Is 1 = 0.9999......

[SilentRunning]

Originally posted by: RossGr

And you don't comprehend that it 0.9999... has no meaning. We can give it a meaning by finding its limit.

Sorry but I have to disagree, look at my proof, I gave it meaning and position without taking limits. Perhaps you need to read the proof and thing about it for a while.

I can combine these inequalities to get

1-.1^ N < .99? < 1+.1^ N

There is only one number which can fit in the middle for which this relationship holds for all N>0 and that
number is 1 therefore we have 1=.999?.


Wow you found the limit of 0.9999....

See you have made the assumption that 0.99999... actually occupies a single point. It is infinity long, that means it will never reach termination. (no termination means it can't actually occupy a point, but it can contain a point)

 

[RossGr]

Wow! you really have no concept of how the number line works. So where is the error in my math? Why is it that I can squeeze that number into a box?

Because it occupies a single point on the number line. Again it is not a matter of walking along dropping another 9 into a slot, it is a matter of 9 existing to infinity.

I have seen people attempt to claim that Pi is infinitly large because it is an irrational number. All irrational numbers have an infinite number of digits, yet they occupy a single point on the number line. The same is true for rational numbers, there is no single number that takes more space on the number line then another, they do not walk arround looking for a home. They have a slot and they stay there.
Infinity by the way is defined as being greater then x where x is a real number.

 

[SilentRunning]

I know exactly how the number line works. You defined a range segments. (or a box as you call it all though how you made a box along a number line I don't know) The segments approach zero length but never reach it. The point that is always within the segments is 1, that is the limit.


I think you have won the stamina award though. I am tired of this thread.

For the unique representation of every real number by a non-terminating decimal, adopt the

Convention: It is forbidden to use non-terminating decimals with the number 9 as period. Then the number 0.5 can only be written as a non-terminating decimal in the unique manner: 0.5000···.

With this convention, every real number is written as a non-terminating decimal fraction in a unique manner, that is, no two distinct non-terminating decimals can represent the same real number.

We emphasise now that this definition actually identifies a real number as a non-terminating decimal fraction.

 

[RossGr]

That seems like a completly logical CONVENTION to me, it does not change the fact that .4999...= .5 it just specifies that we write it the latter way.

 

[silverpig]

Originally posted by: SilentRunning
I know exactly how the number line works. You defined a range segments. (or a box as you call it all though how you made a box along a number line I don't know) The segments approach zero length but never reach it. The point that is always within the segments is 1, that is the limit.


I think you have won the stamina award though. I am tired of this thread.

For the unique representation of every real number by a non-terminating decimal, adopt the

Convention: It is forbidden to use non-terminating decimals with the number 9 as period. Then the number 0.5 can only be written as a non-terminating decimal in the unique manner: 0.5000···.

With this convention, every real number is written as a non-terminating decimal fraction in a unique manner, that is, no two distinct non-terminating decimals can represent the same real number.

We emphasise now that this definition actually identifies a real number as a non-terminating decimal fraction.

Someone decided to use that convention to avoid ambiguity when describing the point 0.5. This is acknowledgement that there is a way to write 0.5 in decimal form other than 0.5 (that being 0.4999...)

Do you see that?

Also:

".999..." does end in a 9 at the infinite spot

ends at infinity huh? Sounds like someone doesn't know what infinite really means.

 

[silverpig]

MadRat: You are confused because you are trying to subtract 0.999... from 1 by using the technique you learned in grade 4 arithmetic. Start at the right and move towards the left. You don't actually have to do this. Try starting from the left and subtract towards the right. I'm sure you would be able to do that. If you try it you will see how the difference is 0.

 

[SilentRunning]

Originally posted by: silverpig

Originally posted by: SilentRunning
I know exactly how the number line works. You defined a range segments. (or a box as you call it all though how you made a box along a number line I don't know) The segments approach zero length but never reach it. The point that is always within the segments is 1, that is the limit.


I think you have won the stamina award though. I am tired of this thread.

For the unique representation of every real number by a non-terminating decimal, adopt the

Convention: It is forbidden to use non-terminating decimals with the number 9 as period. Then the number 0.5 can only be written as a non-terminating decimal in the unique manner: 0.5000···.

With this convention, every real number is written as a non-terminating decimal fraction in a unique manner, that is, no two distinct non-terminating decimals can represent the same real number.

We emphasise now that this definition actually identifies a real number as a non-terminating decimal fraction.

Someone decided to use that convention to avoid ambiguity when describing the point 0.5. This is acknowledgement that there is a way to write 0.5 in decimal form other than 0.5 (that being 0.4999...)

Do you see that?

Yes I see that but it say much more than that. It says no two distinct non-terminating decimals can represent the same real number. So 0.9999... cannot represent one, it must represent something else.


You say that you can divide 1 by 1 and get:

1.000....
0.999....

So if both of those values are equal to one then either value divided by one should return the same result

1.000... divided by 1 = 1.000... or 0.9999...

0.999... divided by 1 = 0.9999... (whoops)

 

[RossGr]

Could you post refernces for you convention quote, I think it needs to be put in context. Though I do not disagree with it and I do not see that it mandates that 1 <> .999...

It specifies which representation shoud be used, not that multible representations do not exist. In fact, since it goes to some length to specify WHICH representation is to be used it is confirming that multiple repersentations EXIST. If there were not multiple repersentations of the same point on the number line such a convention would not be necessary. You would simply use the unique representation for the point you are talking about. But the fact is, multiple representations exist so an author is free to establish such a convention.

Generally an author can establish such a convention for HIS BOOK it does not mean that it is universally used. Axioms and Theorems are universal, conventions are local and may be different in different books or courses.

 

[silverpig]

Originally posted by: SilentRunning

Originally posted by: silverpig

Originally posted by: SilentRunning
I know exactly how the number line works. You defined a range segments. (or a box as you call it all though how you made a box along a number line I don't know) The segments approach zero length but never reach it. The point that is always within the segments is 1, that is the limit.


I think you have won the stamina award though. I am tired of this thread.

For the unique representation of every real number by a non-terminating decimal, adopt the

Convention: It is forbidden to use non-terminating decimals with the number 9 as period. Then the number 0.5 can only be written as a non-terminating decimal in the unique manner: 0.5000···.

With this convention, every real number is written as a non-terminating decimal fraction in a unique manner, that is, no two distinct non-terminating decimals can represent the same real number.

We emphasise now that this definition actually identifies a real number as a non-terminating decimal fraction.

Someone decided to use that convention to avoid ambiguity when describing the point 0.5. This is acknowledgement that there is a way to write 0.5 in decimal form other than 0.5 (that being 0.4999...)

Do you see that?

Yes I see that but it say much more than that. It says no two distinct non-terminating decimals can represent the same real number. So 0.9999... cannot represent one, it must represent something else.


You say that you can divide 1 by 1 and get:

1.000....
0.999....

So if both of those values are equal to one then either value divided by one should return the same result

1.000... divided by 1 = 1.000... or 0.9999...

0.999... divided by 1 = 0.9999... (whoops)

It does not say that at all. It says:

"With this convention... no two distinct non-terminating decimals can represent the same real number."


Basically, it's acknowledging the existence of being able to write a number like 5 as 4.999... It says that the author will use the convention that forbids the writing of a number such as 5 as 4.999... thus making each value have a unique decimal representation. They acknowledge that 0.999... = 1 if you understand what they are saying and why.

 

[SilentRunning]

Originally posted by: RossGr
Could you post refernces for you convention quote, I think it needs to be put in context. Though I do not disagree with it and I do not see that it mandates that 1 <> .999...

It specifies which representation shoud be used, not that multible representations do not exist. In fact, since it goes to some length to specify WHICH representation is to be used it is confirming that multiple repersentations EXIST. If there were not multiple repersentations of the same point on the number line such a convention would not be necessary. You would simply use the unique representation for the point you are talking about. But the fact is, multiple representations exist so an author is free to establish such a convention.

Generally an author can establish such a convention for HIS BOOK it does not mean that it is universally used. Axioms and Theorems are universal, conventions are local and may be different in different books or courses.

Yes multiple representations exist. You can represent 1 as 1.000... and 0.9999... but given 0.9999... as you have said as a just existing number does mean it equals 1.

As I have said you can intentionally divide 1 by itself in such a way that you get the end result 0.9999... but this ignores that there is a remainder. It would not be a number which "just exists" it only exists because of intentional manipulation.

Lets say that someone was to divide one by itself.

1 / 1 = 0.9999999999...

this would go on for hundreds of years, generation after generation. The multiply by 9 and carry the remainder. After several generations of math scholars one finally figures out that 1 can go into 10 ten times not nine. It takes several hundred more years to carry that 10 through all the 9's the previous generations had created. But finally they arrive at the answer 1.000...

You see 1/1 = 0.99999... is not non terminating by definition it is non terminating by choice.

Whereas given the non terminating decimal 0.9999... as a value as you have said it just exists. There was no remainder invovled with its creation.

 

[silverpig]

Originally posted by: SilentRunning

Originally posted by: RossGr
Could you post refernces for you convention quote, I think it needs to be put in context. Though I do not disagree with it and I do not see that it mandates that 1 <> .999...

It specifies which representation shoud be used, not that multible representations do not exist. In fact, since it goes to some length to specify WHICH representation is to be used it is confirming that multiple repersentations EXIST. If there were not multiple repersentations of the same point on the number line such a convention would not be necessary. You would simply use the unique representation for the point you are talking about. But the fact is, multiple representations exist so an author is free to establish such a convention.

Generally an author can establish such a convention for HIS BOOK it does not mean that it is universally used. Axioms and Theorems are universal, conventions are local and may be different in different books or courses.

Yes multiple representations exist. You can represent 1 as 1.000... and 0.9999... but given 0.9999... as you have said as a just existing number does mean it equals 1.

As I have said you can intentionally divide 1 by itself in such a way that you get the end result 0.9999... but this ignores that there is a remainder. It would not be a number which "just exists" it only exists because of intentional manipulation.

Lets say that someone was to divide one by itself.

1 / 1 = 0.9999999999...

this would go on for hundreds of years, generation after generation. The multiply by 9 and carry the remainder. After several generations of math scholars one finally figures out that 1 can go into 10 ten times not nine. It takes several hundred more years to carry that 10 through all the 9's the previous generations had created. But finally they arrive at the answer 1.000...

You see 1/1 = 0.99999... is not non terminating by definition it is non terminating by choice.

Whereas given the non terminating decimal 0.9999... as a value as you have said it just exists. There was no remainder invovled with its creation.

So if you view 0.999... as a progression, and also say there is no remainder, then you must be saying that it "reaches" it's final value.

1

 

[SilentRunning]

Originally posted by: silverpig

Originally posted by: SilentRunning

Originally posted by: RossGr
Could you post refernces for you convention quote, I think it needs to be put in context. Though I do not disagree with it and I do not see that it mandates that 1 <> .999...

It specifies which representation shoud be used, not that multible representations do not exist. In fact, since it goes to some length to specify WHICH representation is to be used it is confirming that multiple repersentations EXIST. If there were not multiple repersentations of the same point on the number line such a convention would not be necessary. You would simply use the unique representation for the point you are talking about. But the fact is, multiple representations exist so an author is free to establish such a convention.

Generally an author can establish such a convention for HIS BOOK it does not mean that it is universally used. Axioms and Theorems are universal, conventions are local and may be different in different books or courses.

Yes multiple representations exist. You can represent 1 as 1.000... and 0.9999... but given 0.9999... as you have said as a just existing number does mean it equals 1.

As I have said you can intentionally divide 1 by itself in such a way that you get the end result 0.9999... but this ignores that there is a remainder. It would not be a number which "just exists" it only exists because of intentional manipulation.

Lets say that someone was to divide one by itself.

1 / 1 = 0.9999999999...

this would go on for hundreds of years, generation after generation. The multiply by 9 and carry the remainder. After several generations of math scholars one finally figures out that 1 can go into 10 ten times not nine. It takes several hundred more years to carry that 10 through all the 9's the previous generations had created. But finally they arrive at the answer 1.000...

You see 1/1 = 0.99999... is not non terminating by definition it is non terminating by choice.

Whereas given the non terminating decimal 0.9999... as a value as you have said it just exists. There was no remainder invovled with its creation.

So if you view 0.999... as a progression, and also say there is no remainder, then you must be saying that it "reaches" it's final value.

1

But because there is no remainder it never quite reaches it, such is the definition of infinity.

Edit thanks for putting my "invovled" in bold really bad typing as I am tired.

Anyway

1/1 = 0.999999..... can be terminated that IS a substantial difference


If we choose to ignore remainders

1/1 = 0.8999999... is just as valid an argument

The difference is that for the first case the remainder approaches 0 and for the second case it approaches
0.1

 

[silverpig]

It can't be terminated at any particular 9 because then it wouldn't equal 1/1.

That's like saying pi = 3.14

It doesn't.

 

[SilentRunning]

Originally posted by: silverpig
It can't be terminated at any particular 9 because then it wouldn't equal 1/1.

That's like saying pi = 3.14

It doesn't.

But you are forgetting the remainder.

 

[silverpig]

Originally posted by: SilentRunning

Originally posted by: silverpig
It can't be terminated at any particular 9 because then it wouldn't equal 1/1.

That's like saying pi = 3.14

It doesn't.

But you are forgetting the remainder.

There is none.

There will always be a remainder for any finite number of iterations of the long division, but we're not dealing with finite terms any more.

 

[element]

Originally posted by: fuzzy bee

Originally posted by: Skoorb
if .99999999999 = 1 then it would be 1 not, .99999999

A Skooby snack for anyone of you that can find a number between 1 and 0.999...

Heck a million skoorby snacks. You can sell them on egay.

 

[element]

Originally posted by: Haircut
Holy crap, this thread is still going?
It's obvious this is getting like the HT thread on the same subject, two sides debating the point and neither refusing to back down.

One side, the side that knows math, put up lots of proofs showing that it is true.
The other side saying that it just can't be or bleating on about numbers such as 0.00....01 which cannot exist but never actually disproving anything that has been said by the other side.

I think we should just give up now, let the people who voted no carry on believing what they want to believe, but please can we let this thread die.

No the challenge now is to get those who don't understand the math to see it anyway, by explaining it in simpler terms for them. Who was it that said "If a scientist can't explain to an 8 year old what he's doing, he is a charlatan"?

Of course 8 year old is an exageration but you get the point I'm sure.

So I ask again for a naysayer to post any number between 1 and .9999...

 

[Skyclad1uhm1]

Originally posted by: element®

Originally posted by: Haircut
Holy crap, this thread is still going?
It's obvious this is getting like the HT thread on the same subject, two sides debating the point and neither refusing to back down.

One side, the side that knows math, put up lots of proofs showing that it is true.
The other side saying that it just can't be or bleating on about numbers such as 0.00....01 which cannot exist but never actually disproving anything that has been said by the other side.

I think we should just give up now, let the people who voted no carry on believing what they want to believe, but please can we let this thread die.

No the challenge now is to get those who don't understand the math to see it anyway, by explaining it in simpler terms for them. Who was it that said "If a scientist can't explain to an 8 year old what he's doing, he is a charlatan"?

Of course 8 year old is an exageration but you get the point I'm sure.

So I ask again for a naysayer to post any number between 1 and .9999...

Ever heard of Hackenstrings?

 

[Haircut]

Originally posted by: Skyclad1uhm1

Originally posted by: element®

Originally posted by: Haircut
Holy crap, this thread is still going?
It's obvious this is getting like the HT thread on the same subject, two sides debating the point and neither refusing to back down.

One side, the side that knows math, put up lots of proofs showing that it is true.
The other side saying that it just can't be or bleating on about numbers such as 0.00....01 which cannot exist but never actually disproving anything that has been said by the other side.

I think we should just give up now, let the people who voted no carry on believing what they want to believe, but please can we let this thread die.

No the challenge now is to get those who don't understand the math to see it anyway, by explaining it in simpler terms for them. Who was it that said "If a scientist can't explain to an 8 year old what he's doing, he is a charlatan"?

Of course 8 year old is an exageration but you get the point I'm sure.

So I ask again for a naysayer to post any number between 1 and .9999...

Ever heard of Hackenstrings?

Wow, that guy was one of my math professors at university

 

[MadRat]

Awww, Hackenstrings. There goes the fallacy of .333...=1/3 again. No, not everyone accepts that just as not everyone accepts .999...=1 as both deny the existence of the remainder after an infinite number of decimal places. Playing with rocks is cute, but it has no relevance to infinity.

If you want to think of 1/3 cleanly then use base 3 numbers. Using base 3 numbers doesn't have the ambiguity of leaving out the remainder after an infinite number of decimal points.

If you want to think of infinity then use base of infinity to determine the equivalent of .999... and you can deal with that undefined remainder. Just because the remainder is undefined in base 10 doesn't mean it has to remain undefined in other numbering systems.

 

[RossGr]

Here is a little thought problem for all you none believers out there.

I think we can all agree that .1 is a nice solid number, not one of those questionable wandering numbers that can never quite find a home, cus the numbers never end.

Well, now let us take the everso friendly .1 and convert it to a binary fraction.

This means we will write it as a sum of powers of 1/2. so in binary 1/2 = .1

1/8 = .001 etc.

The interesting things is that the binary representation of .1 = .0001100110011....

HUT OH! we have a trouble here! suddenly our nice solid number has got up and walked away! Now the dang thing can't seem to find a home! I mean it is an infinitly repeating number! Does it exist? Has .1 suddenly vanished? Merely because I changed number bases! I think not.

Your computer cannot accurately represent .1, it MUST truncate the binary exansion. this is what is called an approximation when the real world steps in and the infinite tail of a number must be chopped off.

The point is there are many ways of representing the same number, an infinitely repeating number can and does represent a fixed point in the number line.

SilentRunning is a caveman with a loaded weapon, he has sufficient knowledge of math to make himself dangerous, unfortunately he does not have the depth of understanding he believes himself to have.

To any who are in doubt, the answer cannot be found here, the answer lies in your nearest University's math department. If you want to understand the real number line you must take at least a Junior level Real Analysis course. This course is the heart of a BS in Math it is one of the courses that is required of ALL math majors, it is key to understanding the basics of mathmatics. Those who have suffered this course can judge who is correct in this thread, others can only guess.

 

[Kyteland]

Originally posted by: Skyclad1uhm1

Ever heard of Hackenstrings?

I actually had a course on game theory which has a unit covering Hackenbush and Hackenstrings. We did this exact same construction of the real numbers but the thing our professor stressed was that while we can make this nice correlation between the two representations, they are not in fact the same thing. The author of your page basically stated that in his paper too.

It should not be thought that the Hackenstrings approach in any way shows that conventional mathematics is `wrong'. Rather, the axioms of Hackenstrings arithmetic are different from those of the real numbers, one consequence being that in Hackenstrings 0.999... < 1. So any proof that 0.999... = 1 must fail when applied to Hackenstrings, which in turn must mean that one of the `different' axioms has been used. In particular the `optical' proofs are making an indirect use of the Archimedean axiom; but this use is rarely made explicit, which makes the proofs misleadingly simple or even inadequate.

So where does this leave us? Depending on the set of axioms you are using to prove your case, it can be shown that either is true. However, Hackenstrings still don't convince me that when using the real number system 0.999...<1. Just look at RossGr's little proof. It is perfectly sound within the definitions of real numbers. I won't accept that Hackenstrings proves 0.999...<1 just as MadRat won't accept that Hackenstrings proves that 0.333...=1/3. It is not the same thing.

 

[SilentRunning]

Originally posted by: Kyteland

Originally posted by: Skyclad1uhm1

Ever heard of Hackenstrings?

I actually had a course on game theory which has a unit covering Hackenbush and Hackenstrings. We did this exact same construction of the real numbers but the thing our professor stressed was that while we can make this nice correlation between the two representations, they are not in fact the same thing. The author of your page basically stated that in his paper too.

It should not be thought that the Hackenstrings approach in any way shows that conventional mathematics is `wrong'. Rather, the axioms of Hackenstrings arithmetic are different from those of the real numbers, one consequence being that in Hackenstrings 0.999... < 1. So any proof that 0.999... = 1 must fail when applied to Hackenstrings, which in turn must mean that one of the `different' axioms has been used. In particular the `optical' proofs are making an indirect use of the Archimedean axiom; but this use is rarely made explicit, which makes the proofs misleadingly simple or even inadequate.

So where does this leave us? Depending on the set of axioms you are using to prove your case, it can be shown that either is true. However, Hackenstrings still don't convince me that when using the real number system 0.999...<1. Just look at RossGr's little proof. It is perfectly sound within the definitions of real numbers. I won't accept that Hackenstrings proves 0.999...<1 just as MadRat won't accept that Hackenstrings proves that 0.333...=1/3. It is not the same thing.

It is all your fault

and.....

SilentRunning is a caveman with a loaded weapon, he has sufficient knowledge of math to make himself dangerous, unfortunately he does not have the depth of understanding he believes himself to have.
--RossGr

Don't assume malice for what stupidity can explain.

Those two tag lines go together so well. RossGr likes his verbal attacks.

 

[bleeb]

Please don't let this thread die!

 

[SilentRunning]

Originally posted by: bleeb
Please don't let this thread die!

It was dead before it ever started


Bleeb, is this the only email you get