Do you think 0.9999999~ =1 , that is infinite 9s.

[VTBoy]

I am trying to see how well people at BABB understand this, because other forums seem to show most people have a poor grasp of this idea.


There is a correct answer to the question. When I posted this in another forum, it was really surpised at how most people answered. So now I want to see how people here accept the correct answer. The correct answer is they are exactly equal. There is no difference between the two.

[earthman2110]

The way it was explained to me:

5/9=0.5555555555555......
6/9=0.66666666666..........
7/9=0.777777777777777....
8/9=0.88888888888..........
9/9=0.999999999999.......

5/5=1.000....
6/6=1.0000....
9/9=1.00000....

[Candy]

I voted no. I don't know why. 8-[

[Lycus]

Also:

1/3 = 0.3333....
0.3333.... * 3 = 0.9999....
Therefore, 0.9999.... = 1

[ZaphodBeeblebrox]

Irony is:

With Nine People, now having answered it, the Poll Numbers, ONLY EQUAL 99%!!!

8-[

[AliCali]

I'm going on what I was taught in Calculus. You must have two points to make a slope, so one point will be at x=1 and the other point will be the limit as x approaches 1, but never reaches it. Thus, assuming x approaches 1 from its lower side, x = 0.999999..., but does not equal 1.

[VTBoy]

I'm going on what I was taught in Calculus. You must have two points to make a slope, so one point will be at x=1 and the other point will be the limit as x approaches 1, but never reaches it. Thus, assuming x approaches 1 from its lower side, x = 0.999999..., but does not equal 1.

Actually according to Calculus it is equal to 1. 0.99999... can be written as a limit. That limit converges onto 1. Thus 0.999999... equals 1.

Also the limit of x as a apporaches 1 is exactly equal to 1, not aproximatly 1.

[sarongsong]

From the numismatic world:
Fineness (http://certifiedmint.com/glossary.htm#p): the purity of a precious metal measured in 1,000 parts of an alloy: a gold bar of .995 fineness contains 995 parts gold and 5 parts of another metal. Example: the American Gold Eagle is .9167 fine, which means it is 91.67% gold. A Canadian Maple Leaf has a fineness of .999, meaning that it is 99.9% pure.

[VTBoy]

From the numismatic world:
Fineness (http://certifiedmint.com/glossary.htm#p): the purity of a precious metal measured in 1,000 parts of an alloy: a gold bar of .995 fineness contains 995 parts gold and 5 parts of another metal. Example: the American Gold Eagle is .9167 fine, which means it is 91.67% gold. A Canadian Maple Leaf has a fineness of .999, meaning that it is 99.9% pure.

We are not talking about 0.999 but 0.9999999... a different number.

[sarongsong]

Oh.

0.9999999~ infinite 9s

[JFM]

Nope!

[A Thousand Pardons]

Yep

I would imagine that anyone who thinks differently has not yet encountered first semester calculus.

[enginelessjohn]

Or is an engineer. Frankly once you're over 0.51, to all intents and purposes you can round it to one. Why? Because it'll vanish into the measurement error when you actually build it....

Cheers
John

[frogesque]

1 is an integer
0.9999999999~ is a real number.

they are not equal.

[kucharek]

Integers are a subset of real numbers. 0.999999~is just a weird way to write 1.
If I ask if π = 3.14159..... you can also not say: No, π is a greek letter and 3.14159....... is a number, so they are not equal.

[Candy]

Yep

I would imagine that anyone who thinks differently has not yet encountered first semester calculus. :oops:

Would anyone be equal to Candy?

[Fortis]

Another way to think about it is to consider the difference between 1 and "0." followed by n 9s. The difference for finite n is given by 10^-n, so in the limit n->infinity, the difference between 1 and 0.9999... tends to zero, hence they are identical. :)

[frogesque]

1 is used in binary (or any other base you care to name), and logic where 0.9999999~ is meaningless. Ok, maybe it's semantics but 0.999999~ is not equal and alike to 1 in all respects.

I do accept that the series (9*10^-1 +9*10-2 + ... + 9*10^-n + ...) >1 as n>infinity but this is only true (= 1) for base 10 calculations.

There is a wider world out there we should all be aware of.

[Candy]

There is a wider world out there we should all be aware of. Like when the ~Sumerians didn't have the number 0? 8-[

[iFire]

I don't think it is equal becuase it istn't. It is close though, and would be acceptable to round... espicially since you have infinate 9s.

[Glom]

Yes. If you don't agree, just ask yourself what you would get if you substracted 9.99999999999999........ from 1.

[Careless]

Yes. If you don't agree, just ask yourself what you would get if you substracted 9.99999999999999........ from 1.
-9?

[Glom]

Exactly, which is the same thing as 1 subtract 10 therefore they are equal.

[Careless]

Exactly, which is the same thing as 1 subtract 10 therefore they are equal.
nice cover

[Candy]

I don't think it is equal becuase it istn't. It is close though, and would be acceptable to round... espicially since you have infinate 9s. That's exactly what I presume the Sumerians said. :-k

I hope with string theory, there is a whole other world opening up to us.

Back to math as we know it.
VVV

[jfribrg]

they are equal. A real number is defined as the limit of a sequence. If two sequences have the same limit, then both of them define the same number.

So there.

[kucharek]

It's as close as you want and so there is no need to round to make it equal to 1.
But this here is a nice excercise on how diffcult it is to grasp the significance of what "limit" means. Am I correct when I say, if 0.9999~ would be not equal 1, then the whole calculus would fall into shreds?

Harald

[Ut]

I don't think it is equal becuase it istn't. It is close though, and would be acceptable to round... espicially since you have infinate 9s.

Close? Close?!?
You understand the concept of infinity, right?

[Moose]

It depends.

Mathematically, they are quite different, but in the real world, the difference is so minor as to be irrelevant.

In computing, the difference depends on whether your ALU rounds or truncates "infinite" floats.

[edit:] I think jfribrg just sold me. They are the same, mathematically, although the why is not an intuitive thing to grasp, I think.

[Spacewriter]

Sure, for small enough values of 1.

:P :P :P :P :P :P

[Candy]

It's as close as you want and so there is no need to round to make it equal to 1.
But this here is a nice excercise on how diffcult it is to grasp the significance of what "limit" means. Am I correct when I say, if 0.9999~ would be not equal 1, then the whole calculus would fall into shreds?

Harald Calculus Theories, as we know them, will fall soon. I know calculus (well from 10 years ago). I just didn't agree with the way it was taught to me. I have had several battles with professors. I was that weird kid in class. :o

There is an explosion with M-Theory about to surface. I don't know this first hand, but I expect it. 8-[

I can't wait. :D

[A Thousand Pardons]

Yep

I would imagine that anyone who thinks differently has not yet encountered first semester calculus. :oops:

Would anyone be equal to Candy?
:)

Just my imagination run amok.

Here's another way of looking at it. An early exercise in limits is to show that a + a*r + a*r^2 + a*r^3 + ... is equal to a/(1 - r). Since .3333... is the notation that we use to represent 3/10 + 3/100 + 3/1000 + 3/10000 + ..., the formula has a = 3/10 and r = 1/10, which means that .3333... is equal to (3/10)/(1 - 1/10), or 3/9, or 1/3, as you'd expect. Using the same formula for .9999...., we get a = 9/10 and r = 1/10, so .9999... = (9/10)/(1-1/10), which is obviously just 1.

[Candy]

Yep

I would imagine that anyone who thinks differently has not yet encountered first semester calculus. :oops:

Would anyone be equal to Candy?
:)

Just my imagination run amok. I know who you are (in every sense of the word). :wink:


Do you not agree with the unknown mathematical beliefs out there?

[Damburger]

It is equal to one. A simple proof

x=0.999r
10x=9.999r
10x-x=9.999r-0.999r
10x-x=9
9x=9
x=1

where 'r' stands for an infinite number of digits.

the fact that they are different numbers doesn't stop them being equal.

[A Thousand Pardons]

Do you not agree with the unknown mathematical beliefs out there?
I agree with everything that is now unknown (I just don't know what it is) :)


the fact that they are different numbers doesn't stop them being equal.
They are different numerals (names, or representations, for numbers), but the same number.

[Candy]

Do you not agree with the unknown mathematical beliefs out there?
I agree with everything that is now unknown (I just don't know what it is) :) There's hope for you yet. :P

[R.A.F.]

Well, I hate to "play" semantics here but...

...Look at the way the question is phrased...

...is .9999~ exactly the equal to one???

Well, it gets very close, but never quite gets there, no matter how many 9's you have...so obviously the answer is NO.

Unless one changes the meaning of the word equal to mean almost but not quite.

[A Thousand Pardons]

Well, I hate to "play" semantics here but...

...Look at the way the question is phrased...

...is .9999~ exactly the equal to one???

Well, it gets very close, but never quite gets there, no matter how many 9's you have...so obviously the answer is NO.

Unless one changes the meaning of the word equal to mean almost but not quite.
Perhaps it is the definition of .999... that is not understood. The ellipsis means it goes out to infinity. It is a shorthand for the limit I described earlier. I think it is clear, and well defined. It is equal to 1.

Otherwise, you'd be saying that .333... is not equal to 1/3 either, and I don't think that we disagree on that, do we?

Do we all agree that .333... equals 1?

[kucharek]

Do we all agree that .333... equals 1?

[Harald, not raising his hand]
:P

[SeanF]

Well, it gets very close, but never quite gets there, no matter how many 9's you have...so obviously the answer is NO.
The problem with this logic is that when you say "no matter how many 9s you have," what you're really thinking is, "no matter when you stop adding 9s." In this case, you never stop adding nines.

0.999... is equal to 1 just like 0.333... is equal to 1/3 (as ATP meant to say! ;) ) and for the same reasons.

[A Thousand Pardons]

[Harald, not raising his hand]
:P
Oops, I meant, 1/3

[kucharek]

[Harald, not raising his hand]
:P
Oops, I meant, 1/3

But 0.333333333... = 1 would be correct in a base 4 system :-)

[R.A.F.]

OK...how about this...

You have 1. you subtract .9999~...the answer will never be exacty equal to zero. It'll be real, real, real, real, real, real~ close. It will be infinitely close, but it will never be exactly equal to zero.

See, it's the words exactly equal that I'm "hung-up" on. For this thought experiment, change the meaning of "exactly equal", and I'll have no problem. :)

[Glom]

You have to realise, R.A.F., that we're taking a limit here. If we take the sum of 10^i from i=1 to infinite, then the number tends to 1. It is a mathematical limit.

[SeanF]

OK...how about this...

You have 1. you subtract .9999~...the answer will never be exacty equal to zero. It'll be real, real, real, real, real, real~ close. It will be infinitely close, but it will never be exactly equal to zero.

See, it's the words exactly equal that I'm "hung-up" on. For this thought experiment, change the meaning of "exactly equal", and I'll have no problem. :)
Well, I'd like for you to answer ATP's question, then. :)

Do you accept that 0.333... is exactly equal to 1/3?

[R.A.F.]

It is a mathematical limit.

I have no problem with the idea that at a certain point it's bascally "it's as close to "one" for all intents and purposes as it will get."

I was just suggesting that the poll question (the "exactly the equal", part) might have been phrased differently. :)

[martin]

It is a mathematical limit.

I have no problem with the idea that at a certain point it's bascally "it's as close to "one" for all intents and purposes as it will get."

I was just suggesting that the poll question (the "exactly the equal", part) might have been phrased differently. :)

This is a question of definition. The usual construction of real number system is by sequences of rational numbers. The two numbers in this poll are equal in conventional mathematics used by everyone because they are defined as equal. In specifics, we can build sequences of rational numbers. Some sequences are Cauchy, this means only that the later elements in the sequence are getting close together in a specific way. We can group together Cauchy sequences in which elements are also getting close to elements in other sequences. Each group is in correspondence with a real number by the definition of real number system. A Cauchy sequence for the real number 1 with no representation in terminating decimal form is 1/2, 2/3, 3/4, 4/5, 5/6...(some are terminating, but others are not).

Anyone can construct any style of number system he likes. It is logical to assign to different Cauchy sequences of rational numbers a different number. But when numbers are defined in this way then the new numbers do not have the same properties as real number system. In specifics, any set which is complete, with total ordering, and with the field axioms of addition and multiplication is in one-to-one correspondence with real numbers. If these two numbers are different, than what is their difference? Zero, or some number which is not zero? If the first, then we are having problems with field axioms. If the second, then we need new numbers also that are not representable in even infinite decimal form, and we still have problems with completeness, field axioms, or total ordering.

So we can define numbers with properties that are having 1.000... and 0.999... being different, but then other properties of real numbers are no longer satisfied, and maybe we have a new forum of BABB in alternative number systems.

Martin

[russ_watters]

I don't think it is equal becuase it istn't. It is close though, and would be acceptable to round... espicially since you have infinate 9s. But if its an infinite sequence, where do you cut it off to round it?

[TrAI]

Hmmm. They are not equal, no matter how many nines you add. The question is if the difference is inside or outside the needed tolerance of what ever you are measuring. Or put another way, can you cut yourself on the difference? :P

Of course people think it is fun to prove that 2+2 is anything else than 4,10 or 11, or to show that 0=1 or what not, but then, they are mad the whole lot of them, humans that is. :wink: :P

[Careless]

and the people who don't think it's 1 aren't addressing damburger's proof?

[SeanF]

The people who don't think it's equal to 1 aren't answering ATP's question, either.

[martin]

Hmmm. They are not equal, no matter how many nines you add. The question is if the difference is inside or outside the needed tolerance of what ever you are measuring. Or put another way, can you cut yourself on the difference? :P

The question is not for finite termination, but infinite repetition. If the numbers are not equal, then what is 1-0.999...? Is it zero, or not equal to zero? If it is not zero, then what is the decimal expansion (maybe infinite) of the difference? In standard real number system, they are equal. If not equal, this is acceptable, someone can be using numbers defined this way, but then these numbers do not satisfy the standard properties of real number system and all laws of mathematics and physics and other disciplines must be reviewed for conformance to an alternative number system.


Of course people think it is fun to prove that 2+2 is anything else than 4,10 or 11, or to show that 0=1 or what not, but then, they are mad the whole lot of them, humans that is. :wink: :P

They can show that 2+2 is not 4 by making an error of proof, or by changing definitions of 2, 4, addition, or equality. Sometimes they are trying to prove 0.999... is equal to something other than 1. If these are not equal, then this is an alternative number system, not the real number system, and the rules of arithmetic and other properties of real number system cannot apply.

Martin

[martin]

Irony is:

With Nine People, now having answered it, the Poll Numbers, ONLY EQUAL 99%!!!

8-[

And now with 45 people it is again 99%. Maybe some will say this proves equality, and others will say it proves inequality.

Martin

[Lycus]

You have 1. you subtract .9999~...the answer will never be exacty equal to zero. It'll be real, real, real, real, real, real~ close. It will be infinitely close, but it will never be exactly equal to zero.
Okay,

1 - 0.99 = 0.01
1 - 0.9999 = 0.0001
1 - 0.999999 = 0.000001
etc. So everytime you add a 9, a zero is added to the difference and the answer always ends with a 1 at the same place that the 9s terminate. However, in 1 - 0.9999~ the number of 9s is infinite. Therefore, the number of zeros in the difference is infinite. Since the 9s never terminate, a 1 is never added to the end. Zero with an infinite number of zeros after the decimal is zero.

1 - 0.9999~ = 0.0000~
0.0000~ = 0
Therefore, 1 = 0.9999~

[martin]

It's as close as you want and so there is no need to round to make it equal to 1.
But this here is a nice excercise on how diffcult it is to grasp the significance of what "limit" means. Am I correct when I say, if 0.9999~ would be not equal 1, then the whole calculus would fall into shreds?

Harald

Probably. Any set with completeness, field axioms, and total ordering is one-to-one in correspondence with real numbers. If we create a new number called 0.999... and say it is not equal to one, then rules of mathematics, physics, and others maybe are in need of changing in the new number system, because properties used throughout all of mathematics are not holding anymore.

Martin

[jfribrg]

I am trying to see how well people at BABB understand this, because other forums seem to show most people have a poor grasp of this idea.


There is a correct answer to the question. When I posted this in another forum, it was really surpised at how most people answered. So now I want to see how people here accept the correct answer. The correct answer is they are exactly equal. There is no difference between the two.

I'm wondering VTBoy, what was the ratio of yes to no on the other boards? Just for laughs, somebody should post the same question on the GLP board. I'm not implying that anyone at GLP would even understand the question, but it could be interesting.

[Glom]

Both Lycus and Damburger have aptly demonstrated that 9.999~=1.

To those who still aren't convinced, you are hung up on the idea that 9.999~ has only a finite number of 9s.

[Careless]

Both Lycus and Damburger have aptly demonstrated that 9.999~=1.

To those who still aren't convinced, you are hung up on the idea that 9.999~ has only a finite number of 9s.
is there a reason you keep writing it that way?

[Glom]

Smeg!

[martin]

Both Lycus and Damburger have aptly demonstrated that 9.999~=1.

To those who still aren't convinced, you are hung up on the idea that 9.999~ has only a finite number of 9s.

The decimal point is in a wrong place - 0.999~.

What is EGBB? 10 nanometres is a very accurate measurement...

Martin

[TrAI]

The question is not for finite termination, but infinite repetition. If the numbers are not equal, then what is 1-0.999...? Is it zero, or not equal to zero? If it is not zero, then what is the decimal expansion (maybe infinite) of the difference? In standard real number system, they are equal. If not equal, this is acceptable, someone can be using numbers defined this way, but then these numbers do not satisfy the standard properties of real number system and all laws of mathematics and physics and other disciplines must be reviewed for conformance to an alternative number system.

The current version of "numerical and mathematical system" is known to be buggy at handling infinites; they are outside the specs given for its development. It was proposed that all mathematical parsers should strip infinites from any formula at run-time, but due to protest from the math-hacker community this proposal was abandoned.

That said, we can not vouch for the systems accuracy when handling infinites, and as such we advice that it should not be attempted. This problem will probably not be solved before a new system is developed, any patches to the current system will likely just make matters worse, but there is little support for developing a new system from the bottom up at this time.
:wink:

[martin]

[Harald, not raising his hand]
:P
Oops, I meant, 1/3

But 0.333333333... = 1 would be correct in a base 4 system :-)

If someone is thinking 0.999~ is not 1, then by changing of base the numbers that are represented change also. If 0.999~ and 1 are different then 0.499~ and 0.5 should be different also, but both are having representation 0.111~ in base three system. So if someone is thinking 0.999~ is not 1, then the systems of different bases do not represent the same sets of numbers.

Martin

[R.A.F.]

Sorry...been gone for a while. :)


Do you accept that 0.333... is exactly equal to 1/3?

Good question. We define 1/3, as being 0.333... But if it were exactly equal to 1/3, then why the need for the repeating 3's?

Defining 1 as being 1 and also defining it as being 0.999... just seems inelegant to me. :)

I'll stop now and concede the point...it's starting to make my head hurt.

[martin]

Sorry...been gone for a while. :)


Do you accept that 0.333... is exactly equal to 1/3?

Good question. We define 1/3, as being 0.333... But if it were exactly equal to 1/3, then why the need for the repeating 3's?

Defining 1 as being 1 and also defining it as being 0.999... just seems inelegant to me. :)

I'll stop now and concede the point...it's starting to make my head hurt.

It is a question of meaning of "equal." You can define 0.999... and 1 as different, but then you cannot have usual laws of real numbers.

Sorry, we do not want to make your head hurt.

Martin

[SeanF]

Sorry...been gone for a while. :)


Do you accept that 0.333... is exactly equal to 1/3?

Good question. We define 1/3, as being 0.333... But if it were exactly equal to 1/3, then why the need for the repeating 3's?
I don't understand your question. The need for the repeating 3s is that if they didn't repeat, it wouldn't be equal to 1/3.


Defining 1 as being 1 and also defining it as being 0.999... just seems inelegant to me. :)
More inelegant than defining 1/3 and 0.333... as the same thing?


I'll stop now and concede the point...it's starting to make my head hurt.
Aw, that's no fun! :)

[martin]

The current version of "numerical and mathematical system" is known to be buggy at handling infinites; they are outside the specs given for its development. It was proposed that all mathematical parsers should strip infinites from any formula at run-time, but due to protest from the math-hacker community this proposal was abandoned.

I do not know what are the bugs you mean, but a common problem is trying to use infinity like a number. The real number system obeys field axioms of addition and multiplication. The extended real number system, with real numbers and positive and negative infinity, violates the field axioms.


That said, we can not vouch for the systems accuracy when handling infinites, and as such we advice that it should not be attempted. This problem will probably not be solved before a new system is developed, any patches to the current system will likely just make matters worse, but there is little support for developing a new system from the bottom up at this time.
:wink:

There is no computational problem here, only mathematical proof. No one can develop a system that includes real numbers, infinity and obeys field axioms, because they are contradictory. We can vouch for accuracy only when using operations formally justified. If we are using results proved under real number system, and then apply these to infinity, then we can be having some bad problems. But equality of 0.999~ and 1 are justified by definitions of real numbers and field axioms. If we are using some other system with different definition of numbers and where field axioms do not hold, then 0.999~ and 1 can be different.

Martin

[VTBoy]

I am trying to see how well people at BABB understand this, because other forums seem to show most people have a poor grasp of this idea.


There is a correct answer to the question. When I posted this in another forum, it was really surpised at how most people answered. So now I want to see how people here accept the correct answer. The correct answer is they are exactly equal. There is no difference between the two.

I'm wondering VTBoy, what was the ratio of yes to no on the other boards? Just for laughs, somebody should post the same question on the GLP board. I'm not implying that anyone at GLP would even understand the question, but it could be interesting.


It was actually 30% to 70%, about the opposite of this place.

[Normandy6644]

I haven't been able to read this whole thread, so forgive me if someone has done this "proof" already. I always found it elegant and pretty cool.

Let n=0.9999....

10n=9.9999...

10n-n=9.9999...-0.9999...

9n=9

n=1

Kinda sneaky, but I still like it. It's akin to the 1/3=.3333... deal.

[ZaphodBeeblebrox]

I am trying to see how well people at BABB understand this, because other forums seem to show most people have a poor grasp of this idea.


There is a correct answer to the question. When I posted this in another forum, it was really surpised at how most people answered. So now I want to see how people here accept the correct answer. The correct answer is they are exactly equal. There is no difference between the two.

I'm wondering VTBoy, what was the ratio of yes to no on the other boards? Just for laughs, somebody should post the same question on the GLP board. I'm not implying that anyone at GLP would even understand the question, but it could be interesting.


It was actually 30% to 70%, about the opposite of this place.

With the Minority coming around, Riight?

Now I Know, Why I LOVE This Place, so Much!

[Damburger]

I haven't been able to read this whole thread, so forgive me if someone has done this "proof" already. I always found it elegant and pretty cool.

Let n=0.9999....

10n=9.9999...

10n-n=9.9999...-0.9999...

9n=9

n=1

Kinda sneaky, but I still like it. It's akin to the 1/3=.3333... deal.

Yeah, I did that one a page or so back. It's not sneaky - it's right. People who don't accept this and other proofs haven't even a vague concept of infinity.

[Glom]

I don't understand your question. The need for the repeating 3s is that if they didn't repeat, it wouldn't be equal to 1/3.

Exactly. 0.333 terminating is approximately 1/3. 0.333...... 3s to infinity is exactly equal to a third. In the same way, 0.999........ 9s to infinity is exactly equal to nine ninths or in other words 1.


What is EGBB? 10 nanometres is a very accurate measurement...

EGBB is the ICAO locator from Birmingham International Airport and NM means nautical miles (as opposed to nm which means nanometres).

[A Thousand Pardons]

If we are using some other system with different definition of numbers and where field axioms do not hold, then 0.999~ and 1 can be different.
Different definitions? Sure. Just the other day, I was eating an 0.999..., and I had to spit out the seeds. I don't do that with 1's.

[VTBoy]

I am trying to see how well people at BABB understand this, because other forums seem to show most people have a poor grasp of this idea.


There is a correct answer to the question. When I posted this in another forum, it was really surpised at how most people answered. So now I want to see how people here accept the correct answer. The correct answer is they are exactly equal. There is no difference between the two.

I'm wondering VTBoy, what was the ratio of yes to no on the other boards? Just for laughs, somebody should post the same question on the GLP board. I'm not implying that anyone at GLP would even understand the question, but it could be interesting.


It was actually 30% to 70%, about the opposite of this place.

With the Minority coming around, Riight?

Now I Know, Why I LOVE This Place, so Much!

Sadly no, the majority was gaining strenght. This was actually around a year ago, I just brought it up now because I was reminded of it in my mathematical proof class when my proffessor showed it to us. But sadly most of them where convinced the proofs are not valid, I showed them the algebraic proof, the calculus proof. They insisted it wasn't valid. Even those with calculus wouldn't accept it, saying incorrect things like limits are only approximations.

[martin]

Yeah, I did that one a page or so back. It's not sneaky - it's right. People who don't accept this and other proofs haven't even a vague concept of infinity.

It can be both sneaky and right :wink: ...but we can see if anyone is proving that (1+10+100+...) is equal to 1/9 by misuse of same method...


EGBB is the ICAO locator from Birmingham International Airport and NM means nautical miles (as opposed to nm which means nanometres).

Sorry, 10 nautical miles is more reasonable then...

Martin

[Fortis]

Of course people think it is fun to prove that 2+2 is anything else than 4,10 or 11, or to show that 0=1 or what not, but then, they are mad the whole lot of them, humans that is. :wink: :P
If you've ever used an old version of Lahey Fortran 77 you would discover that there is a compiler option that would allow you to do that. It allowed you to modify numeric constants, such as "5", that were passed to subroutines. If you had a subroutine, BONKERS(X) that incremented X by one, and then called BONKERS(5), you would find that the result of a PRINT(*,*)5 would be 6. :o :)

[Maksutov]

It would appear R.A.F. et al are getting hung up on infinitesimals. When calculus was later formally founded on limits, these infinitesimal "values" went away. Infinitesimals are not in the realm of the reals.

[Chuck]

If 0.9999999... doesn't equal 1 then there must be a number half way between them, their average. To calculate the average of two numbers you add them and divide the total by 2, that is, (x+y)/2. Let x = 0.9999999... and y = 1. Then x+y = 1.9999999... and 1.9999999... divided by 2 is 0.9999999... (use long division to verify it if you like). Since (x+y)/2 = 0.9999999... and x = 0.9999999... that means that (x+y)/2 = x. Multiply both sides by 2 and you get x+y = 2x. Subtract x from both sides and you get y = x. Since y = 1 and x = 0.9999999... then 0.9999999... equals 1.

[Glom]

Another brilliant proof.

[Moose]

If 0.9999999... doesn't equal 1 then there must be a number half way between them, their average. To calculate the average of two numbers you add them and divide the total by 2, that is, (x+y)/2. Let x = 0.9999999... and y = 1. Then x+y = 1.9999999... and 1.9999999... divided by 2 is 0.9999999... (use long division to verify it if you like). Since (x+y)/2 = 0.9999999... and x = 0.9999999... that means that (x+y)/2 = x. Multiply both sides by 2 and you get x+y = 2x. Subtract x from both sides and you get y = x. Since y = 1 and x = 0.9999999... then 0.9999999... equals 1.

Interesting, but I'm not sure how convincing a proof that 0.9999... = 1 is when it requires one to accept that 0.9999 x 2 is 1.9999... (read: not involving an 8 at the "end" of that infinite chain of 9s) on the very same grounds.

It's a little circular in terms of reasoning, I think.

[A Thousand Pardons]

If 0.9999999... doesn't equal 1 then there must be a number half way between them, their average. To calculate the average of two numbers you add them and divide the total by 2, that is, (x+y)/2. Let x = 0.9999999... and y = 1. Then x+y = 1.9999999... and 1.9999999... divided by 2 is 0.9999999... (use long division to verify it if you like). Since (x+y)/2 = 0.9999999... and x = 0.9999999... that means that (x+y)/2 = x. Multiply both sides by 2 and you get x+y = 2x. Subtract x from both sides and you get y = x. Since y = 1 and x = 0.9999999... then 0.9999999... equals 1.

Interesting, but I'm not sure how convincing a proof that 0.9999... = 1 is when it requires one to accept that 0.9999 x 2 is 1.9999... (read: not involving an 8 at the "end" of that infinite chain of 9s) on the very same grounds.

It's a little circular in terms of reasoning, I think.
You read the proof backwards. It does not assume that 0.999... x 2 is 1.999... -- it says that 1.999... divided by 2 is 0.999..., which is justified by the comment "(use long division to verify it if you like)". You can turn that around to your version of course, but it's not quite the same thing.

[Moose]

I was referring to x + y = 1.9999...,

Oh wait. Okay, I see what I got wrong. It was a sum between 1 and 0.9999... the whole time, not a sum between a pair of 0.9999...s. #-o

Never mind. I'll go back to sleep now.

As for the division, yeah, I've got no argument there, it works the same regardless of your grasp of "infinite".

[A Thousand Pardons]

I was referring to x + y = 1.9999...,

Oh wait. Okay, I see what I got wrong. It was a sum between 1 and 0.9999... the whole time, not a sum between a pair of 0.9999...s. #-o

Now you've done it! There's bound to be someone who objects to 1 plus 0.999... being 1.999... :)

[frogesque]

I was referring to x + y = 1.9999...,

Oh wait. Okay, I see what I got wrong. It was a sum between 1 and 0.9999... the whole time, not a sum between a pair of 0.9999...s. #-o

Now you've done it! There's bound to be someone who objects to 1 plus 0.999... being 1.999... :)

Yep, there's no margin of error stated :lol:

[pghnative]

Hmmm. I see. My kindergarten teacher was wrong. 1 + 1 = 1.999...

[A Thousand Pardons]

Hmmm. I see. My kindergarten teacher was wrong. 1 + 1 = 1.999...
are you sure your kindergarten teacher didn't know that 2 = 1.999... ? :)

[VTBoy]

Not good the No side is gaining more votes. Do people not accept the given proofs.

[Damburger]

OK, as I'm training to be a maths teacher I'm not allowed to let this one drop, so I'll try another method.

Let x=1-0.999r

then x=0.0r1

which is an infinited sequence of zeroes followed by 1.

Now if you multiply x by any integer n, then:

x=0.0rn

i.e the digits of n appended on to the end of an infinite sequence of zeroes.

No matter how large n is, that infinite string of zeroes will still be there, so the number will be lower than any real number you can think of.

The only number that behaves like this is (by definition) zero, so x=0.

[frogesque]

I'm going to go back to the original question, namely:


Do you think 0.9999999~ = 1, that is an infinite 9s.

further qualified by the question at the top of the poll


Do you think 0.9999999~ infinite 9s is exactly the equal to 1.

We have seen various proofs and reasonings (my own included) that at the limit 0.999999~ > 1 for base 10 arithmetic calculation. However the introduction of the word exactly means we have to consider all applications of 0.99999~ and also of 1.

The symbol we use for unity (1) has applications outwith the narowly defined criteria of base 10 numbers and I earlier quoted logic 1 (= true) as a cirumstance where 0.9999~ is meaningless.

On those grounds alone 0.9999~ can not be 1's equal so my answer to the question as posed would have to be. "No".

[Damburger]

I'm going to go back to the original question, namely:


Do you think 0.9999999~ = 1, that is an infinite 9s.

further qualified by the question at the top of the poll


Do you think 0.9999999~ infinite 9s is exactly the equal to 1.

We have seen various proofs and reasonings (my own included) that at the limit 0.999999~ > 1 for base 10 arithmetic calculation. However the introduction of the word exactly means we have to consider all applications of 0.99999~ and also of 1.

The symbol we use for unity (1) has applications outwith the narowly defined criteria of base 10 numbers and I earlier quoted logic 1 (= true) as a cirumstance where 0.9999~ is meaningless.

On those grounds alone 0.9999~ can not be 1's equal so my answer to the question as posed would have to be. "No".

I touched on this above. They can be seen as different numbers, but that doesn't stop them being equal, as quite a few of us have proven.

[martin]

I'm going to go back to the original question, namely:


Do you think 0.9999999~ = 1, that is an infinite 9s.

further qualified by the question at the top of the poll


Do you think 0.9999999~ infinite 9s is exactly the equal to 1.

We have seen various proofs and reasonings (my own included) that at the limit 0.999999~ > 1 for base 10 arithmetic calculation. However the introduction of the word exactly means we have to consider all applications of 0.99999~ and also of 1.

The symbol we use for unity (1) has applications outwith the narowly defined criteria of base 10 numbers and I earlier quoted logic 1 (= true) as a cirumstance where 0.9999~ is meaningless.

On those grounds alone 0.9999~ can not be 1's equal so my answer to the question as posed would have to be. "No".

I touched on this above. They can be seen as different numbers, but that doesn't stop them being equal, as quite a few of us have proven.

If the question is "Are the real numbers represented by 0.999... and 1 equal or not equal," then the answer is "equal." If the question is "Are the symbols 0.999... and 1 exactly equal" then they are not, because we can recognize different symbols. If someone interprets the question this second way, then they can say no. If someone interprets the question the first way and says no, then they are not understanding properties of real numbers. Damburger and others have shown this.

Martin

[A Thousand Pardons]

We have seen various proofs and reasonings (my own included) that at the limit 0.999999~ > 1 for base 10 arithmetic calculation. However the introduction of the word exactly means we have to consider all applications of 0.99999~ and also of 1.

No, we don't. :)

"Exactly" has a particular mathematical meaning, in this context.



The symbol we use for unity (1) has applications outwith the narowly defined criteria of base 10 numbers and I earlier quoted logic 1 (= true) as a cirumstance where 0.9999~ is meaningless.

On those grounds alone 0.9999~ can not be 1's equal so my answer to the question as posed would have to be. "No".

I touched on this above. They can be seen as different numbers, but that doesn't stop them being equal, as quite a few of us have proven.
Not quite the same thing, though. He's saying 1 is not even a number.

*I* touched on that, above (http://www.badastronomy.com/phpBB/viewtopic.php?p=372368#372368).

[Donnie B.]

We have seen various proofs and reasonings (my own included) that at the limit 0.999999~ > 1 for base 10 arithmetic calculation. However the introduction of the word exactly means we have to consider all applications of 0.99999~ and also of 1.

No, we don't. :)

"Exactly" has a particular mathematical meaning, in this context.

Um... it does? As far as I know, "exactly" has no mathematical meaning at all. Equal is equal, meaning "exactly equal" as we might say conversationally.

Look at it the other way: when would you use "equal" in mathematics when the two items were, in fact, "almost equal" or "sort of equal"?

Here's my point of view: you give me an infinite number of nines, and I'll give you a one. :wink:

[Glom]

It is very bad that a full third of the voters have decided to ignore 4 excellent proofs of the fact and instead vote for a side that has provided not a shred of evidence and merely handwaved about accuracies while failing to understand the meaning of the infinitessimal hence implicitly spitting on all of calculus in the process.

[frogesque]

It is very bad that a full third of the voters have decided to ignore 4 excellent proofs of the fact and instead vote for a side that has provided not a shred of evidence and merely handwaved about accuracies while failing to understand the meaning of the infinitessimal hence implicitly spitting on all of calculus in the process.

As far as I'm aware I have calmly given my personal reasons for voting no. Should others disagree with my interpretation of the question posed then they are at liberty to do so. Should anyone wish to debate my interpretation (which I fully admit is somewhat mischievious and tongue in cheek) then I'm happy to do so. I think my stance has been helpful and forced people to consider more deeply the underlying principles on which mathemeatics are based. I know I have learnt a lot by doing so and I hope others have as well.

The poll results (last time I looked it was 40 - 20) are misleading because you are only allowed to vote once and are not allowed to withdraw or modify a vote so it doesn't reflect any 'swing' that might take place during a discussion. Serious debate can still be lighthearted and the poll results are best taken in that vein also.

I have great respect for Isac Newton and the legacy he gave us. I think he would be amused at the concept of spitting on infinitessimals, they are such small targets yet produce truely profound results.

[Fortis]

It is very bad that a full third of the voters have decided to ignore 4 excellent proofs of the fact and instead vote for a side that has provided not a shred of evidence and merely handwaved about accuracies while failing to understand the meaning of the infinitessimal hence implicitly spitting on all of calculus in the process.
I demand a recount. Never abided by this electronic voting anyway. Where's the paper trail? ;) :)

[Glom]

As far as I'm aware I have calmly given my personal reasons for voting no. Should others disagree with my interpretation of the question posed then they are at liberty to do so.

I will take that liberty. :)

You are confusing things by bringing in aspects that were not called for. If the poll was "Do you think 9+9=18?", you, by your logic, would have voted no because in base 12, 9+9=16. You're being a smartaleck.

It is pretty obvious that this question was asked in the context of base 10. By extension, you can take the question to ask about 0.111~=1 in binary and 0.222~=1 in trinary etc.

[TrAI]

It is very bad that a full third of the voters have decided to ignore 4 excellent proofs of the fact and instead vote for a side that has provided not a shred of evidence and merely handwaved about accuracies while failing to understand the meaning of the infinitessimal hence implicitly spitting on all of calculus in the process.

Well, Since you have an infinite string of numbers you also have to have infinite precision, and as long as the string you are processing is within the precision you do not approximate the content. Saying that 0,9...=1 is to use a finite precision.

The 10x proof was nice, but I have a feeling there is something that is a bit of there, I think it is that when you multiply something by 10, you push the decimal point, but you also must add a "0" in the end. Of course usually one can drop this zero, but in 10*0,9... You have the mathematical equivalent to an eternal loop, the string must terminate with a "0", and as you have an infinite amount of zeros you do not get to the point where you are allowed to drop this zero. This problem is more visible when you do for example 0,9...*2 = 1,9...8, you must trail an "8" but never reach the point you can do so...

But then it might just be that I have a hard time with this because I am not... What do you call a math-person that does the more philosophical(?) parts of the art?

[martin]

It is very bad that a full third of the voters have decided to ignore 4 excellent proofs of the fact and instead vote for a side that has provided not a shred of evidence and merely handwaved about accuracies while failing to understand the meaning of the infinitessimal hence implicitly spitting on all of calculus in the process.

Well, Since you have an infinite string of numbers you also have to have infinite precision, and as long as the string you are processing is within the precision you do not approximate the content. Saying that 0,9...=1 is to use a finite precision.

In construction of real numbers, 0.999... is exactly equal to one. No approximation. No finite precision. It is not a matter of saying after 100 million digits we stop and then just round to one. The quantity 0.999... is defined as one. If someone defines it some other way, he can do this, but then his numbers do not obey very basic relations we expect from all number systems. They can define 0.999... as 3908154.2 if they like, but it will not obey rules of arithmetic. This is shown by many here.

There is no number system in which a number exists arbitrarily close to one but not equal to one, but also obeys field axioms (commutivity of addition and multiplication, existence of additive and multiplicative inverse, etc.). If someone constructs a number system without these rules, then he can do what he likes. What is the use of this other number system, nobody knows.

Martin

[VTBoy]

Here is a proof of it with limits. Hope you understand it.

http://home.comcast.net/~jsacto/Proofwith_Limits2.JPG




edit: To fix mistake, thanks normandy pointing it out.

[Normandy6644]

Here is a proof of it with limits. Hope you understand it.

http://home.comcast.net/~jsacto/Proofwith_Limits.JPG

Ooh pretty. :D

Edit: Make sure you start that sum at n=1. :wink:

[TrAI]

In construction of real numbers, 0.999... is exactly equal to one. No approximation. No finite precision. It is not a matter of saying after 100 million digits we stop and then just round to one. The quantity 0.999... is defined as one. If someone defines it some other way, he can do this, but then his numbers do not obey very basic relations we expect from all number systems. They can define 0.999... as 3908154.2 if they like, but it will not obey rules of arithmetic. This is shown by many here.
...

I must admitt that it might very well be me who is unable to see the obvious here(and it is getting quite late here, so I am a bit tired), but to me it seems like the act of making 0,9... into 1 is indivisible from the act of setting a finite number of allowable decimals(that is, setting a finite precision) and rounding to 1, it is the only way to do this without redefining the value of 0,9... to 1, and that is just like defining 0,9... as 3908154,2.

Defining the numbers as anything else is no good really, the numbers are defined by representing some amount of something, and so, if you have 0,9... of cake, you do have 0,9... of cake. Of cause reality has a finite precision, so all of this is just imaginary, but that is no reason to redefine the value, even if it is by an impossibly little amount. I think that this impossibly(infinitly) small amount is the balance you have to carry for invoking the impossibly(infinitly) long string of numbers that an infinite implies...

By the way, in base 2, 0,1... & 1,0... would yield 0,0, I expect. So will not 0,9... & 1,0... be false too?

[TrAI]

Here is a proof of it with limits. Hope you understand it.

http://home.comcast.net/~jsacto/Proofwith_Limits2.JPG




edit: To fix mistake, thanks normandy pointing it out.

I don't know if this was in response to my post or not, as you have not quoted/referenced any nick, but if it was for my benefit, I must apologize and say I do not have the knowledge required to interpret that. Perhaps it is just ignorance of more advanced math that keeps me from seeing the point of saying 0,9...=1... :(

Anyway, I have been thinking about where the problem is in that earlier proof, and I think that the problem is that if the multiplication of 0,9...*10 could run to completion, (0,9...*10)-0,9... < 9, or (0,9...*10)-0,9... = 8,9...1(not really possible, I guess, but I hope what I am thinking of is still understandable)

[A Thousand Pardons]

The 10x proof was nice, but I have a feeling there is something that is a bit of there, I think it is that when you multiply something by 10, you push the decimal point, but you also must add a "0" in the end. Of course usually one can drop this zero, but in 10*0,9... You have the mathematical equivalent to an eternal loop, the string must terminate with a "0", and as you have an infinite amount of zeros you do not get to the point where you are allowed to drop this zero. This problem is more visible when you do for example 0,9...*2 = 1,9...8, you must trail an "8" but never reach the point you can do so...
Here's one way to see your error: try 10 times 1/3, with 1/3 represented as a decimal. Doesn't 10 x 0.3333... equal 3.333... ?

No "trailing zero" required. :)

[krbrown]

if 0.9999999999999.....=1-x then x must be so small it is actualy zero and 1-0=1. therefore 0.9999999999999999999......=1 QED, case solved
8-[

[TrAI]

Here's one way to see your error: try 10 times 1/3, with 1/3 represented as a decimal. Doesn't 10 x 0.3333... equal 3.333... ?

No "trailing zero" required. :)
I know people usually does not think of it this way, they learn that the trailing zeros can be dropped. But you must reach it before you are allowed to do drop it, but with infinite precision and infinite decimals, you will never get there(the multiplication is never fully completed)...

Anyway, 0,3...*10= 3,3...0, you can not have this in reality of cause... I think that 3,3...0-0,3...=2,9...7(this use of over-infinite number of decimals is probably bad math or something :p) not 3 since 0,3... is just to little to be equal to 1/3, it is the same problem as 0,9... is just infinitely smaller than 1. Or put another way, by applying infinite numbers of decimals we are not handling the problem, just pushing it down in size forever, but as long as we allow infinite precision, we can not be allowed to just make 0,9...=1 or 0,3...=1/3

As I see it, when you apply infinitely long numbers to one part of the thing, you must apply them to all the other parts too(kind of like doing the same thing on both sides of an equation), or else you are just approximating...

[A Thousand Pardons]

Anyway, 0,3...*10= 3,3...0, you can not have this in reality of cause...

You mean a zero at the end of an infinite string of 3's? Got that right. :)

However, .333... equals 1/3 and 10 times 1/3 is 3 1/3.


since 0,3... is just to little to be equal to 1/3

o well

As I see it, when you apply infinitely long numbers to one part of the thing, you must apply them to all the other parts too(kind of like doing the same thing on both sides of an equation), or else you are just approximating...
OK.

0.999... equals 1.000...

but that's what we mean, when we write "1". It's 1.000... with infinite precision.

[martin]

In construction of real numbers, 0.999... is exactly equal to one. No approximation. No finite precision. It is not a matter of saying after 100 million digits we stop and then just round to one. The quantity 0.999... is defined as one. If someone defines it some other way, he can do this, but then his numbers do not obey very basic relations we expect from all number systems. They can define 0.999... as 3908154.2 if they like, but it will not obey rules of arithmetic. This is shown by many here.
...

I must admitt that it might very well be me who is unable to see the obvious here(and it is getting quite late here, so I am a bit tired), but to me it seems like the act of making 0,9... into 1 is indivisible from the act of setting a finite number of allowable decimals(that is, setting a finite precision) and rounding to 1, it is the only way to do this without redefining the value of 0,9... to 1, and that is just like defining 0,9... as 3908154,2.

Symbols have meaning only when we give it to them. Everyone here agrees on the meaning of symbols 0, 1, 26, -4, and 2119385. I think everyone also agrees on meaning of fractions. Decimal representations terminating after finite number of digits are also clear. 2119.591 is 2119591/1000.

What is the meaning of an infinite symbol 0.999...? If we have only rational numbers (fractions), then we cannot give meaning to all infinite decimal representations, because not all are rational. Infinite decimal 3.14159265... is pi, but does not correspond to any rational number. Also, square root of two has an infinite decimal representation that can be calculated to any arbitrary length, but square root of two is not rational. So we cannot give meaning to all infinite decimal representations with only rational numbers. So we need real numbers, which are constructed by sequences of rational numbers. When a sequence of rational numbers converges in appropriate sense (meaning later elements are getting close to each other in certain way), then we say this sequence converges to a real number. Many different sequences can converge to the same real number. When constructing real numbers, it is shown that basic field axioms (for example, (a+b)+c == a+(b+c)) extend to real numbers in natural way. We can construct a number system in a different way. For example, we can say sequences (0.9, 0.99, 0.999, 0.9999, ...) and (1.1, 1.01, 1.001, 1.0001, ...) converge to different numbers. But if we define numbers in this way, they will violate basic rules of arithmetic. The book of Rudin "Principles of Mathematical Analysis" explains this in a way that can be understood by many who are good with maths but do not yet understand formal axiomatic construction of real number system.

Now with real numbers constructed, we can assign meaning to infinite decimal representations. What is meaning of 0.999...? If we define it as one, then the rules of basic arithmetic everyone learns in school can apply to infinite decimals just like in finite decimals. If we define this as some real number other than one, then the rules of basic arithmetic that everyone is using do not work on infinite decimals. And if we want to say that 0.999... is real number arbitrarily close to one but not equal to one, then there is a big problem, because no such real number exists. We can construct other number systems with such numbers, but these number systems do not obey laws that everyone thinks numbers should obey. They are called field axioms, but are very simple, only things like commutivity and associativity of addition and multiplication, existence of sums and products, and existence of additive and multiplicative inverses. If someone is wanting a number system in which a number exists which is arbitrarily close to one but not equal to one, then these basic properties of numbers cannot hold.

Only particular numbers have this property of two different representations in base ten system. In decimal system, are 0.4999... and 0.5 equal or not equal? If they are not equal, then what are base two representations for each? Are base two representations same or different? What of 0.0111... and 0.1 in base three system? Are these same or different? If different, what are base ten representations of these numbers?


Defining the numbers as anything else is no good really, the numbers are defined by representing some amount of something, and so, if you have 0,9... of cake, you do have 0,9... of cake. Of cause reality has a finite precision, so all of this is just imaginary, but that is no reason to redefine the value, even if it is by an impossibly little amount. I think that this impossibly(infinitly) small amount is the balance you have to carry for invoking the impossibly(infinitly) long string of numbers that an infinite implies...

How do we know someone has 0.9... of cake unless we agree on definition of 0.9...? Saying 0.9... is 1 is not redefinition, it is original definition, and as many are showing here this definition is consistent with laws of arithmetic. Any other definition in which 0.9... is not equal to 1 cannot satisfy basic laws of arithmetic. It is not approximation to say 0.9... is equal to one.


By the way, in base 2, 0,1... & 1,0... would yield 0,0, I expect. So will not 0,9... & 1,0... be false too?

I am not understanding this example. 0.1... and 1.0 in base two represent same number in real number system.

Martin

[TrAI]

OK.

0.999... equals 1.000...

but that's what we mean, when we write "1". It's 1.000... with infinite precision.

What I meant is something like: 0,9... = 1-0,...1
You loose the balance if you remove 0,...1
But you might remove some infinites, I think: 0,9=1-0,1
:P :wink:

[martin]

Anyway, 0,3...*10= 3,3...0, you can not have this in reality of cause... I think that 3,3...0-0,3...=2,9...7(this use of over-infinite number of decimals is probably bad math or something :p) not 3 since 0,3... is just to little to be equal to 1/3, it is the same problem as 0,9... is just infinitely smaller than 1. Or put another way, by applying infinite numbers of decimals we are not handling the problem, just pushing it down in size forever, but as long as we allow infinite precision, we can not be allowed to just make 0,9...=1 or 0,3...=1/3

Do you believe in following rules?

1. If a and b are numbers, a+b is number.
2. If a and b are numbers, a*b is number.
3. a+b=b+a
4. a*b=b*a
5. (a+b)+c=a+(b+c)
6. (a*b)*c=a*(b*c)
7. 0+x=x
8. 1*x=x
9. 0 is not 1.
10. If a is number, then there is number -a with a+(-a) = 0.
11. If b is number not equal to zero, then there is number 1/b with b*(1/b) = 1.
12. a*(b+c) = a*b+a*c

These are basic laws of arithmetic. Are all correct? If not, which one is wrong?

Martin

[TrAI]

How do we know someone has 0.9... of cake unless we agree on definition of 0.9...? Saying 0.9... is 1 is not redefinition, it is original definition, and as many are showing here this definition is consistent with laws of arithmetic. Any other definition in which 0.9... is not equal to 1 cannot satisfy basic laws of arithmetic. It is not approximation to say 0.9... is equal to one.


The idea was that if you have 0,9 of cake, you are 0,1 of from a full cake, if you have 0,99 of cake you are still missing 0,01, and the natural extension of this is that even with infinites you have to balance the cake you have against the cake you are missing, 0,9... cake is 0,...1 from a full cake. The trouble with infinites is that they are, well, infinite, and so it is all philosophical how you choose to see it.





I am not understanding this example. 0.1... and 1.0 in base two represent same number in real number system.


Hmmm.. Sorry about that, I was trying to do a boolean AND, you know, if the two numbers are the same, an AND operation returns true, if not it returns false.




Do you believe in following rules?

1. If a and b are numbers, a+b is number.
2. If a and b are numbers, a*b is number.
3. a+b=b+a
4. a*b=b*a
5. (a+b)+c=a+(b+c)
6. (a*b)*c=a*(b*c)
7. 0+x=x
8. 1*x=x
9. 0 is not 1.
10. If a is number, then there is number -a with a+(-a) = 0.
11. If b is number not equal to zero, then there is number 1/b with b*(1/b) = 1.
12. a*(b+c) = a*b+a*c

These are basic laws of arithmetic. Are all correct? If not, which one is wrong?

Martin

Hmmm... Yes, They do seem to be right. Which rule is it I might be breaking, in your opinion?

[Grey]

Do you believe in following rules?

1. If a and b are numbers, a+b is number.
2. If a and b are numbers, a*b is number.
3. a+b=b+a
4. a*b=b*a
5. (a+b)+c=a+(b+c)
6. (a*b)*c=a*(b*c)
7. 0+x=x
8. 1*x=x
9. 0 is not 1.
10. If a is number, then there is number -a with a+(-a) = 0.
11. If b is number not equal to zero, then there is number 1/b with b*(1/b) = 1.
12. a*(b+c) = a*b+a*c

These are basic laws of arithmetic. Are all correct? If not, which one is wrong?
Hmm, I'm not sure this set of axioms is complete enough to tackle this problem. To be sure, if you add in a couple that are implicit (that 0 and 1 are numbers would do it, I think)*, these will get you all the rational numbers. But the irrationals aren't encapsulated in these rules, and neither are the transcendentals. If you wanted the irrationals, a new rule (if a is a number, then there is a number b such that b*b = a) should do the job to add them into the set. But if you want the full set of reals, you'd really need an explicit statement that the limit of an infinite set of the operations above is also a number. Otherwise, I could take the rationals as a set of numbers that follow these rules, and 0.999999... isn't even a number, let alone equal to 1!

* Some might argue that rule 9 is sufficient to establish that 0 and 1 are both in the set of numbers, or that their use in rules 7, 8, 10, and 11 is enough. I'd think a better wording for rule 9 would be "0 and 1 are numbers, and are not equal", or some such, and I'd introduce it before any of the other rules that reference them. As long as we're going to the trouble of a complete set of axioms, why not be pedantic?

:D

[krbrown]

What do you get if you remove a currant from a christmas pudding?

I this explains everything.

:-?

[martin]

How do we know someone has 0.9... of cake unless we agree on definition of 0.9...? Saying 0.9... is 1 is not redefinition, it is original definition, and as many are showing here this definition is consistent with laws of arithmetic. Any other definition in which 0.9... is not equal to 1 cannot satisfy basic laws of arithmetic. It is not approximation to say 0.9... is equal to one.


The idea was that if you have 0,9 of cake, you are 0,1 of from a full cake, if you have 0,99 of cake you are still missing 0,01, and the natural extension of this is that even with infinites you have to balance the cake you have against the cake you are missing, 0,9... cake is 0,...1 from a full cake. The trouble with infinites is that they are, well, infinite, and so it is all philosophical how you choose to see it.

Yes, but this is an important point. Anyone can say 0.9... is not equal to 1, this is a question of definition, not reality. But if someone is saying this, then the usual rules of arithmetic cannot apply. This is the price of existence of infinitesmal numbers; these numbers can be defined, but cannot obey same rules as other numbers.





I am not understanding this example. 0.1... and 1.0 in base two represent same number in real number system.


Hmmm.. Sorry about that, I was trying to do a boolean AND, you know, if the two numbers are the same, an AND operation returns true, if not it returns false.

Then I am saying 0.111... and 1 in base two system are equal. I expect you are saying not equal. From above, this is question of definition only, not of reality. But when defining them as not equal, usual rules of mathematics cannot apply.





Do you believe in following rules?

1. If a and b are numbers, a+b is number.
2. If a and b are numbers, a*b is number.
3. a+b=b+a
4. a*b=b*a
5. (a+b)+c=a+(b+c)
6. (a*b)*c=a*(b*c)
7. 0+x=x
8. 1*x=x
9. 0 is not 1.
10. If a is number, then there is number -a with a+(-a) = 0.
11. If b is number not equal to zero, then there is number 1/b with b*(1/b) = 1.
12. a*(b+c) = a*b+a*c

These are basic laws of arithmetic. Are all correct? If not, which one is wrong?

Martin

Hmmm... Yes, They do seem to be right. Which rule is it I might be breaking, in your opinion?

Maybe nothing. These are field axioms. We need also total ordering (every number is positive, negative, or zero, and only one choice) and completeness (or least upper bound property). Every totally ordered field with least upper bound property is in one-to-one correspondence with every other such field. Then the choice is to say 0.999... is equal to one, or some of field axioms must be discarded, or total ordering fails, or least upper bound property fails. Least upper bound property may seem unnecessary, but the definition of integration relies on least upper bound property. If we do not have this property, then we have to find another way to define integrals. (Maybe this is possible, I do not know. But all standard definitions are invalid without least upper bound property.)

But we can first have some fun only with field axioms. If 0.999... is not equal to one, what is 1-0.999..., which must exist by rules 10 and 1. Is it zero, or not zero?

Martin

[martin]

Hmm, I'm not sure this set of axioms is complete enough to tackle this problem. To be sure, if you add in a couple that are implicit (that 0 and 1 are numbers would do it, I think)*, these will get you all the rational numbers. But the irrationals aren't encapsulated in these rules, and neither are the transcendentals. If you wanted the irrationals, a new rule (if a is a number, then there is a number b such that b*b = a) should do the job to add them into the set. But if you want the full set of reals, you'd really need an explicit statement that the limit of an infinite set of the operations above is also a number. Otherwise, I could take the rationals as a set of numbers that follow these rules, and 0.999999... isn't even a number, let alone equal to 1!

Existence of 0 and 1 are required :D

This is definition of field only. Rational numbers are an ordered field, but are not complete. Definition of reals follows by limiting operations, as you say. If we also have total ordering, and completeness (or least upper bound property), then real numbers satisfy these and field axioms. Also, any set with all three (field axioms, completeness, total ordering) is in one-to-one correspondence with real numbers.

Martin

[TrAI]

Maybe nothing. These are field axioms. We need also total ordering (every number is positive, negative, or zero, and only one choice) and completeness (or least upper bound property). Every totally ordered field with least upper bound property is in one-to-one correspondence with every other such field. Then the choice is to say 0.999... is equal to one, or some of field axioms must be discarded, or total ordering fails, or least upper bound property fails. Least upper bound property may seem unnecessary, but the definition of integration relies on least upper bound property. If we do not have this property, then we have to find another way to define integrals. (Maybe this is possible, I do not know. But all standard definitions are invalid without least upper bound property.)

But we can first have some fun only with field axioms. If 0.999... is not equal to one, what is 1-0.999..., which must exist by rules 10 and 1. Is it zero, or not zero?

Martin

Well, the whole problem with this question is that we are invoking an impossibility by using an infinite number of decimals, you can not, for example multiply or divide such a number with another in a straight forward manner, like some of the proofs did. The reason for this is that, as I have indicated before, a math operation like that can only be seen as finished when you have handled all the numbers or decide you have enough accuracy for your needed purpose. But with the goal of infinite number of decimals and infinite accuracy you are never allowed to finish, you are to all intents and purposes stuck on the first cycle of the operation...

Anyway, I was in an earlier post trying to balance the equation, I did this by saying 0,9...= 1-0,...1, that is 0,(infinite number of nines) = 1-(infinite(really infinite-1, but that is still infinite) number of zeros)(one). Just because you are invoking an impossible thing, like an infinite string of nines, I don't see why you should be allowed to add the impossible on only one side of the "=". If you do not put in a balancing infinite, you are kind of saying that any finite like 0,9999=1 is correct too,

This is of course impossible to work, you will never get to the one at the end, but it balances the impossible situation of an infinite number of nines at the other side of the equation, you will never come to a last nine.

But this is the only way we really can get an answer to what, for example 0,9...*2 is. You might think it is just 1,9..., but that is an approximation, since you never allowed the multiplication to finish(something that can not be done in reality). But by looking at how the finite 0,999*2 is solved we can predict that the infinite 0,9...*2, if it could be solved, would end with an "8", so you get 1,9...8, that is (one)(decimal divider)(infinite number of nines)(eight). It is impossible to work in reality, you can't have something at the end of an infinite thing, but it must be done to solve the multiplication accurately.

[martin]

Well, the whole problem with this question is that we are invoking an impossibility by using an infinite number of decimals, you can not, for example multiply or divide such a number with another in a straight forward manner, like some of the proofs did. The reason for this is that, as I have indicated before, a math operation like that can only be seen as finished when you have handled all the numbers or decide you have enough accuracy for your needed purpose. But with the goal of infinite number of decimals and infinite accuracy you are never allowed to finish, you are to all intents and purposes stuck on the first cycle of the operation...

Why not? Many things in mathematics are infinite limits. If we cannot conceive of infinite operation, then there is no such thing as integration.


Anyway, I was in an earlier post trying to balance the equation, I did this by saying 0,9...= 1-0,...1, that is 0,(infinite number of nines) = 1-(infinite(really infinite-1, but that is still infinite) number of zeros)(one). Just because you are invoking an impossible thing, like an infinite string of nines, I don't see why you should be allowed to add the impossible on only one side of the "=". If you do not put in a balancing infinite, you are kind of saying that any finite like 0,9999=1 is correct too,

This is not what I am not saying, that 0.9999=1. What is impossible is to assign to the infinite symbol 0.999... a meaning other than one, and to have resulting number system obey rules of arithmetic. Does the number 1/3 exist? (By field axioms, it must.) It has no finite base 10 representation. Before, you are saying 0.333... is not equal to 1/3. In conventional mathematics, 0.333... is defined as 1/3. Does number 1/3 not exist because infinitely many operations are required to represent it in base 10 system? Before, you are using base 2 for some example. Does existence of numbers depend on which base I am using? Number 1/3 in base 3 has representation 0.1. In base 10, it is only 0.333... If we are doing things this way, then number 1/3 exists in base 3 system, but not in base 10 system.


This is of course impossible to work, you will never get to the one at the end, but it balances the impossible situation of an infinite number of nines at the other side of the equation, you will never come to a last nine.

You can introduce symbol 0.0...1, which does not exist in conventional decimal represnetation system. What is its meaning? Zero, or not zero? If not zero, it will not be obeying certain laws of mathematics.


But this is the only way we really can get an answer to what, for example 0,9...*2 is. You might think it is just 1,9..., but that is an approximation,

This is not approximation, it is exact, if 0.9... and 1.9... are interpreted as in conventional mathematics. If you are saying it is approximation, then you are using numbers which do not obey basic laws of arithmetic.


since you never allowed the multiplication to finish(something that can not be done in reality).

Why not? 0.999...*2=1.999... I have just completed an infinite operation. Who is not allowing me to do this in reality? It is done. Integral of x*x from x=0 to x=1 is 1/3. I have just completed another infinite operation. Mathematics is full of infinite operations. Derivatives are defined as limits. Is derivative of x*x equal to 2*x, or something different from 2*x? When someone is walking across a room, he is completing an infinite operation, because first he is walking 1/2 way, then 3/4 way, then 7/8 way, then 15/16 way, and so on. Can he never come to other side because of infinite number of operations required?


But by looking at how the finite 0,999*2 is solved we can predict that the infinite 0,9...*2, if it could be solved, would end with an "8", so you get 1,9...8, that is (one)(decimal divider)(infinite number of nines)(eight). It is impossible to work in reality, you can't have something at the end of an infinite thing, but it must be done to solve the multiplication accurately.

We do not have an 8 if laws of conventional mathematics apply. Someone can define the symbol 1.9...8, this is no problem. But what is its meaning? 2, or less than 2? If less than two, then mathematics involving these new symbols does not obey rules of conventional mathematics.

With infinite precision, all 0.999...=1 works very well, and cannot be anything else if laws of arithmetic are to apply.

Maybe 1/3 example of A Thousand Pardons is a better example to think on. Does number 1/3 exist? What is its decimal representation?

Martin

[jfribrg]

There are several lessons to be learned by this thread:

1. When you deal with infinities, you have to forget intuition, because it is frequently wrong.

2. It took hundreds of years before there was a generally accepted and correct definition of continuity and limit. This thread gives one an idea of why it took that long.

3. The properties of rational numbers are not the same as real numbers. Don't get them confused.

4. Just because you don't know how to do it (ex. multiply two irrational numbers) doesn't mean it is impossible.

4. 0.99999..... really is = 1.00000000 and does not depend on how you look at it.

[martin]

There are several lessons to be learned by this thread:

1. When you deal with infinities, you have to forget intuition, because it is frequently wrong.

2. It took hundreds of years before there was a generally accepted and correct definition of continuity and limit. This thread gives one an idea of why it took that long.

3. The properties of rational numbers are not the same as real numbers. Don't get them confused.

4. Just because you don't know how to do it (ex. multiply two irrational numbers) doesn't mean it is impossible.

4. 0.99999..... really is = 1.00000000 and does not depend on how you look at it.

<bold added>

Additional lesson is 4+1=5 :D

Martin

[TrAI]

Why not? Many things in mathematics are infinite limits. If we cannot conceive of infinite operation, then there is no such thing as integration.

It is not that one can not have something that would give an infinite answer, or run infinite iterations, but when applying such things one must impose some limit or one will never get an answer. It is like computing pi or creating a fractal, one imposes some limit on it, so it is not really infinite, even if it has the potential to be.


This is not what I am not saying, that 0.9999=1. What is impossible is to assign to the infinite symbol 0.999... a meaning other than one, and to have resulting number system obey rules of arithmetic. Does the number 1/3 exist? (By field axioms, it must.) It has no finite base 10 representation. Before, you are saying 0.333... is not equal to 1/3. In conventional mathematics, 0.333... is defined as 1/3. Does number 1/3 not exist because infinitely many operations are required to represent it in base 10 system? Before, you are using base 2 for some example. Does existence of numbers depend on which base I am using? Number 1/3 in base 3 has representation 0.1. In base 10, it is only 0.333... If we are doing things this way, then number 1/3 exists in base 3 system, but not in base 10 system.
0,9... is an infinite string of numbers, 0,9...=0,9..., saying that it is equal to one is redefining the basic way that we represent the idea of an decimal number. Starting the number with "0," says that it must be less than 1 no matter what we do with it.



You can introduce symbol 0.0...1, which does not exist in conventional decimal represnetation system. What is its meaning? Zero, or not zero? If not zero, it will not be obeying certain laws of mathematics.
It is more than zero but infinitely small. It is not breaking the laws of mathematics, as no mathematics is applied to it, it is just a way to represent an idea, i do realize it is an impossible value, but the nature of the way we represent numbers explicitly states that any number starting with "0," must be less than 1. The thing is that if I write "0,9..." this is a symbolic representation of a value, but it bases itself on some basic rules, that is, 0 is the symbol for nothing, "," is a symbol that says anything after it is to be counted as less than one whole unit of what ever we are representing, so we do not even have to step up to mathematics, to say that "0,9..." is not less than one is redefining the very basics of how we represent the idea of a number...



This is not approximation, it is exact, if 0.9... and 1.9... are interpreted as in conventional mathematics. If you are saying it is approximation, then you are using numbers which do not obey basic laws of arithmetic.

Why not? 0.999...*2=1.999... I have just completed an infinite operation. Who is not allowing me to do this in reality? It is done. Integral of x*x from x=0 to x=1 is 1/3. I have just completed another infinite operation. Mathematics is full of infinite operations. Derivatives are defined as limits. Is derivative of x*x equal to 2*x, or something different from 2*x? When someone is walking across a room, he is completing an infinite operation, because first he is walking 1/2 way, then 3/4 way, then 7/8 way, then 15/16 way, and so on. Can he never come to other side because of infinite number of operations required?

0,9...*2 = 0,9... + 0,9..., right? Well, to do an addition you must start at the very end of the number and add the smallest place, and work up from there. This is impossible with an infinite string of numbers because of the very fact that they are infinite, but what we do know is that the first 9+9 of any string of "9" will give "8" and "1" to carry, that means that 0,9... + 0,9... =1,9... the string of "9"s does not end, and so you never get the "8" that is necessary for this answer to be correct, 0,9... + 0,9... is less than 1,9... by an infinitely small amount, but as you are allowing infinites in the length of your answer you must accept that you can have something infinite, and so something that is infinitely small must also be allowed. It is this idea I try to represent by using the combinations like "1,9...8"


We do not have an 8 if laws of conventional mathematics apply. Someone can define the symbol 1.9...8, this is no problem. But what is its meaning? 2, or less than 2? If less than two, then mathematics involving these new symbols does not obey rules of conventional mathematics.
It is not really anything to mathematics, it is a way to represent the idea of a value that contain an infinite number of numerals. It is not a symbol I define as anything, it is a combination of discrete symbols that still retain their individual meanings. "1" means the value 1, "," means that the following is less than a whole, the "9" is the value 9 placed at the 0,1 point(9*(10^-1)), "..." indicates an infinite continuing string of 9's, but ending with an "8"(of course, an infinite string does not end, but the infinite nature of the string makes it impossible to get the "full" answer)


With infinite precision, all 0.999...=1 works very well, and cannot be anything else if laws of arithmetic are to apply.

Anything less than one must be less than one. And the use of a decimal point/comma explicitly states that the following string of numbers is not enough to make one whole of what ever...



Maybe 1/3 example of A Thousand Pardons is a better example to think on. Does number 1/3 exist? What is its decimal representation?
Martin

1/3 is a number, yes, though it is just to bad that it can not be represented in decimal form in base 10, the problem is that 0.3... is almost, but not quite enough, if it was you would not have an infinite number of decimals, even the infinite number of "3"s is an approximation of the value of 1/3, the trick is we just hides this fact by pushing the amount of inaccuracy down for infinity, but it is still there, so it doesn't matter in anything but a philosophical debate like the one we are having, since reality have finite precision.

[Glom]

It is not that one can not have something that would give an infinite answer, or run infinite iterations, but when applying such things one must impose some limit or one will never get an answer. It is like computing pi or creating a fractal, one imposes some limit on it, so it is not really infinite, even if it has the potential to be.

You have entered the realm of the mathematical. What has physical limitation got to do with anything?
"TrAI"]0,9... is an infinite string of numbers, 0,9...=0,9..., saying that it is equal to one is redefining the basic way that we represent the idea of an decimal number. Starting the number with "0," says that it must be less than 1 no matter what we do with it.

By extending the decimal to infinity, the error reduces to zero, hence when the limit is taken, the equality is achieved. It is the ability to theorise infinity that makes us better than the animals. When chickens come up with calculus, then we'll renegotiate our dietary habits.


It is more than zero but infinitely small.

But zero is infinitely small. The difference tends to zero as the limit is taken.


1/3 is a number, yes, though it is just to bad that it can not be represented in decimal form in base 10, the problem is that 0.3... is almost, but not quite enough, if it was you would not have an infinite number of decimals, even the infinite number of "3"s is an approximation of the value of 1/3, the trick is we just hides this fact by pushing the amount of inaccuracy down for infinity, but it is still there, so it doesn't matter in anything but a philosophical debate like the one we are having, since reality have finite precision.

Exactly. The inaccuracy becomes infinitessimal, which means it becomes zero.

[A Thousand Pardons]

0,9... is an infinite string of numbers, 0,9...=0,9..., saying that it is equal to one is redefining the basic way that we represent the idea of an decimal number.



to say that "0,9..." is not less than one is redefining the very basics of how we represent the idea of a number...

No, it is yourself that is redefining the basic way. 0.999... means (limit as n approaches infinity of the sum of 9/10 through 9/10^n) which is known to be exactly 1.

[SeanF]

0,9...*2 = 0,9... + 0,9..., right? Well, to do an addition you must start at the very end of the number and add the smallest place, and work up from there.
Nope. We're taught to do it that way because it's easier (as it avoids the issue of changing digits that have already been calculated), but it is quite possible to add two numbers by going through the digits from left to right.

In fact, if you're asked to add 1057 and 1028 in your head, I bet the first thing you think is "two thousand . . ." I know I do.

And when the digits of the two numbers repeat ad infinitum, then the digits of the sum will also, and it can be shown exactly how they will repeat without calculating each repetition separately. Adding two numbers with infinite repeating 3s results in a sum with infinite repeating 6s. Adding two numbers with infinite repeating 9s results in a number with infinite repeating 9s. There is no 8, because there's never a digit that won't have a "carried-over" 1 from the next digit.


This is impossible with an infinite string of numbers because of the very fact that they are infinite, but what we do know is that the first 9+9 of any string of "9" will give "8" and "1" to carry, that means that 0,9... + 0,9... =1,9... the string of "9"s does not end, and so you never get the "8" that is necessary for this answer to be correct
You say, "the first 9+9 of any string . . .," but since you're talking about adding from the smallest digit, you really mean the last 9+9. In an infinite repeating decimal, there is no last, therefore there's no need for an 8 to make the answer "correct."

[TrAI]

Nope. We're taught to do it that way because it's easier (as it avoids the issue of changing digits that have already been calculated), but it is quite possible to add two numbers by going through the digits from left to right.

In fact, if you're asked to add 1057 and 1028 in your head, I bet the first thing you think is "two thousand . . ." I know I do.

You don't usually have to do it from the end, no, you can use shortcuts or approximate the answer and correct the approximation as you go, but it is still doing the same, for small numbers in the head it is possibly a more natural way to do it. The addition of two infinite numbers is a bit trickier, you see, it can not be done, not even if you had infinite time. Just saying 0,9...+ 0,9... =1,9... is not doing the math, if it was, we would all be stuck in an infinite loop. But I think it is wrong to just drop the fact that there should be an 8 at the end of the answer of any addition of all nines just because you never get there. I know it seems strange to think of having something at the end of something infinite, as it is infinite, but it is just the way I was trying to show that by the very nature of using an infinite there will be something like an approximation being done. I am not sure how to word the thing, but "approximation" is close, even if it is not something that is done intentionally, it is just unavoidable.



And when the digits of the two numbers repeat ad infinitum, then the digits of the sum will also, and it can be shown exactly how they will repeat without calculating each repetition separately. Adding two numbers with infinite repeating 3s results in a sum with infinite repeating 6s. Adding two numbers with infinite repeating 9s results in a number with infinite repeating 9s. There is no 8, because there's never a digit that won't have a "carried-over" 1 from the next digit.

I know that you can never get to the end of the string of numbers, that is the whole problem, an calculation with infinite numbers is never over, what we do in this thread is possible only because of the infinite nature of it, we can just say "well, it is infinite, so that we never really compute it all doesn't matter, we can predict that it goes on for ever"


You say, "the first 9+9 of any string . . .," but since you're talking about adding from the smallest digit, you really mean the last 9+9. In an infinite repeating decimal, there is no last, therefore there's no need for an 8 to make the answer "correct."
I did of course mean the last, yes...

I am just saying the same over again, aren't I... Ah well, I guess I never will see how 0,9... =1 can be the accurate solution... :( But thanks to all of you, for trying to explain it to me.

[mutineer]

The ONLY difference between 0.999 ... recurring and 1
is PRECISELY THE FACT that 1 IS the LARGER number,
by an infinitely small amount.

What is the difference between zero and an infinitely small amount?
Well, for one: you can divide by an infinitely small amount, ordinarily
producing an infinitely large quotient. You may never divide by zero.

And that's OFFICIAL !!! [-X

[martin]

The ONLY difference between 0.999 ... recurring and 1
is PRECISELY THE FACT that 1 IS the LARGER number,
by an infinitely small amount.

What is the difference between zero and an infinitely small amount?
Well, for one: you can divide by an infinitely small amount, ordinarily
producing an infinitely large quotient. You may never divide by zero.

And that's OFFICIAL !!! [-X

Kindly spell out how basic rules of mathematics work in this alternative system with infinitely small numbers not equal to zero. In particular, show how these infinitely small and infinitely large quantities obey the field axioms I have set out before. Since it is OFFICIAL you must know how to do this.

Martin

[A Thousand Pardons]

I know it seems strange to think of having something at the end of something infinite, as it is infinite, but it is just the way I was trying to show that by the very nature of using an infinite there will be something like an approximation being done.
It does not seem strange to me. It seems wrong, to say that there is something at the end of it when there is no end.

That contradiction alone should get you started in the right direction.

The ONLY difference between 0.999 ... recurring and 1
is PRECISELY THE FACT that 1 IS the LARGER number,
by an infinitely small amount.

What is the difference between zero and an infinitely small amount?
Well, for one: you can divide by an infinitely small amount, ordinarily
producing an infinitely large quotient. You may never divide by zero.

And that's OFFICIAL !!! [-X
Not anymore, not for at least a hundred years. :)

You divide by finite amounts, which would be zero at the limit.

1 is not the larger number, is the contention. I'm not sure why you would say that is a fact.

[mutineer]

The ONLY difference between 0.999 ... recurring and 1
is PRECISELY THE FACT that 1 IS the LARGER number,
by an infinitely small amount.

What is the difference between zero and an infinitely small amount?
Well, for one: you can divide by an infinitely small amount, ordinarily
producing an infinitely large quotient. You may never divide by zero.

And that's OFFICIAL !!! [-X

Kindly spell out how basic rules of mathematics work in this alternative system with infinitely small numbers not equal to zero. In particular, show how these infinitely small and infinitely large quantities obey the field axioms I have set out before. Since it is OFFICIAL you must know how to do this.

Martin

One of the basic rules of mathematics (often a troublesome one) not included in your list is that division by zero is forbidden.

However, there is nothing wrong with the statement that a cake may be divided into an infinitely large number of infinitely small crumbs.

[VTBoy]

The ONLY difference between 0.999 ... recurring and 1
is PRECISELY THE FACT that 1 IS the LARGER number,
by an infinitely small amount.

What is the difference between zero and an infinitely small amount?
Well, for one: you can divide by an infinitely small amount, ordinarily
producing an infinitely large quotient. You may never divide by zero.

And that's OFFICIAL !!! [-X

Kindly spell out how basic rules of mathematics work in this alternative system with infinitely small numbers not equal to zero. In particular, show how these infinitely small and infinitely large quantities obey the field axioms I have set out before. Since it is OFFICIAL you must know how to do this.

Martin

One of the basic rules of mathematics (often a troublesome one) not included in your list is that division by zero is forbidden.

However, there is nothing wrong with the statement that a cake may be divided into an infinitely large number of infinitely small crumbs.

Wrong, an infinitly small number is equal to zero. One of the basic rules of mathematics, and thats Official. [-X

[martin]

The ONLY difference between 0.999 ... recurring and 1
is PRECISELY THE FACT that 1 IS the LARGER number,
by an infinitely small amount.

What is the difference between zero and an infinitely small amount?
Well, for one: you can divide by an infinitely small amount, ordinarily
producing an infinitely large quotient. You may never divide by zero.

And that's OFFICIAL !!! [-X

Kindly spell out how basic rules of mathematics work in this alternative system with infinitely small numbers not equal to zero. In particular, show how these infinitely small and infinitely large quantities obey the field axioms I have set out before. Since it is OFFICIAL you must know how to do this.

Martin

One of the basic rules of mathematics (often a troublesome one) not included in your list is that division by zero is forbidden.

Division is multiplication by multiplicative inverse. This is included on the list as combination of multiple rules. Note the exception of zero.


However, there is nothing wrong with the statement that a cake may be divided into an infinitely large number of infinitely small crumbs.

Yes, if you do not care if ordinary rules of mathematics apply, you can invent numbers arbitrarily close to zero if you like. You can create a whole new mathematical system based on this. But it will not obey rules set out above (field axioms), total ordering (every number is positive, negative, or zero, and only one of these), and completeness (the limit of every converging sequence of real numbers is also real numbers). Every system of numbers with these three properties is in one-to-one correspondence with real numbers, and real numbers do not have infinitely small numbers not equal to zero. If you do not care about these things on which all of modern mathematics is based, then you can do what you like. But every result of mathematics, physics, and everything else based on these rules is invalid in some other system not obeying these rules.

If you can show how infinitely small numbers obey field axioms, completeness, and total ordering, I want to see. Real numbers obey these three things. Please show how a number system with infinitely small numbers and infinitely large numbers also satisfies these rules. It cannot be done.

On dividing cake into infinitely small pieces, problems like this are subject of measure theory, with libraries full of books. All of these books are avoiding construction of infinitely small numbers not equal to zero, because with these numbers nothing in mathematics works. Infinitely small numbers are out of mathematics for centuries now. You can have them if you like, but then all results of modern mathematics are wrong in such a number system.

Martin

[mutineer]

However, there is nothing wrong with the statement that a cake may be divided into an infinitely large number of infinitely small crumbs.
Well, I think my argument is the more succinct!
I note that you do not claim that the number of crumbs would be less than infinite :D
or, indeed, more :o
I agree that modern mathematics finds it difficult to deal with the problem of how many fewer than infinity would remain if I ate some of the crumbs. I should not for a moment wish to challenge you on that matter. 8)

[Severian]

Just a brief note - there is a system called the hyperreals which includes infinitesimals and infinities. However, as Martin notes, this isn't what we typically use. There is even some version of calculus for it.

However, the question here is referring to the real numbers. As some one said earlier, this representation of 1 is somewhat of a pathological example, which means we have to turn back to definitions. Which means we have to know the definitions (equivalence classes of Cauchy sequences of rational numbers). Under the definition, there is no debate; the number is 1.

edit - Earlier I didn't mention what Cauchy sequences I was talking about

[jfribrg]

Number 11 on Martin's list deals with the division by zero. In field terms, the additive identity element of a field cannot have a multiplicative inverse. If it did have an inverse, then there would be other field elements that do not have an inverse. Real numbers satisfy the field axioms that Martin mentioned, and so therefore cannot allow division by zero. Infinity is not an element of the field of real numbers. There are an infinite number of reals, but infinity is not one of them.

There also seems to be an infinite number of invalid "proofs" that 0.99999... is not = 1.0000000

This all goes back to the definition of what a real number is. Don't think in terms of points on a number line because the term "point" is ill-defined. Think in terms of limits of an infinite sequence, which is what the calculus is based on.

[russ_watters]

Why not? Many things in mathematics are infinite limits. If we cannot conceive of infinite operation, then there is no such thing as integration.

It is not that one can not have something that would give an infinite answer, or run infinite iterations, but when applying such things one must impose some limit or one will never get an answer. It is like computing pi or creating a fractal, one imposes some limit on it, so it is not really infinite, even if it has the potential to be. That isn't true. You've done calculus the easy and the long way, haven't you? Either way, the derivative of x^2 is exactly equal to 2x.

The whole point of calculus is fixing the problem of doing an infinite number of iterations (see: Newton's method of numerical integration) and allows you to calculate such things exactly.

[martin]

There also seems to be an infinite number of invalid "proofs" that 0.99999... is not = 1.0000000

:D


This all goes back to the definition of what a real number is. Don't think in terms of points on a number line because the term "point" is ill-defined. Think in terms of limits of an infinite sequence, which is what the calculus is based on.

Yes, I think this is a source of many problems. People are having intuition about the meaning of 0.999..., but cannot define it rigourously...

Martin

[Fortis]

I know that you can never get to the end of the string of numbers, that is the whole problem, an calculation with infinite numbers is never over, what we do in this thread is possible only because of the infinite nature of it, we can just say "well, it is infinite, so that we never really compute it all doesn't matter, we can predict that it goes on for ever"
In mathematics it isn't at all uncommon to express a number as a sum of an infinite series. Consider Sin(pi), which is an infinite series of powers of another infinite series, (i.e. the expansion of pi) We're generally happy, though, to state that Sin(Pi)=0.

Another example is the geometric series, y = 1 + x + x^2 +x^3..., etc.

Because the series is infinite, you can write

(y - 1) = x*y

and solve for y in terms of x, i.e.

y = 1 / (1 - x).

Hope this helps. :)

[martin]

I know that you can never get to the end of the string of numbers, that is the whole problem, an calculation with infinite numbers is never over, what we do in this thread is possible only because of the infinite nature of it, we can just say "well, it is infinite, so that we never really compute it all doesn't matter, we can predict that it goes on for ever"
In mathematics it isn't at all uncommon to express a number as a sum of an infinite series. Consider Sin(pi), which is an infinite series of powers of another infinite series, (i.e. the expansion of pi) We're generally happy, though, to state that Sin(Pi)=0.

Another example is the geometric series, y = 1 + x + x^2 +x^3..., etc.

Because the series is infinite, you can write

(y - 1) = x*y

and solve for y in terms of x, i.e.

y = 1 / (1 - x).

Hope this helps. :)

I add only that a series must converge in appropriate sense to manipulate in this manner. For geometric series, this is only with -1<x<1. For other values, we cannot use this method. Without this restriction, maybe someone here is proving 0=1 or something like this. The infinite power series for sin(x) converges for all x, so we can use any number.

Martin

[Fortis]

I know that you can never get to the end of the string of numbers, that is the whole problem, an calculation with infinite numbers is never over, what we do in this thread is possible only because of the infinite nature of it, we can just say "well, it is infinite, so that we never really compute it all doesn't matter, we can predict that it goes on for ever"
In mathematics it isn't at all uncommon to express a number as a sum of an infinite series. Consider Sin(pi), which is an infinite series of powers of another infinite series, (i.e. the expansion of pi) We're generally happy, though, to state that Sin(Pi)=0.

Another example is the geometric series, y = 1 + x + x^2 +x^3..., etc.

Because the series is infinite, you can write

(y - 1) = x*y

and solve for y in terms of x, i.e.

y = 1 / (1 - x).

Hope this helps. :)

I add only that a series must converge in appropriate sense to manipulate in this manner. For geometric series, this is only with -1<x<1. For other values, we cannot use this method. Without this restriction, maybe someone here is proving 0=1 or something like this. The infinite power series for sin(x) converges for all x, so we can use any number.

Martin
Yup, we don't want to allow such abominations such as
1 + 1 - 1 + 1 - 1 + 1... :o ;)

Actually I think that there is more to the sine example. Even if you don't know anything about Pi, as the function goes from positive to negative at non-zero values of x, there clearly must be a value for which this infinite power series is exactly zero for a non-zero value of x, i.e. an infinite sum is identically equal to zero. You may not know what that value of x is, but at least you know that it exists. (This is a bit rambling, but our newborn is teaching me the effects of sleep deprivation. ;) )

[Disinfo Agent]

The ONLY difference between 0.999 ... recurring and 1
is PRECISELY THE FACT that 1 IS the LARGER number,
by an infinitely small amount.

What is the difference between zero and an infinitely small amount?
Well, for one: you can divide by an infinitely small amount, ordinarily
producing an infinitely large quotient. You may never divide by zero.

And that's OFFICIAL !!! [-X
It's possible to define infinitesimal numbers (it's not necessary or standard in mathematics, though!). However, that doesn't solve the matter.

Even within nonstandard analysis, if the difference between 0.999... and 1 is infinitely small, then either they are the same real number, or one of them is not real. See pages 35 - 36 of this textbook chapter (http://www.math.wisc.edu/%7Ekeisler/chapter_1.pdf) (pdf), especially remark (3) after definition 1.

Now, 0.999... is usually defined as the limit of the sequence (0.9, 0.99, 0.999, ...), and limits by definition only comprise the standard (i.e. real) part of a hyperreal (see page 16 of this chapter (http://www.math.wisc.edu/%7Ekeisler/chapter_3.pdf) (pdf)). Therefore, 0.999... must be real, and thus equal to 1.

On the construction of hyperreals. (http://mathforum.org/dr.math/faq/analysis_hyperreals.html)

[martin]

Actually I think that there is more to the sine example. Even if you don't know anything about Pi, as the function goes from positive to negative at non-zero values of x, there clearly must be a value for which this infinite power series is exactly zero for a non-zero value of x, i.e. an infinite sum is identically equal to zero. You may not know what that value of x is, but at least you know that it exists. (This is a bit rambling, but our newborn is teaching me the effects of sleep deprivation. ;) )

This is a good example. But only if we are accepting standard axioms of mathematics. Maybe some of the people here are saying the value of sin function for pi is infinitely close to zero but not equal. :(

Martin

[martin]

Now, 0.999... is usually defined as the limit of the sequence (0.9, 0.99, 0.999, ...), and limits by definition only comprise the standard (i.e. real) part of a hyperreal (see page 16 of this chapter (http://www.math.wisc.edu/%7Ekeisler/chapter_3.pdf) (pdf)). Therefore, 0.999... must be real, and thus equal to 1.

On the construction of hyperreals. (http://mathforum.org/dr.math/faq/analysis_hyperreals.html)

From this book, it seems hyperreals form totally ordered field, but do not have completeness property. So properties relying on completeness (like existence of supremum and infinum) are lost. But I only read for a few minutes...

Martin

[Richard of Chelmsford]

Duuurrr!! :cry: :cry: :cry:

I failed maths at school four times, then bought a calculator.

So why are these two figures equal?? :oops:

[Fortis]

Duuurrr!! :cry: :cry: :cry:

I failed maths at school four times, then bought a calculator.

So why are these two figures equal?? :oops:
In short, because in the limit that the number of 9s after the decimal point tends to infinity (i.e. the value tends to what it really is), the difference between it and 1 tends to zero. :)

[§rv]

Since I have only read the first and last pages of this thread (laziness is prevalent), I don't know if my question has been asked already.

If 0.999* continues to infinity then wouldn't it be more than 1?

[Zjm7891]

Whats larger?
0.999999...
or
1.000000...

Now, 1 is a SEPARATE value from a number infinitly close to 1. I don't see how you guys can say they are the SAME

simple example

lim 1-x/1-x
x -> 1

The answer as you get infinitly close to 1 will be 1 but the moment you reach 1 your value is undefined (however, it is a removeable discontinuity but a discontinuity all the same)

[VTBoy]

Whats larger?
0.999999...
or
1.000000...

Now, 1 is a SEPARATE value from a number infinitly close to 1. I don't see how you guys can say they are the SAME

simple example

lim 1-x/1-x
x -> 1

The answer as you get infinitly close to 1 will be 1 but the moment you reach 1 your value is undefined (however, it is a removeable discontinuity but a discontinuity all the same)

You are implying that
lim (1-x)/(1-x) < 1, but that is not true. Since it is equal to one.
x---> 1

A simple fact is according to the axioms of the reals 0.99999... is equal to 1.


Look at

lim (1-x)/(2-x) = 1, it is not approximatly 1, it is 1. It converges onto 1.
x--->infinity
Very sad that more and more people are voting not equal when they are equal.

[Zjm7891]

Yes, the LIMIT is equal to 1..

However, the problem lies in the fact that if we try to use 1, we infact cannot complete the equation as dividing by 0 is not possible. Therefore we must use a value infinitly CLOSE to 1 to solve the equation (Or simplify but thats different)

[VTBoy]

Yes, the LIMIT is equal to 1..

However, the problem lies in the fact that if we try to use 1, we infact cannot complete the equation as dividing by 0 is not possible. Therefore we must use a value infinitly CLOSE to 1 to solve the equation (Or simplify but thats different)

You are misunderstanding what a limit is. The limit isn't just very close to 1, it is one. A value infinitly close to 1 is equal to 1. No difference. My proof of 0.999999=1 used limits, and is very valid. There is a reason why it uses equal signs, because they are all equal to each other. This is my proof.

http://home.comcast.net/~jsacto/Proofwith_Limits2.JPG

[Zjm7891]

If that IS the case, then why can you not use 1? Could it be that you are infering from the fact that as you approach infinitly close to 1 the function approaches infinitly close to 1 aswell therefore you make a presumtion, while perfectly valid, that the function must be equal to 1 at that point?

The function can be determined at 1. However, you can determine what it should be at 1 by testing a value as it is approached. Therefore the values of the function are different. If what your saying is true, then why at 1 is the function undefined but at 1 it equals 1?

[VTBoy]

If that IS the case, then why can you not use 1? Could it be that you are infering from the fact that as you approach infinitly close to 1 the function approaches infinitly close to 1 aswell therefore you make a presumtion, while perfectly valid, that the function must be equal to 1 at that point?

The function can be determined at 1. However, you can determine what it should be at 1 by testing a value as it is approached. Therefore the values of the function are different. If what your saying is true, then why at 1 is the function undefined but at 1 it equals 1?

If it didn't equal 1 then math as we know crumples, or atleast the axioms of the reals do, since according to the axioms of the real it must equal 1. If you want to make your own little axioms and base up you can, but modern math is done using real and the axioms of the real. Also in this "new" math you have created things like Calculus would no longer work since you are invaldating them.

[mutineer]

If 0.999* continues to infinity then wouldn't it be more than 1?I enjoyed that mutinous thought! :D But it’s not actually that much dafter than the idea that 0.99999… and 1 are equal.

Any fraction beginning zero-point-whatever is, so to speak, by definition less than unity. Those who propose that 0.99999… and 1 are equal are merely showing that they are EQUIVALENT FOR COMPUTATIONAL PURPOSES. Of course they are! And to any number of decimal points of accuracy anyone is ever likely to require. I do not dispute it, in fact I still insist upon it. As I said before, the ONLY difference is that 1 is greater by an infinitely small amount.

Until the thought police outlaw infinitesimal thoughts that will remain the case! :evil:


Very sad that more and more people are voting not equal when they are equal.It is sad that the simple truth cannot always be perceived – especially when the truth is simpler than the falsehood.


math as we know crumplesSteady on, old chap! :o Don't get alarmed! Keep on taking the tablets! :o

[Fortis]

Any fraction beginning zero-point-whatever is, so to speak, by definition less than unity.
Why? Where is it defined to be the case? Do you agree that the infinite series
1 + 1/2 + 1/4 + 1/8 + 1/16 + ... = 2 ?

The difference between the truncated form of 0.999... and 1 tends to zero as the number of digits increases. So in the limit that the number of digits becomes infinite, the difference becomes zero. (As has been mentioned previously, "What number falls between 0.999... and 1?")



Those who propose that 0.99999… and 1 are equal are merely showing that they are EQUIVALENT FOR COMPUTATIONAL PURPOSES. Of course they are! And to any number of decimal points of accuracy anyone is ever likely to require. I do not dispute it, in fact I still insist upon it. As I said before, the ONLY difference is that 1 is greater by an infinitely small amount.

It is a standard mathematical result that they are identical, not just equivalent for computational purposes. Thinking about infinite sums can seem tricky at first, but (as long as they are convergent) they are well defined beasts.



Until the thought police outlaw infinitesimal thoughts that will remain the case! :evil:
Have a thought about the following problem.

"What is the integral of x between 0 and 1?" Well, it's just a triangle so it's obviously 1/2.

Now ask yourself what the integral of x^2 is between 0 and 1.
This is defined in terms of the limit of infinite sum, and the result is 1/3. Not approximately equal to 1/3 (or equivalent to 1/3 for computational purposes), but equal to 1/3.

If limits of infinite series didn't exist, then an equals sign would be a rare sight in anything other than simple algebra. ;) :)

[VTBoy]

If 0.999* continues to infinity then wouldn't it be more than 1?I enjoyed that mutinous thought! :D But it’s not actually that much dafter than the idea that 0.99999… and 1 are equal.

Any fraction beginning zero-point-whatever is, so to speak, by definition less than unity. Those who propose that 0.99999… and 1 are equal are merely showing that they are EQUIVALENT FOR COMPUTATIONAL PURPOSES. Of course they are! And to any number of decimal points of accuracy anyone is ever likely to require. I do not dispute it, in fact I still insist upon it. As I said before, the ONLY difference is that 1 is greater by an infinitely small amount.

Until the thought police outlaw infinitesimal thoughts that will remain the case! :evil:


Very sad that more and more people are voting not equal when they are equal.It is sad that the simple truth cannot always be perceived – especially when the truth is simpler than the falsehood.


math as we know crumplesSteady on, old chap! :o Don't get alarmed! Keep on taking the tablets! :o

The fact is if we couldn't do the limit of infinites and covergence was forbiden then Calculus as we know it would fall apart. We do math based on the reals and the axiom of the reals. In all fields of modern math, and the axioms used 0.999999999...=1. So please tell me what is this new field where 0.9999999... does not equal 1 and what are the axioms that define this field. Again thins like Calculus would not work in this new field. It is not merly equal for computational purposes, according to the axioms of the real they are equal in all senses.

[martin]

If 0.999* continues to infinity then wouldn't it be more than 1?I enjoyed that mutinous thought! :D But it’s not actually that much dafter than the idea that 0.99999… and 1 are equal.

Let us see who is "daft" and who is not, and who below is "thought police" and who is not.


Any fraction beginning zero-point-whatever is, so to speak, by definition less than unity.

Maybe in your definition. Not in definition of the real number system.


Those who propose that 0.99999… and 1 are equal are merely showing that they are EQUIVALENT FOR COMPUTATIONAL PURPOSES.

Maybe some are showing this. I am saying they are equal by definition.


As I said before, the ONLY difference is that 1 is greater by an infinitely small amount.

Yes, by definition of real numbers, 1 is greater by an infinitely small amount called zero. So they are equal. In mathematical system you are using, I do not know...


Until the thought police outlaw infinitesimal thoughts that will remain the case! :evil:

When you are saying real number system is "daft," who is thought police?

I am saying from beginning that anyone can create alternate number system if he likes. But alternate number system in which 0.999... is not equal to one disobeys many basic axioms of mathematics. This is price to pay. There are no thought police preventing you from using system of mathematics with infinitesmal numbers. What can you do with these, I do not know...field of mathematics has rejected infinitesmals long time ago because they are not useful for anything and causing all kinds of problems.



math as we know crumplesSteady on, old chap! :o Don't get alarmed! Keep on taking the tablets! :o

I do not need any tablets for this, I can use real numbers where 0.999... and one are equal, there is no problem. People who are using alternate number systems where 0.999... and one are not equal and cannot do integration, maybe they are needing some tablets...When thought police are outlawing real number system because it is "daft," maybe I am needing some tablets also...

If you are strong believer that 0.999... and 1 are different, maybe you can develop mathematical system based on this belief and show us how useful it is? Someone before is citing a book on hyperreal system with infinitesmal numbers (but even this person is saying 0.999... is equal to one). This should be good starting point.

Martin 8)

[Disinfo Agent]

Now, 1 is a SEPARATE value from a number infinitly close to 1.
Not necessarily. A number infinitely close to another may simply be the same number.


However, the problem lies in the fact that if we try to use 1, we infact cannot complete the equation as dividing by 0 is not possible. Therefore we must use a value infinitly CLOSE to 1 to solve the equation (Or simplify but thats different)
It's not an equation; it's an expression. So we don't solve it; we compute it.

You chose an example where the function is not defined for x=1. But, in general, limits also make sense when the function is already defined for x=1. If the function is continuous, then the limit as x->1 will equal f(1). Note: equal, not approximate. E.g.:

f(x)= (x-1)/(x+1) has the limit 0 as x->1.

Also, f(1)=0.


Until the thought police outlaw infinitesimal thoughts that will remain the case! :evil:
As I wrote in the previous page, infinitesimals (when properly defined, not just manipulated intuitively) do not agree with what you're saying.


The problem, in my opinion, is that you guys are trying to use your intuition to figure out what 0.999~ is. You're imagining the number being written down, one 9 after another 9. Since there will always be nines, how can the number suddenly become 1.000~?

But this is a case where intuition fails us. (Note that we're dealing with infinitely many nines!) When we do define 0.999~ in a rigorous way, we come to the conclusion that it's the same number as 1 after all. "0.999~" and "1" are just two ways of writing the same number, just as "2/4" and "3/6" are two ways of writing 0.5.

P.S.

I do not know...field of mathematics has rejected infinitesmals long time ago because they are not useful for anything and causing all kinds of problems.

Actually, infinitesimals (now that we've found ways of defining them rigorously) can make some proofs shorter and more intuitive. :)

[A Thousand Pardons]

If that IS the case, then why can you not use 1? Could it be that you are infering from the fact that as you approach infinitly close to 1 the function approaches infinitly close to 1 aswell therefore you make a presumtion, while perfectly valid, that the function must be equal to 1 at that point?

I had to go back to make sure which function you were talking about, but as near as I can tell, we are not saying that the function is equal to one, we are saying that the limit is equal to one. Big difference.

And the definition of 0.999... is the limit as the number of 9's grows to infinity. That is, it is equal to one.

[mutineer]

Do you agree that the infinite series
1 + 1/2 + 1/4 + 1/8 + 1/16 + ... = 2 ?But THAT is about COMPUTATION.

The fraction 0.999 recurring gives numerical expression to the idea of an amount less than unity by an infinitely small amount. Ordinary people understand the idea of an infinitely small amount – insofar as it is within human comprehension. They understand the idea of a cake being divided into an infinite number of infinitely small crumbs. They cannot imagine it, but they can understand it as a limiting case.

And what is wrong with that? Does anyone wish to BAN such thoughts? Are they a threat to the integrity of mathematics? Would you really rather not contemplate that cake? Does it make you feel uneasy? Does the very idea of the infinitesimal as the reciprocal of infinity make you shudder?

Numbers have their highest and defining expression in the realm of human psychology wherein lies their creation. In their purest form they cannot be overridden by any particular set of axioms necessary to support calculus, trigonometry, or other form of applied computation.


If you are strong believer that 0.999... and 1 are different, maybe you can develop mathematical system based on this belief and show us how useful it is?Asking this question shows that you have completely missed the point. I am not talking about "useful". I have already INSISTED that from the point of view of COMPUTATION they two numbers are equivalent.

Now then, if a new integer were discovered, say, more than 836 but less than 837 . . . ! THAT is the sort of thing that would really threaten mathematics. It might turn out to account both for the precession of Mercury and for the anomalous acceleration of Pioneer 11. :D But probably not. On the whole, it would be a devastating blow – both psychologically and mathematically! :evil: Let us hope it never happens. 8)

[Damburger]

The fraction 0.999 recurring gives numerical expression to the idea of an amount less than unity by an infinitely small amount. Ordinary people understand the idea of an infinitely small amount – insofar as it is within human comprehension. They understand the idea of a cake being divided into an infinite number of infinitely small crumbs. They cannot imagine it, but they can understand it as a limiting case.


If 'ordinary people' believe that 0.999r=1, then ordinary people are simply wrong.

Common sense is no substitute for mathematical proof.

[Disinfo Agent]

The fraction 0.999 recurring gives numerical expression to the idea of an amount less than unity by an infinitely small amount.
Maybe you got that idea, but it's not what mathematicians mean by that expression. Unless by infinitely small you mean "zero".


Ordinary people understand the idea of an infinitely small amount – insofar as it is within human comprehension. They understand the idea of a cake being divided into an infinite number of infinitely small crumbs.They cannot imagine it, but they can understand it as a limiting case.
So why do you have a problem with 0.999~ as a limit?


Now then, if a new integer were discovered, say, more than 836 but less than 837 . . . ! THAT is the sort of thing that would really threaten mathematics.
Alas, mathematicians can prove that there is no such integer, just as they can prove that 0.999~ and 1 are one and the same.

[A Thousand Pardons]

Do you agree that the infinite series
1 + 1/2 + 1/4 + 1/8 + 1/16 + ... = 2 ?But THAT is about COMPUTATION.

Yahbut, 0.999... refers to the infinite series, 9/10 + 9/100 + 9/1000 + ..., which is equal to 1.


And what is wrong with that? Does anyone wish to BAN such thoughts? Are they a threat to the integrity of mathematics? Would you really rather not contemplate that cake? Does it make you feel uneasy? Does the very idea of the infinitesimal as the reciprocal of infinity make you shudder?
No, but having dealt with it for forty years, I know the dangers involved.

No one on the BABB is trying to ban thoughts, that I know of. :)


The fraction 0.999 recurring gives numerical expression to the idea of an amount less than unity by an infinitely small amount. Ordinary people understand the idea of an infinitely small amount – insofar as it is within human comprehension. They understand the idea of a cake being divided into an infinite number of infinitely small crumbs. They cannot imagine it, but they can understand it as a limiting case.


If 'ordinary people' believe that 0.999r=1, then ordinary people are simply wrong.

Common sense is no substitute for mathematical proof.
Having read some of your previous posts (http://www.badastronomy.com/phpBB/viewtopic.php?p=373030#373030), I'm pretty sure you meant "If 'ordinary people' believe that 0.999r not equal 1, then ordinary people are simply wrong," is that correct?

[pghnative]

Do you agree that the infinite series
1 + 1/2 + 1/4 + 1/8 + 1/16 + ... = 2 ?But THAT is about COMPUTATION.Hmmmm. So you agree that 1 + 1/2 + 1/4 + 1/8 + ... = 2. Subtracting 1 from each side gives:

1/2 + 1/4 + 1/8 + 1/16+1/32 + etc... = 1. Presumably you agree with that also.

So let's see. 1/2 + 1/4 = 0.75.

Adding 1/8 (0.125) gives 0.875
Continuing for seven more terms gives 1.9990 (I rounded this)
Continuing for 20 more terms gives 1.9999999991 (also rounded)

Do you see the pattern? Continuing for an infinte number of terms gives 1.999... (that is 1.999 with the 9's infinitely repeating.)

So if you agree that 1 + 1/2 + 1/4 + 1/8 (etc) = 2, then apparently you agree that 0.999... = 1.

Imagine that. And it's proven via COMPUTATION!! :D

[Disinfo Agent]

Or how about this? :)

Imagine a particle moving along the x-axis, in the interval [0, 1], from 0 to 1. When the particle reaches the abscissa x=0.9, it has only a tenth of a unit left to go. When it reaches x=0.99, it has only a hundredth of a unit left to go. When it reaches x=0.999, it has only a thousandth of a unit left to go... When it reaches 0.999...9 = 0.9 * (1 + 0.1 + 0.01 + ... + 0.1^n), it has only a 10^(n+1)-th of a unit left to go.

By adding enough nines, each submultiple of the unit is eventually covered. When you place all the nines after the decimal point, all the submultiples are covered, which means the particle has reached its destination.

(This is not a proof, just a motivation.)

[Edited wrong word.]

[martin]

I do not know...field of mathematics has rejected infinitesmals long time ago because they are not useful for anything and causing all kinds of problems.

Actually, infinitesimals (now that we've found ways of defining them rigorously) can make some proofs shorter and more intuitive. :)

You are right, I am speaking wrongly on this point :D But infinitesmals are not behaving like people here are saying...

Martin

[martin]

Do you agree that the infinite series
1 + 1/2 + 1/4 + 1/8 + 1/16 + ... = 2 ?But THAT is about COMPUTATION.

It is about definition. Infinite series are defined by limits in conventional mathematics. In your mathematics, I do not know, because you are not telling us...


The fraction 0.999 recurring gives numerical expression to the idea of an amount less than unity by an infinitely small amount. Ordinary people understand the idea of an infinitely small amount – insofar as it is within human comprehension. They understand the idea of a cake being divided into an infinite number of infinitely small crumbs. They cannot imagine it, but they can understand it as a limiting case.

Ordinary people here are having great difficulty with limits. Ordinary people are not explaining rules followed by infinitesmal numbers. Disinfo has posted link to book about hyperreal system containing infinitely small numbers. In hyperreal system, 0.999... is equal to one.


And what is wrong with that? Does anyone wish to BAN such thoughts?

I do not want to ban such thoughts. I am not even saying they are "daft," as you are saying people who understand properties of real number system are daft.


Are they a threat to the integrity of mathematics? Would you really rather not contemplate that cake?

If someone is using non-standard number system, this is no threat to integrity of mathematics at all. Maybe it is even useful for some things. Mathematicians are contemplating cake at all times in lebesgue measure theory, in way always consistent with properties of real numbers and with no infinitely small number different from zero. Which part of lebesgue measure theory do you think is wrong? Is lebesgue measure theory "daft"?

Mathematicians are contemplating that cake at all times. Book of billingsley Probability and Measure is good reference. No infinitesmal numbers can be found there. Or would you really rather not contemplate that cake?


Does it make you feel uneasy? Does the very idea of the infinitesimal as the reciprocal of infinity make you shudder?

Who is shuddering? Hyperreal system cited by Disinfo has this. I am not shuddering on this. And in hyperreal system 0.999... is equal to one. Do properties of real numbers make you shudder? Do properties of hyperreal numbers make you shudder?



If you are strong believer that 0.999... and 1 are different, maybe you can develop mathematical system based on this belief and show us how useful it is?Asking this question shows that you have completely missed the point. I am not talking about "useful". I have already INSISTED that from the point of view of COMPUTATION they two numbers are equivalent.

Making this statement shows that you have completely missed the point. I am not talking about computation, I am talking about definition. What are rules followed by infinitely small numbers different from zero? I am always asking this question, and no one is answering. Please tell us how these numbers are behaving. Many people have posted correct definition of real numbers, others are saying these people are "daft" but they are not providing any alternate definition or showing us rules of new mathematical system. Someone else is posting hyperreal number system, and still 0.999... is equal to one. Others are saying we are "daft," but they are not telling us anything about mathematics that would not be daft.


Now then, if a new integer were discovered, say, more than 836 but less than 837 . . . ! THAT is the sort of thing that would really threaten mathematics. It might turn out to account both for the precession of Mercury and for the anomalous acceleration of Pioneer 11. :D But probably not. On the whole, it would be a devastating blow – both psychologically and mathematically! :evil: Let us hope it never happens. 8)

Anyone can define new number between 836 and 837, and call it "integer." This is no threat to mathematics at all, even if he is calling everyone not using this number "daft."

Martin

[VTBoy]

If 0.99999... did not equal exactly 1 then many of the ideas and concepts in Astronomy and Physics must be questioned, since Astronomy and Physics rely heavily on Calculus, and if 0.9999999... is only equal to 1 for computational purposes then Calculus is not valid for application and using it in Physics and Astronomy is no longer valid.

[Normandy6644]

Do you agree that the infinite series
1 + 1/2 + 1/4 + 1/8 + 1/16 + ... = 2 ?But THAT is about COMPUTATION.

I don't agree with this at all. The fact that you're doing an infinite sum means by definition it isn't computation since you could never computer all the terms. That's why limits and series are so powerful, because they give the correct result (albeit non-intuitive and slightly weird) without the work of an infinite number of terms!

[Fortis]

I still think that the fact that no number exists between 0.9999... and 1 should suggest to people that they are one and the same. :)

[Chuck]

Anyone who believes that 0.99999... is less than 1 can test out the theory. Lower your head and run toward a brick wall. Before your head can get to the wall it has to go 9/10 of the distance, and then 9/10 of what remains, and then 9/10 of what remain after that, continuing forever. When you're done your head will still be an infinitesimal distance from the wall so this test is perfectly safe.

[ZaphodBeeblebrox]

Anyone who believes that 0.99999... is less than 1 can test out the theory. Lower your head and run toward a brick wall. Before your head can get to the wall it has to go 9/10 of the distance, and then 9/10 of what remains, and then 9/10 of what remain after that, continuing forever. When you're done your head will still be an infinitesimal distance from the wall so this test is perfectly safe.

OW!!!!

WHAT THE [Bad Word Deleted] [Bad Word Deleted] WAS I THINKING???

Ok, I Concede the Point ...

[martin]

Anyone who believes that 0.99999... is less than 1 can test out the theory. Lower your head and run toward a brick wall. Before your head can get to the wall it has to go 9/10 of the distance, and then 9/10 of what remains, and then 9/10 of what remain after that, continuing forever. When you're done your head will still be an infinitesimal distance from the wall so this test is perfectly safe.

OW!!!!

WHAT THE [Bad Word Deleted] [Bad Word Deleted] WAS I THINKING???

Ok, I Concede the Point ...

Be careful. President of galaxy should not hurt his head like this...

Martin

[ZaphodBeeblebrox]

Anyone who believes that 0.99999... is less than 1 can test out the theory. Lower your head and run toward a brick wall. Before your head can get to the wall it has to go 9/10 of the distance, and then 9/10 of what remains, and then 9/10 of what remain after that, continuing forever. When you're done your head will still be an infinitesimal distance from the wall so this test is perfectly safe.

OW!!!!

WHAT THE [Bad Word Deleted] [Bad Word Deleted] WAS I THINKING???

Ok, I Concede the Point ...

Be careful. President of galaxy should not hurt his head like this...

Martin

Eh ...

No worries, it was only My Right One.

Me NEVER Tinks wid Dat One ...

[gzhpcu]

You all can come up with all the fancy math you like, but it will never change the fact that 0.9999... and 1 are infinitely close, but never, never equal. Instead of using math, try and use logic.

typo

[VTBoy]

You all can come up with all the fancy math you like, but it will never change the fact that 0.9999... and 1 are infinitely close, but never, never equal. Instead of using math, try and use logic.

typo

WoW. To ignore math is to discredit astronomy, physics, and countless others sciences. If you ignore mathematical facts things like Nancy PX may really exist. Also logic tells us that it is equal to 1.

[Careless]

You all can come up with all the fancy math you like, but it will never change the fact that 0.9999... and 1 are infinitely close, but never, never equal. Instead of using math, try and use logic.

typo
Math is, generally speaking, logical. Proofs are, by definition, logical.
What you're appealing to is not logic, but common sense

[Normandy6644]

You all can come up with all the fancy math you like, but it will never change the fact that 0.9999... and 1 are infinitely close, but never, never equal. Instead of using math, try and use logic.

typo
Math is, generally speaking, logical. Proofs are, by definition, logical.
What you're appealing to is not logic, but common sense

I'll echo that. If .9999... is anything other than 1, a lot of things are wrong. And they're not.

[A Thousand Pardons]

I still think that the fact that no number exists between 0.9999... and 1 should suggest to people that they are one and the same. :)
I like this Fortis. Maybe we should ask what the average of 0.999... and 1.000... is? :)

I guess that's kinda what Chuck was doing (http://www.badastronomy.com/phpBB/viewtopic.php?p=372461#372461).

You all can come up with all the fancy math you like, but it will never change the fact that 0.9999... and 1 are infinitely close, but never, never equal. Instead of using math, try and use logic.
Try to do the math above, and see what you come up with. :)

It's not very fancy but it seems to work.

[Eroica]

I still think that the fact that no number exists between 0.9999... and 1 should suggest to people that they are one and the same. :)
=D> Nice one!

[gzhpcu]

You all can come up with all the fancy math you like, but it will never change the fact that 0.9999... and 1 are infinitely close, but never, never equal. Instead of using math, try and use logic.

typo

WoW. To ignore math is to discredit astronomy, physics, and countless others sciences. If you ignore mathematical facts things like Nancy PX may really exist. Also logic tells us that it is equal to 1.

You must be kidding. Math is fine, but not to be taken literally. Math brought us 10 spatial dimensions, vibrating strings and colliding branes. Math is OK for approximations. Not for exactitude. Certainly not in this case... :)

[Fortis]

You all can come up with all the fancy math you like, but it will never change the fact that 0.9999... and 1 are infinitely close, but never, never equal. Instead of using math, try and use logic.

typo

WoW. To ignore math is to discredit astronomy, physics, and countless others sciences. If you ignore mathematical facts things like Nancy PX may really exist. Also logic tells us that it is equal to 1.

You must be kidding. Math is fine, but not to be taken literally. Math brought us 10 spatial dimensions, vibrating strings and colliding branes. Math is OK for approximations. Not for exactitude. Certainly not in this case... :)
The above is theoretical physics, not pure mathematics. 1=0.9999..., however, is pure mathematics and is a rigorous, and exact, result. :)

[Fortis]

I still think that the fact that no number exists between 0.9999... and 1 should suggest to people that they are one and the same. :)
I like this Fortis. Maybe we should ask what the average of 0.999... and 1.000... is? :)

I guess that's kinda what Chuck was doing (http://www.badastronomy.com/phpBB/viewtopic.php?p=372461#372461).
My comment was really just re-expressing (though in a slightly more general way) what Chuck had said. It's still a real killer for those who believe that the two numbers are different. It'd take a very sharp knife to slice up the real number line and leave 1.0 and 0.9999... on either side. :)

[mutineer]

There’s nane sae blind as those that canna see.A lot of people obviously find it almost impossible to contemplate the infinitely large and infinitely small in LOGICAL terms. Mathematics is merely the name we give to various tools, which enable us to manipulate numbers in order to perform computations. These tools have rules, and we employ logic to follow the rules – but logic ultimately stands above mathematics, because the rules of the latter are only as good as they need to be for the task in hand.

Now infinity is a very difficult thing to think about, although it is also very mundane. For example, the concept of forever invokes the concept of infinity – but we use the word in everyday conversation without contemplating its awesomeness.

Try to follow the following mental exercise. Imagine as a given fact that a certain event will take place at an indefinite point of time in the future, and that beyond that we possess no further information. Suppose also that the future lasts forever. Then the chance of that event taking place on May 23 784896758435579AD is infinity-to-one against. In fact, the chances of it happening as soon as that are also infinity-to-one against.

And yet, the chance of the event happening on that date are (as far as we know) the same as it happening on any other. And that chance is not zero. However infinitely small that chance, it will not go away – even though time lasts forever. That’s right, isn’t it? Is there anyone who claims that because time lasts forever the chance is zero? Clearly not, because the sum of even an infinity of zeros is still zero – and it is a given that the sum is 1.

It is the same with the string of nines. 0.9, 0.99, 0.999, 0.9999 . . . with every nine we add the discrepancy between the fraction and unity is diminished. Even though the string lasts forever, the discrepancy cannot be vanquished. The logic of this position is unassailable. Anyone who thinks differently has not properly got their brain around the concept of infinity. R.A.F. and gzhpcu are right to have an unshakeable confidence in what the application of logic tells them.

What is so disappointing is that others who resort to mathematical “proofs” that the two numbers are equal do not at least appreciate that they are faced with a dilemma. They chuck away what logic must tell them (if they have seriously thought about the matter) in the face of mere KNOWLEDGE about the application of mathematical techniques, such as calculus. Worse, some of them obviously think themselves superior to ordinary people who, addressing the matter in terms of logic unaided by mathematics, are accused of INTUITION.

For practical purposes of computation, 0.999 … recurring is equal to 1, as I have insisted from the start. That is why the rules of mathematics treat the two values as indistinguishable. They may safely do so. Any number of so-called proofs can therefore be derived to show that the two numbers are equal. There is nothing surprising about this. I fully understand the “proofs” that have been presented. They are mere examples of self-delusion. A LOGICAL PROOF TRUMPS A MATHEMATICAL PROOF.

[gzhpcu]

Anyone who thinks differently has not properly got their brain around the concept of infinity. R.A.F. and gzhpcu are right to have an unshakeable confidence in what the application of logic tells them.

Since this I think is the first time R.A.F. and I ever agreed on ânything, it must undoubtedly serve as a further irrefutable proof that 0.9999... does not equal 1! :D

[martin]

You all can come up with all the fancy math you like, but it will never change the fact that 0.9999... and 1 are infinitely close, but never, never equal. Instead of using math, try and use logic.

typo

I am saying the same thing from beginning. In real number system there is no number infinitely close to one but not equal to one. Real number system is developed rigourously through methods of logic. So people who are saying real number system has no such number are using logic. Many people are posting definition of real number system. Disinfo is posting information about hyperreal number system, developed by rigourous analytic logic. In hyperreal number system 0.999... is equal to one.

Someone can develop alternate number system from real numbers and hyperreals, and maybe in this number system 0.999... is not equal to one. But no one claiming this is posting any definition of this number system, rules which it obeys, or anything like this. They are not using math or logic. They are using only fuzzy intuition.

Martin

[martin]

What is so disappointing is that others who resort to mathematical “proofs” that the two numbers are equal do not at least appreciate that they are faced with a dilemma. They chuck away what logic must tell them (if they have seriously thought about the matter) in the face of mere KNOWLEDGE about the application of mathematical techniques, such as calculus. Worse, some of them obviously think themselves superior to ordinary people who, addressing the matter in terms of logic unaided by mathematics, are accused of INTUITION.

So the person who is saying others are "daft" is complaining that others think themselves superior?


For practical purposes of computation, 0.999 … recurring is equal to 1, as I have insisted from the start. That is why the rules of mathematics treat the two values as indistinguishable. They may safely do so. Any number of so-called proofs can therefore be derived to show that the two numbers are equal. There is nothing surprising about this. I fully understand the “proofs” that have been presented. They are mere examples of self-delusion. A LOGICAL PROOF TRUMPS A MATHEMATICAL PROOF.

I am saying always the same thing. Real number system has no such number. Hyperreal number system posted by Disinfo has infinitely small numbers, and still 0.999... and 1 are equal. Maybe you have some other number system, but so far you are not telling us its definition or rules it follows, or why real number system is "daft." Where is your logical proof of the daftness of real number system? What are rules of your number system? I am asking this over and over, and no one is saying...

Martin

[Stregone]

An infinite amount of something is nothing like an infinitly repeating decimal. 0.9... never gets larger than 1 even though the 9's go on infinitly(not indefinitly). An infinite number of apples or years is infinitly larger than 1. And infinity is infinite, it doesn't even end if the universe comes to an end. It keeps going, infinitly, forever no matter what.

So, if you are infinitly close to something then you are already there.

[VTBoy]

You all can come up with all the fancy math you like, but it will never change the fact that 0.9999... and 1 are infinitely close, but never, never equal. Instead of using math, try and use logic.

typo

WoW. To ignore math is to discredit astronomy, physics, and countless others sciences. If you ignore mathematical facts things like Nancy PX may really exist. Also logic tells us that it is equal to 1.

You must be kidding. Math is fine, but not to be taken literally. Math brought us 10 spatial dimensions, vibrating strings and colliding branes. Math is OK for approximations. Not for exactitude. Certainly not in this case... :)

You are saying we can only find approximations using math. If everything in Science is nothing more than an approximation then we must question it, since being an approximation it could be very wrong.

[gzhpcu]

You are saying we can only find approximations using math. If everything in Science is nothing more than an approximation then we must question it, since being an approximation it could be very wrong.

Not necessarily wrong, but just an approximation. Look at the many open questions in physics, for example. From QM to SS and now to M-theory (10 spatial dimensions, 5 dimensional branes, open and closed vibration strings, for Pete's sake...) and cosmology is not much better off with inflationary universe, dark energy, etc.

[mutineer]

Where is your logical proof of the daftness of real number system? What are rules of your number system?
Ah, Martin! Rules, rules, rules . . . but I know what it’s like in Singapore. (A professor at NTU is an old friend.) Numbers have a natural meaning and value over and above any rules that need to be applied for purposes of computation.

But I apologize for wagging my finger in my first post, and using the word ‘daft’ in the next. Thing is, I did want to provoke a reaction and keep this thread going. Reading it gives interesting insights into the way people think. I suspect that there may have been a slight tendency for more numerate BABBers to vote what I consider to be the wrong way. (Beyond the average, I mean.) A year of two ago I posted an IQ question where getting the right answer showed a negative correlation with intelligence. All interesting stuff for a retired psychologist.

[snowcelt]

Where is your logical proof of the daftness of real number system? What are rules of your number system?
Ah, Martin! Rules, rules, rules . . . but I know what it’s like in Singapore. (A professor at NTU is an old friend.) Numbers have a natural meaning and value over and above any rules that need to be applied for purposes of computation.

But I apologize for wagging my finger in my first post, and using the word ‘daft’ in the next. Thing is, I did want to provoke a reaction and keep this thread going. Reading it gives interesting insights into the way people think. I suspect that there may have been a slight tendency for more numerate BABBers to vote what I consider to be the wrong way. (Beyond the average, I mean.) A year of two ago I posted an IQ question where getting the right answer showed a negative correlation



with intelligence. All interesting stuff for a retired psychologist.

What! Are you saying that whatever someone says it it wrong because they are too smart? Or too dumb?

Just when I thought it was safe to return to the library.

[mutineer]

The Learned reach for their learning before they reach for their logic. Remember that, O snowcelt, when you emerge from the library even more Learned than before!

[A Thousand Pardons]

Try to follow the following mental exercise. Imagine as a given fact that a certain event will take place at an indefinite point of time in the future, and that beyond that we possess no further information. Suppose also that the future lasts forever. Then the chance of that event taking place on May 23 784896758435579AD is infinity-to-one against. In fact, the chances of it happening as soon as that are also infinity-to-one against.

And yet, the chance of the event happening on that date are (as far as we know) the same as it happening on any other. And that chance is not zero. However infinitely small that chance, it will not go away – even though time lasts forever. That’s right, isn’t it? Is there anyone who claims that because time lasts forever the chance is zero? Clearly not, because the sum of even an infinity of zeros is still zero – and it is a given that the sum is 1.

This is an interesting approach--and you don't have to go out to infinity to have the same issues crop up. Just imagine taking a random real number from zero to one. What is the probability of any single number? It is zero, and yet, if you chose a number, there will be a number chosen.

Mathematically, that is expressed by a probability distribution. The graph of this distribution is a square one unit high, from zero to one. That it is flat on top represents the notion that it is a uniform probability distribution. It is a square, and its area is one (as is true for all probability distributions.)

What is the area of that square if you leave off a single point? Say, (1,0)? Since the mathematical area of a point is zero, leaving off the point still allows the remaining area to be equal to one. What about leaving off the boundaries entirely--the four lines at the top, bottom, left, and right? Still, the area of a line is zero, so the area that is left is still one.

Now, just leave off the right most line.

And take the sum of the successive rectangles whose widths are 9/10 and 9/100 and 9/1000... The sum of all of those will fill that square, without the rightmost line. Every other point will be in one of those rectangles. The sum of their area will still be one.

That's the mathematical (and logical) argument.

[Fortis]

And just to show that it isn't just a few of us on this board that are happy with 0.999...=1.
http://mathforum.org/dr.math/faq/faq.0.9999.html
(It even has a few references that you can look up.)
:)

[martin]

Where is your logical proof of the daftness of real number system? What are rules of your number system?
Ah, Martin! Rules, rules, rules . . . but I know what it’s like in Singapore. (A professor at NTU is an old friend.) Numbers have a natural meaning and value over and above any rules that need to be applied for purposes of computation.

But I apologize for wagging my finger in my first post, and using the word ‘daft’ in the next. Thing is, I did want to provoke a reaction and keep this thread going. Reading it gives interesting insights into the way people think. I suspect that there may have been a slight tendency for more numerate BABBers to vote what I consider to be the wrong way. (Beyond the average, I mean.) A year of two ago I posted an IQ question where getting the right answer showed a negative correlation with intelligence. All interesting stuff for a retired psychologist.

What is right answer to this question depends on definition. This is first sentence in my first post.

I am saying only following:

1. In real number system, 0.999... is equal to one. This follows from application of rules of logic and standard definition of real number system.

2. Real number system is not only number system one can be inventing. Maybe in some other number system, 0.999... is not equal to one. But if people are saying this, what is system they are using? How do numbers in such system behave?

Others are saying inequality of 0.999... and 1 is "fact." If someone is saying this, then I do not know what numbers they are using, because in real number system this is not so. So what is number system they are using? How do numbers in this system behave? How can I say if 0.999... and 1 are equal or not equal if they are not using standard definition, but are not telling me what alternate definition is?

Martin

[gzhpcu]

Instead of pedantically coming up with quotations of definitions and formulas, just try and use your heads for a change: 0.9999..... is infinitely close to, but never equal to 1.

This does not seem to be so difficult to understand, but apparently it is.

[martin]

Try to follow the following mental exercise. Imagine as a given fact that a certain event will take place at an indefinite point of time in the future, and that beyond that we possess no further information. Suppose also that the future lasts forever. Then the chance of that event taking place on May 23 784896758435579AD is infinity-to-one against. In fact, the chances of it happening as soon as that are also infinity-to-one against.

And yet, the chance of the event happening on that date are (as far as we know) the same as it happening on any other. And that chance is not zero. However infinitely small that chance, it will not go away – even though time lasts forever. That’s right, isn’t it? Is there anyone who claims that because time lasts forever the chance is zero? Clearly not, because the sum of even an infinity of zeros is still zero – and it is a given that the sum is 1.

This is an interesting approach--and you don't have to go out to infinity to have the same issues crop up. Just imagine taking a random real number from zero to one. What is the probability of any single number? It is zero, and yet, if you chose a number, there will be a number chosen.

Mathematically, that is expressed by a probability distribution. The graph of this distribution is a square one unit high, from zero to one. That it is flat on top represents the notion that it is a uniform probability distribution. It is a square, and its area is one (as is true for all probability distributions.)

What is the area of that square if you leave off a single point? Say, (1,0)? Since the mathematical area of a point is zero, leaving off the point still allows the remaining area to be equal to one. What about leaving off the boundaries entirely--the four lines at the top, bottom, left, and right? Still, the area of a line is zero, so the area that is left is still one.

Now, just leave off the right most line.

And take the sum of the successive rectangles whose widths are 9/10 and 9/100 and 9/1000... The sum of all of those will fill that square, without the rightmost line. Every other point will be in one of those rectangles. The sum of their area will still be one.

That's the mathematical (and logical) argument.

This is subject of lebesgue measure theory. The big problem people are always making mistakes on is that real interval from 0 to 1 cannot be constructed from points even with infinite sequence of operations. This is shown by diagonal proof (I think from cantor, but maybe someone else). Proof requires strange manipulation to avoid problems from equality of 0.999... and one, but since people here are not believing in this equality, maybe diagonal proof can be simplified for them. :D

Apparent contradiction of uniform probability distribution on interval of real line but zero probability of each point is discussed in detail in measure theory books like of billingsley I am citing before. Difference between countable and non-countable sets is resolution.

Martin

[martin]

Instead of pedantically coming up with quotations of definitions and formulas, just try and use your heads for a change: 0.9999..... is infinitely close to, but never equal to 1.

This does not seem to be so difficult to understand, but apparently it is.

I am saying from beginning it is question of definition of number system. This does not seem to be so difficult to understand, but apparently it is.

In real number system, 0.999... and one are equal as shown by many people in many ways here. You tell others to try using their heads, try using your head instead. Are proofs of absence of infinitesmal in real number system wrong? If so, point out errors. Are you rejecting real number system and using some other number system instead? If so, what is definition of other number system and how are numbers behaving in this system? Show us, since we are pedants who are not using heads. But you will never show us, you will only make more insults.

Martin

[gzhpcu]

We have evidently reached a mexican standoff here. No way you can convince me, and no way I can convince you. Our brains are wired differently. Calculus does not convince me. It is convenient for calculus to treat them as equal, just as it is convenient for QM to treat particles as points, or SS as strings. But it ain't so. :D

[A Thousand Pardons]

Apparent contradiction of uniform probability distribution on interval of real line but zero probability of each point is discussed in detail in measure theory books like of billingsley I am citing before. Difference between countable and non-countable sets is resolution.
No, that makes no difference--as you can have a uniform probability distribution on the rationals, which are countable.

[Fortis]

Instead of pedantically coming up with quotations of definitions and formulas, just try and use your heads for a change: 0.9999..... is infinitely close to, but never equal to 1.

This does not seem to be so difficult to understand, but apparently it is.
I think you'll find that we are using our heads. :)
So, what number do you think comes between 0.9999... and 1? ;)

[Chuck]

1 divided by 3 is 0.33333...

0.99999... divided by 3 is 0.33333...

Use long division to verify these if you like.

If 0.99999... doesn't equal 1 then either we have two different numbers that have the same quotient when divided by 3 or we have 0.33333... not being equal to itself. These two conditions are both impossible in current mathematics so 0.99999... must equal 1.

[gzhpcu]

Instead of pedantically coming up with quotations of definitions and formulas, just try and use your heads for a change: 0.9999..... is infinitely close to, but never equal to 1.

This does not seem to be so difficult to understand, but apparently it is.
I think you'll find that we are using our heads. :)
So, what number do you think comes between 0.9999... and 1? ;)

Easy enough: an infinitismally small number... :D

[Normandy6644]

Instead of pedantically coming up with quotations of definitions and formulas, just try and use your heads for a change: 0.9999..... is infinitely close to, but never equal to 1.

This does not seem to be so difficult to understand, but apparently it is.
I think you'll find that we are using our heads. :)
So, what number do you think comes between 0.9999... and 1? ;)

Easy enough: an infinitismally small number... :D

But what are the roles of infinitessimals in calculus? :P

[gzhpcu]

1 divided by 3 is 0.33333...

0.99999... divided by 3 is 0.33333...

Use long division to verify these if you like.

If 0.99999... doesn't equal 1 then either we have two different numbers that have the same quotient when divided by 3 or we have 0.33333... not being equal to itself. These two conditions are both impossible in current mathematics so 0.99999... must equal 1.

No, 1 divided by 3 is not 0.333333.... wrong assumption. Only an approximation. Same problem as we are discussing here.

[gzhpcu]

Instead of pedantically coming up with quotations of definitions and formulas, just try and use your heads for a change: 0.9999..... is infinitely close to, but never equal to 1.

This does not seem to be so difficult to understand, but apparently it is.
I think you'll find that we are using our heads. :)
So, what number do you think comes between 0.9999... and 1? ;)

Easy enough: an infinitismally small number... :D

But what are the roles of infinitessimals in calculus? :P

These are convenient rules for calculus. They do not apply to the real world.

[gzhpcu]

I'll put it another way: just like in cosmology the Planck time creates problems for our physics (and has allowed the proposal of the inflationary universe to circumvent problems with the BB), math conveniently circumvents problems by assuming that 0.99999.... equals 1.

[Grey]

Instead of pedantically coming up with quotations of definitions and formulas, just try and use your heads for a change: 0.9999..... is infinitely close to, but never equal to 1.

This does not seem to be so difficult to understand, but apparently it is.
I think you'll find that we are using our heads. :)
So, what number do you think comes between 0.9999... and 1? ;)

Easy enough: an infinitismally small number... :D

No, he's not asking what number is the difference of the two. He's asking what number comes between them? That is, what is a number larger than 0.999... but smaller than 1?

[Normandy6644]

Instead of pedantically coming up with quotations of definitions and formulas, just try and use your heads for a change: 0.9999..... is infinitely close to, but never equal to 1.

This does not seem to be so difficult to understand, but apparently it is.
I think you'll find that we are using our heads. :)
So, what number do you think comes between 0.9999... and 1? ;)

Easy enough: an infinitismally small number... :D

But what are the roles of infinitessimals in calculus? :P

These are convenient rules for calculus. They do not apply to the real world.

How do you figure? That's negating everything that math does.

I see your argument though. You're saying mathematically, they are equal. However, since you can never add up an infinite number of terms, you never actually reach one. While I agree with you there, saying that if someone set out to add up the terms he would never reach one, I still disagree since I say that the rules developed by mathematics do apply to the real world, even if you personally could never apply them.

[Chuck]

1 divided by 3 is 0.33333...

0.99999... divided by 3 is 0.33333...

Use long division to verify these if you like.

If 0.99999... doesn't equal 1 then either we have two different numbers that have the same quotient when divided by 3 or we have 0.33333... not being equal to itself. These two conditions are both impossible in current mathematics so 0.99999... must equal 1.

No, 1 divided by 3 is not 0.333333.... wrong assumption. Only an approximation. Same problem as we are discussing here.
Dividing 1 by 3 and dividing 0.99999... by 3 both produce a decimal point followed by an infinite string of threes. Are you saying that one of them doesn't? If they don't which one doesn't and what does it produce?

[Fortis]

We have evidently reached a mexican standoff here. No way you can convince me, and no way I can convince you. Our brains are wired differently. Calculus does not convince me. It is convenient for calculus to treat them as equal, just as it is convenient for QM to treat particles as points, or SS as strings. But it ain't so. :D
It isn't just a convenience, it is a logical consequence of the way that real numbers are constructed. As Martin has pointed out, you could build your numbers up from a different set of rules, (and have 0.999... NE 1.0), but they wouldn't be the set of real numbers. :)

[VTBoy]

We have evidently reached a mexican standoff here. No way you can convince me, and no way I can convince you. Our brains are wired differently. Calculus does not convince me. It is convenient for calculus to treat them as equal, just as it is convenient for QM to treat particles as points, or SS as strings. But it ain't so. :D

It is not just convient for Calculus to treat them as equal, because in the set of all reals they are equal to one. If they didn't equal to one then the Axioms are the Reals is invalid, and Calculus is invalid. Having 0.999999...=1 does apply to real world just like Calculus.

Just wondering have you and all of the people saying no taken any courses in Mathematical Proof, Abstract Mathematics, Topology, ect. What is the highest level math course you have taken.

[Donnie B.]

1 divided by 3 is 0.33333...

0.99999... divided by 3 is 0.33333...

Use long division to verify these if you like.

If 0.99999... doesn't equal 1 then either we have two different numbers that have the same quotient when divided by 3 or we have 0.33333... not being equal to itself. These two conditions are both impossible in current mathematics so 0.99999... must equal 1.

No, 1 divided by 3 is not 0.333333.... wrong assumption. Only an approximation. Same problem as we are discussing here.
Astounding. Absolutely astounding. You're actually arguing that 1/3 is NOT 0.3... ? Perhaps you do not understand what the symbol 0.3... represents?

Let's try to approach this from a different angle. Is everyone here comfortable with the concept of imaginary numbers? They are defined by the equation i = (sqrt)(-1).

Now, it's easy to show that no real number, when multiplied by itself, produces a negative number. Yet it has proven to be incredibly fruitful to imagine (get it?) that such a number exists and go from there.

So I propose that we begin by making this definition of the symbol '#':

0.9... < # < 1

That is, I'm defining # as a number between 0.9... and 1. Now that we have this number, what are the implications? Is there anything we can use it for?

Note that I agree with, and am trying to illustrate, martin's oft-repeated point that if such a number exists, we are talking about something other than the conventional real number system. Perhaps martin, ATP, or someone else would like to run with this. Or maybe one of our contrarians has some ideas...

[Side note, and I hope it doesn't offend anyone, but am I the only one who hears Charlie Chan's voice when reading martin's posts? :wink: I really enjoyed those old movies, though I fear they are not politically correct by today's standards.]

[russ_watters]

Instead of pedantically coming up with quotations of definitions and formulas, just try and use your heads for a change: 0.9999..... is infinitely close to, but never equal to 1.

This does not seem to be so difficult to understand, but apparently it is.
I think you'll find that we are using our heads. :)
So, what number do you think comes between 0.9999... and 1? ;)

Easy enough: an infinitismally small number... :D That's a cop-out and I think you know it. You're just being argumentative. But if you really, honestly believe you are right, identify that number. If you can't identify that specific number, you must concede you are wrong.

I'll help you out a little though: in another forum, there is a guy who argues that there is another type of number that most mathematicians don't acknowledge exists. It looks like this:

0.000....1

That's an infinite number of zeroes followed by a 1. How's that work? (yes, guys, I do know the answer...)

[Richard of Chelmsford]

Duuurrr!! :cry: :cry: :cry:

I failed maths at school four times, then bought a calculator.

So why are these two figures equal?? :oops:
In short, because in the limit that the number of 9s after the decimal point tends to infinity (i.e. the value tends to what it really is), the difference between it and 1 tends to zero. :)

Hello Fortis.

But what about chaos theory?

[Richard of Chelmsford]

Keep on taking the tablets! :o

That's a quote from Spike Milligan.

[A Thousand Pardons]

We have evidently reached a mexican standoff here. No way you can convince me, and no way I can convince you. Our brains are wired differently. Calculus does not convince me. It is convenient for calculus to treat them as equal, just as it is convenient for QM to treat particles as points, or SS as strings. But it ain't so. :D
You talk as if there were a real object that we are talking about.


These are convenient rules for calculus. They do not apply to the real world.
We're not talking about the real world, we're talking about the world where 0.999... goes out forever. The world of math.

[gzhpcu]

This thread shows how people can be brainwashed by a educational system to accept statements from "reliable" sources without thinking things out themselves.

Look, one more time:

0.99999999999999999............

is not equal to

1.000000000000000000.........

If you think you are looking at the same number, I think you need glasses...

:D

[Normandy6644]

This thread shows how people can be brainwashed by a educational system to accept statements from "reliable" sources without thinking things out themselves.

Look, one more time:

0.99999999999999999............

is not equal to

1.000000000000000000.........

If you think you are looking at the same number, I think you need glasses...

:D

It's not being brainwashed, it's just math. What's wrong with the infinite series approach? Or the defining n=.9999... and working from there? I'm sure there are even other ways to prove it. Is it being brainwashed when you're using the tools you were taught in class?

[Severian]

This thread shows how people can be brainwashed by a educational system to accept statements from "reliable" sources without thinking things out themselves.

Look, one more time:

0.99999999999999999............

is not equal to

1.000000000000000000.........

If you think you are looking at the same number, I think you need glasses...

:D

That's your argument? That they *look* different? They are two ways of writing down the same real number. Do you also think that 1/2 and 2/4 are different numbers because they look different?

Do you understand the construction of the real numbers from the rationals?

A sequence of rational numbers (fractions of integers) is a function from the positive integers to the rationals; we denote such a function by x_1, x_2, etc where each x_i is rational; I will use the notation (x_n) or similar to talk about a sequence collectively.

A sequence is Cauchy if for any rational number r > 0, there is some positive integer N so that for any n , m > N , the absolute value of the difference of x_n and x_m is less than r, in symbols |x_n -x_m| < r (intuitively, the terms get "closer together" as you move farther down the sequence).

We now say that any two Cauchy sequences (x_n) and (y_m) are equivalent if for any rational r >0, there is a positive integer N such that |x_n -y_m| < r for any n , m > N (intuitively, the sequences are getting closer to each other in addition to the terms getting closer together as you move down the sequence).

Now we define a real number to be an equivalence class of Cauchy sequences of rationals; that is to say a real number is a set of all Cauchy sequences which are equivalent. We may map the rationals into the real numbers by way of the map which sends a rational r to the equivalence class of the Cauchy sequence x_n=r for all n.

An infinite decimal expansion, then, is the equivalence class of the Cauchy sequence of rationals which result from truncating the infinite expansion; i.e. 0.999999.... with infinitely many 9's is actually the set of all Cauchy sequences of rational numbers which are equivalent (with the above definition) to the Cauchy sequence whose n^th term is given by
x_n = (10^n - 1)/10^n = 0.9999..99 (terminating at n nines).

This is very clearly equivalent to the Cauchy sequence
y_n = 1 for all n
This sequence (y_n) is our representation for the rational number 1.

We say that two real numbers are equal if they are the same equivalence class of Cauchy sequences of rational numbers; that is to say the equivalence class of (a_n) is equal to the equivalence class of (b_m) if the two sequences are equivalent.

It isn't that we've been brainwashed; we just happen to know the definitions. If you don't like this definition of the real numbers, then you are free to invent your own definition, just as we'll be free to ignore your definition because it doesn't have one of the nice properties of the real numbers that we currently use.

[russ_watters]

This thread shows how people can be brainwashed by a educational system to accept statements from "reliable" sources without thinking things out themselves.

Look, one more time:

0.99999999999999999............

is not equal to

1.000000000000000000.........

If you think you are looking at the same number, I think you need glasses...

:D Once again, that's a cop out and you know it: subtract one from the other and tell me the number you get.

There is a word for being purposely obtuse and argumentative in an internet forum.

[gzhpcu]

This thread shows how people can be brainwashed by a educational system to accept statements from "reliable" sources without thinking things out themselves.

Look, one more time:

0.99999999999999999............

is not equal to

1.000000000000000000.........

If you think you are looking at the same number, I think you need glasses...

:D Once again, that's a cop out and you know it: subtract one from the other and tell me the number you get.

There is a word for being purposely obtuse and argumentative in an internet forum.

You think I am being purposely obtuse, but I am not. We have reached a Mexican standoff and better leave it at that. People have made up their minds. I can not convince you and you can not convince me.

I sincerely do not accept equality. Just like for me 1/3 = 0.3333.... is not exact.

But probably the it has to do with the concept of infinity which causes headaches everywhere from cosmology to math.

[gzhpcu]

An afterthought: my apologies to those who think I am purposely being argumentative. Not so. Just standing up for what I believe is correct. But I will drop out of the thread... Mexican standoff... :D

[lti]

this is retarded!

i thought the people frequenting this site were intelligent. but 40% of u are proving to have a tenuous grasp of maths.

ofcourse 0.9 recurring is equal to 1

they are exaxctly the same in real life and in maths (last time i checked those two werent mutually exclusive)

there is an INFINITE number of 9s! not just lots and lots. inifinite!

[Fortis]

Instead of pedantically coming up with quotations of definitions and formulas, just try and use your heads for a change: 0.9999..... is infinitely close to, but never equal to 1.

This does not seem to be so difficult to understand, but apparently it is.
I think you'll find that we are using our heads. :)
So, what number do you think comes between 0.9999... and 1? ;)

Easy enough: an infinitismally small number... :D
No. That number is less than 0.999...

I'm looking for one that comes between 0.999... and 1, i.e. I'm looking for x, where 0.999... < x < 1. :)

[Disinfo Agent]

Instead of pedantically coming up with quotations of definitions and formulas, just try and use your heads for a change: 0.9999..... is infinitely close to, but never equal to 1..
Well, I don't believe it. Prove it to me.


I sincerely do not accept equality. Just like for me 1/3 = 0.3333.... is not exact.
Why do you say it's not exact?


Look, one more time:

0.99999999999999999............

is not equal to

1.000000000000000000.........

If you think you are looking at the same number, I think you need glasses...
Tell me, are the expressions (x+a)^2 and x^2 + 2*x*a + a^2 equal or different?

[gzhpcu]

I'm looking for one that comes between 0.999... and 1, i.e. I'm looking for x, where 0.999... < x < 1.

OK. It is =.000000000000000000........
However it is not equal to 0.

[gzhpcu]

Tell me, are the expressions (x+a)^2 and x^2 + 2*x*a + a^2 equal or different?
equal.

Man, I hate continuing this discussion, but what the heck. And for the last time I am not doing this to irritate anyone. It is my conviction. You do not have to try to prove anything to me, you can just ignore me... :)

Prove it to you?

Evidently I can not. Because when I look at
0.999999999999.........
and
1.00000000000000....
I see two different numbers, irregardless of an infinite number of 9's and 0's.

[gzhpcu]

this is retarded!

i thought the people frequenting this site were intelligent. but 40% of u are proving to have a tenuous grasp of maths.

ofcourse 0.9 recurring is equal to 1

they are exaxctly the same in real life and in maths (last time i checked those two werent mutually exclusive)

there is an INFINITE number of 9s! not just lots and lots. inifinite!

Thanks for the compliment. This reenforces my conviction that I am right. :D

[Disinfo Agent]

A lot of people obviously find it almost impossible to contemplate the infinitely large and infinitely small in LOGICAL terms.
You're obviously having trouble realising that you're one of those people.

Tell me, munineer, since your understanding of the infinite is so much better than ours, how much is 1+1/2+1/3+1/4+ ... ?


Try to follow the following mental exercise. Imagine as a given fact that a certain event will take place at an indefinite point of time in the future, and that beyond that we possess no further information. Suppose also that the future lasts forever. Then the chance of that event taking place on May 23 784896758435579AD is infinity-to-one against. In fact, the chances of it happening as soon as that are also infinity-to-one against.

And yet, the chance of the event happening on that date are (as far as we know) the same as it happening on any other. And that chance is not zero. However infinitely small that chance, it will not go away – even though time lasts forever. That’s right, isn’t it?
Call that nonzero chance c. Then the chance that the event will take place some time in the future should be c + c + c + ..., right?
The trouble is that this sum is greater than 1!


0.9, 0.99, 0.999, 0.9999 . . . with every nine we add the discrepancy between the fraction and unity is diminished. Even though the string lasts forever, the discrepancy cannot be vanquished. The logic of this position is unassailable.
Not at all. The discrepancy cannot be vanquished if you stop at any finite iteration. But if the string "lasts forever", then you don't stop, do you?


What is so disappointing is that others who resort to mathematical “proofs” that the two numbers are equal do not at least appreciate that they are faced with a dilemma.
How do you know that we don't? Does your intuition of infinities extend to mind-reading?


For practical purposes of computation, 0.999 … recurring is equal to 1, as I have insisted from the start. That is why the rules of mathematics treat the two values as indistinguishable. They may safely do so. Any number of so-called proofs can therefore be derived to show that the two numbers are equal. There is nothing surprising about this.
Really? You find nothing surprising about arriving at a contradiction?!
Many of the proofs presented by other posters were aimed at showing you that allowing 0.999... not to be 1 leads to logical contradictions.
You see no problem with that?

[Candy]

This subject has now drifted over to FWIS. #-o

[Disinfo Agent]

Tell me, are the expressions (x+a)^2 and x^2 + 2*x*a + a^2 equal or different?
equal.
Which means that two mathematical expressions don't have to look identical to be equal, right?...


Man, I hate continuing this discussion, but what the heck.
You're free to leave at any time.


And for the last time I am not doing this to irritate anyone. It is my conviction.
I think those who are disagreeing with you realise that. I don't remember anyone accusing you of trying to irritate anyone.


Prove it to you?

Evidently I can not.
Exactly: you cannot. Now ponder about that for a while. And re-read my first reply in this post.

Corrected phrase.

[Disinfo Agent]

this is retarded!

i thought the people frequenting this site were intelligent. but 40% of u are proving to have a tenuous grasp of maths.

ofcourse 0.9 recurring is equal to 1

they are exaxctly the same in real life and in maths (last time i checked those two werent mutually exclusive)

there is an INFINITE number of 9s! not just lots and lots. inifinite!
[-X
Read the forum rules in the FAQ.

The question we're discussing can be confusing for people with a limited (albeit solid) mathematical background.

[Disinfo Agent]

Let's try to approach this from a different angle. Is everyone here comfortable with the concept of imaginary numbers? They are defined by the equation i = (sqrt)(-1).

Now, it's easy to show that no real number, when multiplied by itself, produces a negative number. Yet it has proven to be incredibly fruitful to imagine (get it?) that such a number exists and go from there.

That's not a very good definition of imaginary numbers, though. :)



I'm looking for one that comes between 0.999... and 1, i.e. I'm looking for x, where 0.999... < x < 1.

OK. It is =.000000000000000000........
However it is not equal to 0.
Mutineer wrote a while ago (http://www.badastronomy.com/phpBB/viewtopic.php?p=375591#375591) that an infinite sum of zeroes is still zero. Are you disagreeing with him?

Edited.

[A Thousand Pardons]

An afterthought: my apologies to those who think I am purposely being argumentative. Not so. Just standing up for what I believe is correct. But I will drop out of the thread... Mexican standoff... :D
That's the third time that you have used that phrase. You should either drop out of the discussion, or defend your position.


I'm looking for one that comes between 0.999... and 1, i.e. I'm looking for x, where 0.999... < x < 1.

OK. It is =.000000000000000000........
However it is not equal to 0.
That number is definitely not between 0.999... and 1. :)



Tell me, are the expressions (x+a)^2 and x^2 + 2*x*a + a^2 equal or different?
equal.

Disinfo Agent's point is that two expressions can look different, and yet still be equal. All it takes is a bit of mathematical manipulation--in either case.


Man, I hate continuing this discussion, but what the heck. And for the last time I am not doing this to irritate anyone. It is my conviction. You do not have to try to prove anything to me, you can just ignore me... :)

Prove it to you?

Evidently I can not. Because when I look at
0.999999999999.........
and
1.00000000000000....
I see two different numbers, irregardless of an infinite number of 9's and 0's.
If they are two different real numbers, then there should be a third number that is between them. Doesn't that also stand to reason?

[russ_watters]

An afterthought: my apologies to those who think I am purposely being argumentative. Not so. Just standing up for what I believe is correct. [emphasis added] The reason it appears to me that you are being purposely argumentative is that you're not making logical/mathematical arguments, you just keep stating what you believe.

That said, though it appeared at first your misunderstanding was with what an infinite series is, now it looks far more basic: you don't understand the concept of "equal". As demonstrated, two numbers don't need to look the same to be equal.

[Severian]

gzhpcu, I posted one of the commonly used definitions of the real numbers (you may also define with Dedekind cuts and I'm sure others, but they are all equivalent). You seem to find this definition lacking, since you don't agree with one of the consequences.

So, what is a real number? You must have a clear idea, since you argue with some conviction that the real number represented by 0.999.... is not the real number 1. In fact, you seem to believe that there is a real number between the two (which in the standard system would be an immediate consequence from their not being equal, but I'm not sure in your system). What is the problem with the definition that I gave, and what is your definition? Further, what do you mean when you say 0.99999.....? Can you give an explicit decimal representation of a real number strictly between 0.99999.... and 1? Under your definition, is the set of real numbers complete? Is it even a metric space? What is the distance between 0.9999.... and 1?

[gzhpcu]

An afterthought: my apologies to those who think I am purposely being argumentative. Not so. Just standing up for what I believe is correct. [emphasis added] The reason it appears to me that you are being purposely argumentative is that you're not making logical/mathematical arguments, you just keep stating what you believe.

That said, though it appeared at first your misunderstanding was with what an infinite series is, now it looks far more basic: you don't understand the concept of "equal". As demonstrated, two numbers don't need to look the same to be equal.

OK, my last post in this thread. Just have to state I dô understand the concept of equal.

But 0.9999999999..... and 1 are exactly the same representation. In your examples, the representation was different.

Peace. :D

[Disinfo Agent]

But 0.9999999999..... and 1 are exactly the same representation. In your examples, the representation was different.
They are both in decimal notation. But they're not identical symbols, are they? (If they were, no one would doubt that they stood for the same number.) That's what we meant, obviously.

[Tom Mazanec]

I learned this in high school, not sure if it was up to calculus yet, might have been algebra.

[pghnative]

Is it just me, or is this thread sort of like a relativity thread, but with a half-dozen Sam5's.

[Normandy6644]

Is it just me, or is this thread sort of like a relativity thread, but with a half-dozen Sam5's.

Ah memories. :D

[jfribrg]

This subject has now drifted over to FWIS. #-o

Forgive me for not being up to date on my acronyms. What is FWIS?

[SeanF]

This subject has now drifted over to FWIS. #-o

Forgive me for not being up to date on my acronyms. What is FWIS?

From Where I Stand (http://loresinger.com/FWIS/index.php).

It's another bulletin board with slightly more relaxed rules of conduct than this one. :)

[Glom]

Slightly more relaxed? It's a flame pit!

[Normandy6644]

Slightly more relaxed? It's a flame pit!

Don't scare everyone else away! :D

[AstroRockHunter]

Consider:

1/(1.0000...-0.99999...)

Is the answer:

a) Undefined
b) Infinity

edited for clarity

[Disinfo Agent]

Whom are you asking?

[Normandy6644]

Whom are you asking?

I think those who don't believe it equals one.

[Donnie B.]

Well, I don't believe it equals one. I'd vote for undefined, with that zero in the denominator.

[Bad jcsd]

Consider:

1/(1.0000...-0.99999...)

Is the answer:

a) Undefined
b) Infinity

edited for clarity

In the real numbers the answer is undefined as in the real numbers (as indeed in any non-trivial field) divison by zero cannot be defined.

so to repharse your question:

Given the field axioms the expression 1/1-0.999.. is guaranteed to be defined IF 0.999.. is not equal to 1 so what is this number (remebering that infinity is not and cannot be part of the field of real numbers).

[Normandy6644]

Well, I don't believe it equals one. I'd vote for undefined, with that zero in the denominator.

Wait, so you do think that .9999....=1, or you don't?

[Disinfo Agent]

Well, I don't believe it equals one. I'd vote for undefined, with that zero in the denominator.
Picky, picky, picky... :)
Normandy6644's "it" was "0.999~".