[split] 0.999... equals 1

[Tiberius]

1. You cannot get more accurate, because you could not divide 1 evenly into three. 1/3=.3^ is an approximation based on that we cannot get any closer without overshooting one, but neither can we reach one.

If we can't get closer without overshooting, we are at the number itself. This again is the definition of different numbers.

2. 3 X 3 does = 9, and .9^ still doesn't equal one. I do not see why I would have to doubt this.

1/3 = 0.3r

What you don't understand is that 0.3r * 3 equals 0.9r, so I must assume that you do not believe 3 * 3 = 9.

3. If that is not mathematically possible, then infinity was not possible to begin with Infinity + 1 is still infinity... it is just an added one on top of the initial concept.

Infinity is a concept, one that you don't seem to understand. Infinity is not a finite number; it is limitless, it has no end. You cannot add 1 to infinity since it has no limit. 0.9r is an infinitely long number; it has no end. It is a limitless string of 9s. There is nothing you can add to it to make it equal 1 (there is no gap for anything to be put in), therefore it is already equal to 1 (as I have shown).

I do not see how you reckon that .9^ is any more valid than the .0^1 long gap .9^ requires to equal one. There are an infinite amount of zeros between that decimal and that one... just as there is an infinite amount of nines following the decimal.

An infinite amount cannot have an end. It is limitless. The concept of "an infinite amount of zeroes" followed by a 1 means there is an end to infinity, which is a contradiction.

Establishment does not make something 'right', and mathematical establishment has been proved wrong before. Some of us understand the concept of endless... and justly realize that endless .9^ by definition means that it never becomes one. If .9^ were equal to one... then .0^1 would be equal to zero. Those numbers are separated from both one and zero by a mere .0^1... but that does not mean that the .0^1 is not there.

Again, .0^1 is a logically and mathematically impossible number. You cannot have an infinite string of zeros with a 1 on the end. Infinity has no limits, ergo there is no end for the 1 to be placed on.

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They aren't the same number because there's always a gap to infinity. They cannot meet.

Evidently you don't understand the concept of infinity if you think there are gaps. Infinity is limitless.

Right - 'amount'. By their logic - I could give you 10,000 x 99p in exchange for your £10,000 and be £100 in profit. Wanna do some money exchanges?

99p is not an infinitely long number. Your example is complete nonsense, and not related to mine in any way since you do not use infinities.

Decimal numbers can't always be accurate. This proves it.

Decimal notation can write any kind of number. That is their purpose. All this proves is that you don't understand mathematical concepts like infinity.

Pi is a decimal summation so is more accurate than 22/7

You just said decimals can't always be accurate, and you use the phrase "so is more accurate", meaning you know that decimals are more accurate (why say "so" otherwise). Either you accept that decimals are always accurate and accept the proof, or reject mathematics

[fr0d0]

They aren't the same number because there's always a gap to infinity. They cannot meet.

Evidently you don't understand the concept of infinity if you think there are gaps. Infinity is limitless.

And limitlessly 'gappy' - otherwise it'd be 1

To infinity the gap gets smaller, but never ever converges. Or are you saying that it does converge?

Right - 'amount'. By their logic - I could give you 10,000 x 99p in exchange for your £10,000 and be £100 in profit. Wanna do some money exchanges?

99p is not an infinitely long number. Your example is complete nonsense, and not related to mine in any way since you do not use infinities.

So I rounded up. Isn't that what this is about?

Decimal numbers can't always be accurate. This proves it.

Decimal notation can write any kind of number. That is their purpose. All this proves is that you don't understand mathematical concepts like infinity.

I understand infinity perfectly well I think. I don't understand this for the reasons given. I find it interesting. the fat lady hasn't sung yet (unless I missed it! )

Pi is a decimal summation so is more accurate than 22/7

You just said decimals can't always be accurate, and you use the phrase "so is more accurate", meaning you know that decimals are more accurate (why say "so" otherwise). Either you accept that decimals are always accurate and accept the proof, or reject mathematics

"can't always" allows for "is more accurate" on occasion.

[Eilonnwy]

So I rounded up. Isn't that what this is about?

Noooo! This is about infinities, not rounding up. You just proved you don't understand infinity at all.

I understand infinity perfectly well I think. I don't understand this for the reasons given. I find it interesting. the fat lady hasn't sung yet (unless I missed it! )

LOL! It has, you completely missed it. It sung when the entire mathematical community came to a consensus on the issue.

[Tiberius]

And limitlessly 'gappy' - otherwise it'd be 1

To infinity the gap gets smaller, but never ever converges. Or are you saying that it does converge?

It's not limitlessly gappy, that's the point. If it were limitlessly "gappy", it wouldn't be an infinite string of 9s.

So I rounded up. Isn't that what this is about?

No.

I understand infinity perfectly well I think. I don't understand this for the reasons given. I find it interesting. the fat lady hasn't sung yet (unless I missed it! )

Evidently you don't. You think that there can be a finite gap between 0.9r and 1, when 9 is the highest digit in the decimal system, and 0.9r has an infinite string of them. There is no number you can add to 0.9r that turns it into 1. Saerules said 0.0r1 is that number, but 0.0r1 is a mathematical impossibility because it is an infinite (i.e. endless) string of 0s with a 1 on the end. An endless string with something on the end. You might not see the contradiction mathematically, but formally you can. An endless string cannot (by definition) have an end. There is no end for Saerules' magic 1 to go.

"can't always" allows for "is more accurate" on occasion.

Yet they provably do. There is no value that cannot be expressed as a decimal number, be it irrational or rational. The decimal might be infinitely long, but it is still a valid number.

[fr0d0]

So I rounded up. Isn't that what this is about?

Noooo! This is about infinities, not rounding up. You just proved you don't understand infinity at all.

Come out of the playground Eely. Us adults are allowed to play inside

I understand infinity perfectly well I think. I don't understand this for the reasons given. I find it interesting. the fat lady hasn't sung yet (unless I missed it! )

LOL! It has, you completely missed it. It sung when the entire mathematical community came to a consensus on the issue.

And I accept that Eel. I don't understand it. Am I not allowed to question and try to understand

[Violet]

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1. You cannot get more accurate, because you could not divide 1 evenly into three. 1/3=.3^ is an approximation based on that we cannot get any closer without overshooting one, but neither can we reach one.

If we can't get closer without overshooting, we are at the number itself. This again is the definition of different numbers.

2. 3 X 3 does = 9, and .9^ still doesn't equal one. I do not see why I would have to doubt this.

1/3 = 0.3r

What you don't understand is that 0.3r * 3 equals 0.9r, so I must assume that you do not believe 3 * 3 = 9.

3. If that is not mathematically possible, then infinity was not possible to begin with Infinity + 1 is still infinity... it is just an added one on top of the initial concept.

Infinity is a concept, one that you don't seem to understand. Infinity is not a finite number; it is limitless, it has no end. You cannot add 1 to infinity since it has no limit. 0.9r is an infinitely long number; it has no end. It is a limitless string of 9s. There is nothing you can add to it to make it equal 1 (there is no gap for anything to be put in), therefore it is already equal to 1 (as I have shown).

I do not see how you reckon that .9^ is any more valid than the .0^1 long gap .9^ requires to equal one. There are an infinite amount of zeros between that decimal and that one... just as there is an infinite amount of nines following the decimal.

An infinite amount cannot have an end. It is limitless. The concept of "an infinite amount of zeroes" followed by a 1 means there is an end to infinity, which is a contradiction.

Establishment does not make something 'right', and mathematical establishment has been proved wrong before. Some of us understand the concept of endless... and justly realize that endless .9^ by definition means that it never becomes one. If .9^ were equal to one... then .0^1 would be equal to zero. Those numbers are separated from both one and zero by a mere .0^1... but that does not mean that the .0^1 is not there.

Again, .0^1 is a logically and mathematically impossible number. You cannot have an infinite string of zeros with a 1 on the end. Infinity has no limits, ergo there is no end for the 1 to be placed on.

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They aren't the same number because there's always a gap to infinity. They cannot meet.

Evidently you don't understand the concept of infinity if you think there are gaps. Infinity is limitless.

Right - 'amount'. By their logic - I could give you 10,000 x 99p in exchange for your £10,000 and be £100 in profit. Wanna do some money exchanges?

99p is not an infinitely long number. Your example is complete nonsense, and not related to mine in any way since you do not use infinities.

Decimal numbers can't always be accurate. This proves it.

Decimal notation can write any kind of number. That is their purpose. All this proves is that you don't understand mathematical concepts like infinity.

Pi is a decimal summation so is more accurate than 22/7

You just said decimals can't always be accurate, and you use the phrase "so is more accurate", meaning you know that decimals are more accurate (why say "so" otherwise). Either you accept that decimals are always accurate and accept the proof, or reject mathematics

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So i take it, if .3^*3=.9^... and 3*3=9... then 3*3= ten? ^_^ lol ^_^

If you can't get any closer without overshooting... it means that you cannot evenly distribute the 1 into that many pieces. 3's and 4's, 6's, 7's 8's, and 9's do not perfectly go into 10... and I do not see why they should perfectly go into one either. What you do not understand is that this 'proven' math is complete nonsense.

You tied a concept into a number, I tied a concept into a number. Your .9^ is no more valid than .0^1. This number exists only because .9^ is not one until you add a number with the = # of 0's as you have nines... and then put a one following them. If you have limitless nines... you have limitless zeros. But no matter how many nines you get, you will always need that minute .0^1 to reach one. By the definition of infinite nines: your nines will never reach one.

There is no contradiction. There is a limitless amount of zeros, one zero less than the amount of nines, every digit of the way. And wherever your ever-moving last 9 in .9^ is: there is a 1 occupying the equivalent place in the 0^1. You are always adding your digits on at the end of your equation... but I am inserting my infinite zeros from the beginning.

You mock my understanding of infinity... when you keep rounding infinite nines to one ^_^ Infinite nines will infinitely never reach 1, just as infinite .3^ will infinitely never reach .34. (The point, is that the gap is NOT finite... the gap is ENDLESS! )

[Tiberius]

You don't accept it, since you don't accept 0.1r is a valid decimal form of the fraction 1/9, or that 0.3r is a valid decimal form of the fraction 1/3.

If you don't accept these, I can understand completely why you wouldn't accept that 0.9r = 1. Perhaps you should read the Wikipedia article. It cleared everything up for others, so it should for you.

Possibly a humourous note, but as Randall Monroe (creator of xkcd) noted, the only debate on Wikipedia concerning 0.9r = 1 is whether the page on 0.9r should be merged with the article on 1.

[fr0d0]

And limitlessly 'gappy' - otherwise it'd be 1

To infinity the gap gets smaller, but never ever converges. Or are you saying that it does converge?

It's not limitlessly gappy, that's the point. If it were limitlessly "gappy", it wouldn't be an infinite string of 9s.

So does it converge?

I understand infinity perfectly well I think. I don't understand this for the reasons given. I find it interesting. the fat lady hasn't sung yet (unless I missed it! )

Evidently you don't. You think that there can be a finite gap between 0.9r and 1, when 9 is the highest digit in the decimal system, and 0.9r has an infinite string of them. There is no number you can add to 0.9r that turns it into 1. Saerules said 0.0r1 is that number, but 0.0r1 is a mathematical impossibility because it is an infinite (i.e. endless) string of 0s with a 1 on the end. An endless string with something on the end. You might not see the contradiction mathematically, but formally you can. An endless string cannot (by definition) have an end. There is no end for Saerules' magic 1 to go.

I agree (how can't I?) I can see nothing would go on the end. Only 9's are possible. But this is surely illustrating the weakness of the digital representation of a whole. 1 is simple. 1/3 is simple. 0.999... & 0.333... are inaccurate representations of those.

"can't always" allows for "is more accurate" on occasion.

Yet they provably do. There is no value that cannot be expressed as a decimal number, be it irrational or rational. The decimal might be infinitely long, but it is still a valid number.

I get that it's a valid number and works for maths. I think it's the convergence thing I have a problem with logically. If 0.999... & 1 never converge how are they the same?

[Tiberius]

So i take it, if .3^*3=.9^... and 3*3=9... then 3*3= ten? ^_^ lol ^_^

No. How on earth does that follow?

If you can't get any closer without overshooting... it means that you cannot evenly distribute the 1 into that many pieces. 3's and 4's, 6's, 7's 8's, and 9's do not perfectly go into 10... and I do not see why they should perfectly go into one either. What you do not understand is that this 'proven' math is complete nonsense.

No it doesn't. It means your value is equal to the target. 3,4,6,7,8,9 don't go perfectly into 10, but they do go into 10. Perfection isn't what we are arguing about here. I never claimed that 3 went into one perfectly (firstly because it's impossible since 3 is larger than 1) but it does go into 1...0.3r times.

You tied a concept into a number, I tied a concept into a number. Your .9^ is no more valid than .0^1. This number exists only because .9^ is not one until you add a number with the = # of 0's as you have nines... and then put a one following them. If you have limitless nines... you have limitless zeros. But no matter how many nines you get, you will always need that minute .0^1 to reach one. By the definition of infinite nines: your nines will never reach one.

Your concept is invalid. You cannot have a 1 on the end of an infinite amount of 0s. An infinite amount of 0s has no end for the 1 to go on.

By the definition of infinite nines, there can be no possible gap between 0.9r and 1, and numbers that have no gap between them are equal.

There is no contradiction. There is a limitless amount of zeros, one zero less than the amount of nines, every digit of the way. And wherever your ever-moving last 9 in .9^ is: there is a 1 occupying the equivalent place in the 0^1. You are always adding your digits on at the end of your equation... but I am inserting my infinite zeros from the beginning.

"one zero less" is a limit, thus nulling your argument that this is a limitless amount of zeros. I'm not always adding digits. You can't add digits to an infinite number...it's already infinite.

You mock my understanding of infinity... when you keep rounding infinite nines to one ^_^ Infinite nines will infinitely never reach 1, just as infinite .3^ will infinitely never reach .34. (The point, is that the gap is NOT finite... the gap is ENDLESS! )

I'm not rounding at all. This has nothing to do with rounding. I'm mocking your understanding of infinity because you keep making amateurish mistakes when dealing with the concept, including saying your example is limitless and then proceeded to put a limit on it.

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So does it converge?

Yes, in the same way 2 converges with 2 by its very definition. 0.9 approaches one, as you add more 9s, the number gets closer to one. 0.9r is 1, since there is no gap between them. As I've said before, numbers that have no gap are the same, because there is nothing you can add to them to make them closer. 0.9r + 0 = 1 in other words, which simplifies to 0.9r = 1.

I agree (how can't I?) I can see nothing would go on the end. Only 9's are possible. But this is surely illustrating the weakness of the digital representation of a whole. 1 is simple. 1/3 is simple. 0.999... & 0.333... are inaccurate representations of those.

I never said it wasn't simple, but it's accurate. I fail to see any reason why a complex number is an inaccurate one. Pi is a complex number, but is perfectly accurate, more so than any fraction. Fractions try to represent numbers by comparison to other numbers (i.e. 1/3 is a third of 1, comparing 3 to 1). Decimals state their actual values as a representation of digits.

I get that it's a valid number and works for maths. I think it's the convergence thing I have a problem with logically. If 0.999... & 1 never converge how are they the same?

Like I said, they converge because they are the same thing.

[fr0d0]

Ok I can dismiss fractions. I can't get the convergence thing still tho'. You may as well call it 1 because there's nothing to add to it to make it 1. But why not just call it '1'?

Thinking about that I can see why you could say it's one because there's nothing you can add to it, but conversely, we also know that 0.9r can never logically converge with one.

You could also claim that 2 - 0.9r = 1

Hmm ..I guess I get it

Thanks for taking the time to explain it.