? I already explained that I understand it.

If 'No' then: Did you actually read the explanation in my previous post, yes or no?

I know you've read them, but have you comprehended any of my posts? Not just in this thread, but out of every post of mine you've ever read. We've been over this so many times: You presume too much or simply fail to understand what I'm saying.

I'm going to assume that means "No, but I'm too afraid to admit that I was wrong so I'm going to say 'yes' and qualify it so it makes it sound like I still know what I'm talking about".
....Because that's honestly what it sounds like to me.

For future reference, there is nothing wrong with being wrong sometimes. Admitting it is the mature thing to do. Everyone makes mistakes.

I know you've read them, but have you comprehended any of my posts?

You stated something which was incorrect I gave you valid reasoning why it was wrong. You then ignored my posts stating you "already understood".
There is not much there to "comprehend" about them except that you are just trying to get out of admitting a mistake.

Not just in this thread, but out of every post of mine you've ever read. We've been over this so many times: You presume too much or simply fail to understand what I'm saying.

I don't "presume too much". I don't make assumptions, unless those assumptions are necessary because of lack of information or context.

I've quoted what you have actually stated. I then gave an argument based on how that statement was incorrect.
You then gave absolutely no argument in return and therefore failed to support your statement.

There is nothing else to understand here. Don't try to make me out to look like 'a bad guy' or some kind of idiot just because I was just explaining that you were wrong about something. Just admit to your mistakes and move on. It's the proper thing to do.

I'm going to assume that means "No, but I'm too afraid to admit that I was wrong so I'm going to say 'yes' and qualify it".
Because that's honestly what it sounds like to me.

The fact that you took everything I said and turned it into that says to me that you have negative motives here, or you really did not understand what I was saying. I'm stating that I view it as illogical under specific circumstances.

For future reference, there is nothing wrong with being wrong sometimes. Everyone makes mistakes.

You clearly don't know me, because I have no issue admitting that I'm wrong or ignorance about something. The simple fact is that, in this case, I'm not wrong. To clarify, neither are you.

I've quoted what you have actually stated. I then gave an argument based on how that statement was incorrect.
You then gave absolutely no argument in return and therefore failed to support your statement.

Why would I give an argument for it? What would that argument even be?

There is nothing else to understand here. Don't try to make me out to look like 'a bad guy' or some kind of idiot just because I was just explaining that you were wrong about something. Just admit to your mistakes and move on. It's the proper thing to do.

You completely misunderstood what I said then have the audacity to not only assert that I was wrong, but also that I should admit to it because it's the right thing to do? Cosmicspore, you should take your own advice. But fear not, for I don't demand or even ask that you admit that you were the one who made a mistake. I'm much better than that.

I did read.... That's why I pointed out that you are wrong.
....Read more carefully for yourself next time.

As I said, you actually believe there is a space between 0.999~ and 1. This space, even if you call it "infinitely small", does not actually exist. There is no space. It is nonexistent.
This is why you are confused, and why you're wrong that it is an inconsistency. You're the one making the inconsistency. It does not exist otherwise.

Math is internally consistent in every way. You really have absolutely no argument.
You could argue that math is inconsistent with reality if you'd like, but that is not what you are doing.

This is what I can't quite grasp... 0.9999... repeating has infinite number of 9's, and with each 9, the number gets slightly closer to 1. So, the space between the numbers would grow infinitely small, but you can't say there is no space. Likewise, it's convienient in every way in math to simply call 0.999... 1, but it technically doesn't.

Also, is this proof valid?

Let A = 0.9999... repeating, and let's assume that A also = 1. Let B = 0.0000...1 \neq 0.

(A + B)= 1
1 - (A + B)= 1 - (1)
1 - (A + B)= 0
(1 - A) + B = 0
(1 - 0.99...) + B = 0
(1 - 1) + B = 0
B = 0 *contradiction*

This is what I can't quite grasp... 0.9999... repeating has infinite number of 9's, and with each 9, the number gets slightly closer to 1. So, the space between the numbers would grow infinitely small, but you can't say there is no space. Likewise, it's convienient in every way in math to simply call 0.999... 1, but it technically doesn't.

Also, is this proof valid?

Let A = 0.9999... repeating, and let's assume that A also = 1. Let B = 0.0000...1 \neq 0.

(A + B)= 1
1 - (A + B)= 1 - (1)
1 - (A + B)= 0
(1 - A) + B = 0
(1 - 0.99...) + B = 0
(1 - 1) + B = 0
B = 0 *contradiction*

No, your proof is not valid. You cannot have a 1 at the end of 0.0 repeating because by definition the zeros go one forever. Not only that, but by definition, .999~ is equal to 1. B is = to 0, as A+B =1, and A = .999~ There is no contradiction.

No, your proof is not valid. You cannot have a 1 at the end of 0.0 repeating because by definition the zeros go one forever. Not only that, but by definition, .999~ is equal to 1. B is = to 0, as A+B =1, and A = .999~ There is no contradiction.

I see. Well, thanks. By that logic, though, wouldn't it be impossible for 0.999... to be equal to 1, since eventually it would have to be rounded up, and that's impossible since the 9's go on forever?

I see. Well, thanks. By that logic, though, wouldn't it be impossible for 0.999... to be equal to 1, since eventually it would have to be rounded up, and that's impossible since the 9's go on forever?

I had an interesting proof for it, but alas, it's in my Algebra notebook which I didn't bring home from school.

Sorry, 0.0...1 is not a valid notation for an actual number. The ellipsis marks represent an infinte series. You cannot have something "after" an infinite series. You might as well have said "Let B = 0.01boston01" for all the sense that makes.

This is what I can't quite grasp... 0.9999... repeating has infinite number of 9's, and with each 9, the number gets slightly closer to 1. So, the space between the numbers would grow infinitely small, but you can't say there is no space.

You're thinking of it as a true 'series' of numbers, which it is not.
As a whole, it is a 'complete' number. The 'whole' of the infinity in this case is equal to 1.
This infinity only 'repeats' for purposes of writing it for decimal representation.

The problem is that you're imaging the number 'expanding' and 'approaching 1' but it never does so... There is no 'expansion' and therefore no 'space' between the two numeric representations. Taken as a whole, the entirety of the infinity expressed (0.9999~) is equal to 1 simply because it is.

The representation as an infinity is nothing more than an error in representation. It is a conceptual misrepresentation of the actual value.
"3/3" represents the 'whole' of the value in a better way and does not involve the same problem in representation.
3/3 of course being equal to 1.

The true problem relies in the fact that people were always taught that 1/3 = 0.33333~ and so you simply 'accept' the fact that these infinities exist, when they truly do not.

Take a piece of paper and cut it into 1/3.... Do you actually see an infinity of 0.333~ of that paper? Of course not. You see 1/3 of the paper.
The "0.3333~" decimal value is a conceptual misrepresentation used only within mathematics. It's real-life counterpart is only "1/3".

As I explained to Nerevar, if you care to 'piece it back together' using an "infinitely small" number (0.0000~1) which is equivalent to nothing (0), then you can find a secondary mathematical proof that 0.999~ = 1... But this requires two unnecessary assumptions:
1. That infinitely small numbers exist.
2. That a 0.000~1 equals 0.

These are two assumptions which can be ignored by mathematics, because there is proof that 0.99~ = 1 without them.

Likewise, it's convienient in every way in math to simply call 0.999... 1, but it technically doesn't.

No, No... That is incorrect. Technically, it truly DOES.
If you cut a paper into thirds, and then put them back together, you still have ONE full-size piece of paper.

The concept is incredibly simple. The difficulty is that people were trained and educated to understand the problem one way, and then this phenomena changes the way people must understand it.

By the way... I'm not trying to discourage anyone from trying to philosophically understand why an infinity equals 1...
If you want to think about it as something greater than simply a 'writing error' be my guest and feel free to conjecture. I've done it plenty of times before myself.
It's quite an interesting phenomena.

But it should be understood that this is a clear fact of mathematics. There should be no confusion about this.
"0.999~ = 1" in the same way "1 = One". They are merely two representations of the same number.

Feel free to wonder 'why'.... but just don't say it isn't true...

OK thanks. Please let us know when you've finished having every math book out there rewritten to accomodate your proclamation.

C'mon Cosmicpsore, you gotta read more carefully than that.

-The Power to Destroy-I did read.... That's why I pointed out that you are wrong.

....Read more carefully for yourself next time.

As I said, you actually believe there is a space between 0.999~ and 1. This space, even if you call it "infinitely small", does not actually exist. There is no space. It is nonexistent.

This is why you are confused, and why you're wrong that it is an inconsistency. You're the one making the inconsistency. It does not exist otherwise.

Math is internally consistent in every way. You really have absolutely no argument.

You could argue that math is inconsistent with reality if you'd like, but that is not what you are doing.

-The Power to Destroy-Well, then why are you arguing that it is inconsistent then? This is perfectly consistent.

If 0.0000~1 (an infinitely small number) did indeed exist, and was equal to 0 (nothing)...

Then this is true and consistent:

0.9999~ = 1

0.000~1 = 0

0.9999~ + 0.000~1 = 1

1 + 0 = 1

Therefore: 0.999~ + 0 = 1

Finally: 0.999~ = 1

This is implying the existence of "infinitely small" values. These 'values', however, are typically ignored because they have no true value in mathematics, and no relationship to reality.

As you can see from above, we still reach the same conclusion as we would without it... Simply because it literally has no value.

This is why there is no "space" between 0.999~ and 1. There is literally no value between them. Therefore it is logically and internally consistent.

And unless you can prove the existence of an actual value for "infinitely small" numbers, then they will never have meaning within mathematics.

-The Power to Destroy-You said it was a "logical inconsistency".

If you understood what I've stated to you, then you would also understand that it is not and that you were wrong.

Since you have neither admitted to this error, or explained that you understand in a way that would be acceptable for me to see you have accepted this error.... then I can only assume you needed more explanation.

As well, I repeated nothing. Apparently you are still refusing to read people's posts.... again.

Just answer me two things.....

Do you still believe '0.999~ = 1' is a "logical inconsistency", yes or no?

If 'No' then: Did you actually read the explanation in my previous post, yes or no?

It is because of the way I am imagining it. If you graph it, 0.99999 never actually reaches 1.

Or you just didn't understand what I was saying.

What error was there to admit to? I already explained that I understand it.

You're repeating the same augmentative consensus, that 1=0.9999~. And, actually, I have seen that specific argument used before.

In a certain way, yes.

I know you've read them, but have you comprehended any of my posts? Not just in this thread, but out of every post of mine you've ever read. We've been over this so many times: You presume too much or simply fail to understand what I'm saying.

-The Power to Destroy-I'm going to assume that means "No, but I'm too afraid to admit that I was wrong so I'm going to say 'yes' and qualify it so it makes it sound like I still know what I'm talking about".

....Because that's honestly what it sounds like to me.

For future reference, there is nothing wrong with being wrong sometimes. Admitting it is the mature thing to do. Everyone makes mistakes.

You stated something which was incorrect I gave you valid reasoning why it was wrong. You then ignored my posts stating you "already understood".

There is not much there to "comprehend" about them except that you are just trying to get out of admitting a mistake.

I don't "presume too much". I don't make assumptions, unless those assumptions are necessary because of lack of information or context.

I've quoted what you have actually stated. I then gave an argument based on how that statement was incorrect.

You then gave absolutely no argument in return and therefore failed to support your statement.

There is nothing else to understand here. Don't try to make me out to look like 'a bad guy' or some kind of idiot just because I was just explaining that you were wrong about something. Just admit to your mistakes and move on. It's the proper thing to do.

Nothing else to see here folks, move along.

The fact that you took everything I said and turned it into that says to me that you have negative motives here, or you really did not understand what I was saying. I'm stating that

Iview it as illogical under specific circumstances.You clearly don't know me, because I have no issue admitting that I'm wrong or ignorance about something. The simple fact is that, in this case, I'm not wrong. To clarify, neither are you.

That never happened. It's one thing to misunderstand, but you just fabricated that. Don't do that again, please.

This is a bold thing to say for someone who doesn't assume (and thus presume). It's reasons like this why it's hard to take you seriously.

Why would I give an argument for it? What would that argument even be?

You completely misunderstood what I said then have the audacity to not only assert that I was wrong, but also that I should admit to it because it's the right thing to do? Cosmicspore, you should take your own advice. But fear not, for I don't demand or even ask that you admit that you were the one who made a mistake. I'm much better than that.

I agree. There never was anything to see -- well, maybe a showman with a broken arm.

-The Power to Destroy-This is what I can't quite grasp... 0.9999... repeating has infinite number of 9's, and with each 9, the number gets slightly closer to 1. So, the space between the numbers would grow infinitely small, but you can't say there is

nospace. Likewise, it's convienient in every way in math to simply call 0.999... 1, but it technically doesn't.Also, is this proof valid?

Let A = 0.9999... repeating, and let's assume that A also = 1. Let B = 0.0000...1 \neq 0.

(A + B)= 1

1 - (A + B)= 1 - (1)

1 - (A + B)= 0

(1 - A) + B = 0

(1 - 0.99...) + B = 0

(1 - 1) + B = 0

B = 0 *contradiction*

No, your proof is not valid. You cannot have a 1 at the end of 0.0 repeating because by definition the zeros go one forever. Not only that, but by definition, .999~ is equal to 1. B is = to 0, as A+B =1, and A = .999~ There is no contradiction.

I see. Well, thanks. By that logic, though, wouldn't it be impossible for 0.999... to be equal to 1, since eventually it would have to be rounded up, and that's impossible since the 9's go on forever?

I had an interesting proof for it, but alas, it's in my Algebra notebook which I didn't bring home from school.

I'll get it tomorrow, I promise!

Considering the fact that all repeating decimals are rational numbers, and that all rational numbers are real numbers, then 0.9... is a real number.

Sorry, 0.0...1 is not a valid notation for an actual number. The ellipsis marks represent an infinte series. You cannot have something "after" an infinite series. You might as well have said "Let B = 0.01boston01" for all the sense that makes.

You're thinking of it as a true 'series' of numbers, which it is not.

As a whole, it is a 'complete' number. The 'whole' of the infinity in this case is equal to 1.

This infinity only 'repeats' for purposes of writing it for decimal representation.

The problem is that you're imaging the number 'expanding' and 'approaching 1' but it never does so... There is no 'expansion' and therefore no 'space' between the two numeric representations. Taken as a whole, the entirety of the infinity expressed (0.9999~) is equal to 1 simply because it is.

The representation as an infinity is nothing more than an error in representation. It is a conceptual misrepresentation of the actual value.

"3/3" represents the 'whole' of the value in a better way and does not involve the same problem in representation.

3/3 of course being equal to 1.

The true problem relies in the fact that people were always taught that 1/3 = 0.33333~ and so you simply 'accept' the fact that these infinities exist, when they truly do not.

Take a piece of paper and cut it into 1/3.... Do you actually see an infinity of 0.333~ of that paper? Of course not. You see 1/3 of the paper.

The "0.3333~" decimal value is a conceptual misrepresentation used only within mathematics. It's real-life counterpart is only "1/3".

As I explained to Nerevar, if you care to 'piece it back together' using an "infinitely small" number (0.0000~1) which is equivalent to nothing (0), then you can find a secondary mathematical proof that 0.999~ = 1... But this requires two unnecessary assumptions:

1. That infinitely small numbers exist.

2. That a 0.000~1 equals 0.

These are two assumptions which can be ignored by mathematics, because there is proof that 0.99~ = 1 without them.

No, No... That is incorrect. Technically, it truly DOES.

If you cut a paper into thirds, and then put them back together, you still have ONE full-size piece of paper.

The concept is incredibly simple. The difficulty is that people were trained and educated to understand the problem one way, and then this phenomena changes the way people must understand it.

But... 0_o

How this?

1 = 0.(9)

It's weird... (Logically)

But it's possible.

Typo. Forums doesn't take every charsets.

If you want to think about it as something greater than simply a 'writing error' be my guest and feel free to conjecture. I've done it plenty of times before myself.

It's quite an interesting phenomena.

But it should be understood that this is a clear fact of mathematics. There should be no confusion about this.

"0.999~ = 1" in the same way "1 = One". They are merely two representations of the same number.

Feel free to wonder 'why'.... but just don't say it isn't true...

a= 0.(9)10

a= 9.(9)10

a-a= 9.(9) - 0.(9)9

a= 9a= 10.(9) = 1

Still weird...

Damn you, that was the post I was going to post! CURSES!

Oh well, no more posting mathematical proofs for me