Is 1 = 0.9999......

[bleeb]

Originally posted by: Kyteland

Originally posted by: bleeb
haven't you been reading about the difference sizes of infinity???

In this particular solution, i'm using the aleph-NULL infinity set, instead of the aleph-ONE infinity set...

Um, I think you're using that wrong. Go back and reread those pages about cardinality.

*hint* just because your number ends in 1 and there was an impressive word that had a 1 in it (aleph-1) on that page doesn't imply any correlation between the two.

hahh yeah I know.. i'm just joking around with this one.

 

[MadRat]

Originally posted by: juiio

Originally posted by: bleeb
The difference is 1 - 0.9999.... = 0.00000.....(infinity)....00001.

QED

0.9999 != 1

There is no last digit. Hence infinity.

No last digit, but there is the little factor called a limit. The limit being infinity.

 

[MadRat]

Originally posted by: TuxDave
dieeee..... omg... how can this thread still be here. Esp since the 0.999...=1 group has already won. Hehehe... but madrat's math does seem rather interesting. Kinda bends traditional thinking. I heard one idea that all parallel lines will intersect each other at infinity. And all lines will end up at infinity. This is probably where Madrat's getting his ideas from because if a point CAN exist at infinity, the 0.000...001 can exist... otherwise.. no.

The poll indicates .999...<>1 by popular vote. Actually I don't believe parrallel lines intersect by definition, but yes lines do reach their limits at infinity if you follow the idea of infinity as a limit. Everything metric has a limit, with infinity being a valid limit. They don't necessarily "end" at the infinite length, they are nonetheless defined by the limit of infinity.

 

[jonmullen]

Originally posted by: flood
1/3 = 0.3333...

1/3 + 1/3 + 1/3 = 1
0.3333... + 0.3333... + 0.3333... = 0.9999...

1 = 0.9999...

This is the one I have always seen

 

[RossGr]

. Everything metric has a limit, with infinity being a valid limit. They don't necessarily "end" at the infinite length, they are nonetheless defined by the limit of infinity.

First we need to understand Mathematically just what the concept of limit means.

Let X(n) represent some sequence of Real numbers indexed by the integers.

To say L= lim (n->infinity) X(n) means that there exists an integer M and an arbitrary real number d>0 such that for all n> M

| L - X(n) | < d

Now Let us see what happens if we try to use infinity as a limit

We have L = infinity

| infinity - X(n) | = |infinity| This by the definition of infinity posted in my link above.

but also from the definition of infinity , infinity > x for all real x. According to the definition of a limit there must exist an M for which |x(n) -L| < d such an M does not exist, likewise there in NO real number greater then infinity thus, clearly by the definition of limit, infinity can NOT be a limit.

A function or sequence of numbers which tends to infinity is said to INCREASE WITHOUT BOUND or to be UNLIMITED. It is generally expresses as "the limit does not exist". So Infinity is not, and cannot be a limit.

The word metric is also carefully defined in a manner which makes your use of it incorrect. The basic requirement of a metric is that it be greater then or equal to 0 there is no upper bound, so it is not even a bit clear what you are trying to say. You are attempting to use NON MATHEMATICAL definitions in a mathematical context. This simply does not work. Your philosophical definition of infinity as a much place in a discussion of mathematics as my mathematical definition has in dicussing a spirtiual infinity, or some other philosophical context. I am aware enough of the difference to beable to use the correct term in the correct context. Are you, Madrat?

 

[MadRat]

What is clear to you and what is clear to over 50% of the people here are two different opinions. I like how you use one tiny offshoot of philosophy, the discipline of mathematics, to draw parameters around this debate. This opinion of yours has already been noted and duly dismissed by a simple majority. Popular opinion has always been the prevailing mood, and frankly your argument lost in results of the poll.

The limit of your equation is clearly infinity. By definition the idea that it extends into infinity does place a restraint on your value. To have no limit means that its limit is infinite, therefore it has a limit of infinity. Just because you view infinity with an interpretation that infinity can not be a limit is no relevance to the argument. Nor does your argument that one must introduce mathematics into this debate simply because in your opinion this is a math equation have any merit. This is a philosophical argument over values, so the difference in our opinions in math become irrelevant.

OT: As for your earlier insults, I'm still waiting for you to be a man. When you want to apologize I'll be waiting.

 

[smartt]

Look, if 1=a + 0 and 1 + 0 = 1 then a must be 1 and 1 = 1. Geesh!

 

[speed01]

Uhhh......sure.......

 

[bleeb]

0.9999...!=1

 

[RossGr]

Originally posted by: MadRat
What is clear to you and what is clear to over 50% of the people here are two different opinions. I like how you use one tiny offshoot of philosophy, the discipline of mathematics, to draw parameters around this debate. This opinion of yours has already been noted and duly dismissed by a simple majority. Popular opinion has always been the prevailing mood, and frankly your argument lost in results of the poll.

The limit of your equation is clearly infinity. By definition the idea that it extends into infinity does place a restraint on your value. To have no limit means that its limit is infinite, therefore it has a limit of infinity. Just because you view infinity with an interpretation that infinity can not be a limit is no relevance to the argument. Nor does your argument that one must introduce mathematics into this debate simply because in your opinion this is a math equation have any merit. This is a philosophical argument over values, so the difference in our opinions in math become irrelevant.

OT: As for your earlier insults, I'm still waiting for you to be a man. When you want to apologize I'll be waiting.

Limit of what equation? Did you not understand my last post, by the basic definition of limit, infinity cannot be a limit. So please drop that misconception. When something tends to infinity it is said to increase without limit. Sort of opposite of the concept of a limit. So if we are not talking about a mathematical concept what are we talking about?

What meaning does .999... have outside of math? It is mearly three curvy things and three dots. Perhaps you need to start a different thread about the fundamental philosophical nature of curvy things and dots, that seems more your speed. Why do you even contribute to a mathematical disscussion if you cannot or will not talk math?

 

[MadRat]

So far you've dismissed any counter-argument for supralogical reasons so I gave you a taste of your own medicine. Doesn't make alot of sense to you I see. Well neither does your use of some of your own arguments. You've gone outside the logic of the nature of debate to try to prove your point. Since you lack debating skills then how are we to trust your sense of argument? Thats rhetorical.

Its already been demonstrated that your argument is based on absolutes that other opinions do not share. Your argument fails in the sense that it deals with what your mentor and colleagues believe, not in what they know. (There is no way to prove anything absolute when dealing with infinity.) A mere fifty years ago the math community was far more divided and there were alot of ways to look at answers. In your eyes, today, there is only one way to look at the answer. Somewhere along the lines of time we've seem to have lost some sense of progress. Or perhaps that is just because you've closed your mind to the alternative ways of approaching a solution.

You've chosen a meaning of Real numbers that makes .999... real for you. I've chosen one, using the same exact meaning, that makes it invalid. Your proof that it is Real uses the same exact reason to make .000...1 invalid, but in your case it makes .999... all the more valid. I've pointed out the flaw in that argument, which by any definition of logic your conclusion is wrong. The simplest form of the argument is that either both the largest and smallest decimals exist, based on the idea of inifinite position, or neither exists.

Either way its still consensus by opinion. And the opinion of the majority has said that .999...<>1, which makes it so.

 

[Haircut]

You've chosen a meaning of Real numbers that makes .999... real for you. I've chosen one, using the same exact meaning, that makes it invalid. Your proof that it is Real uses the same exact reason to make .000...1 invalid, but in your case it makes .999... all the more valid. I've pointed out the flaw in that argument, which by any definition of logic your conclusion is wrong. The simplest form of the argument is that either both the largest and smallest decimals exist, based on the idea of inifinite position, or neither exists.

0.999 is a real number and 0.000...1 is not.
See here for a brief explanation why.

As you say, if we can have an infinite position then both largest and smallest decimals exist, if we can't have an infinite decimal then neither exist.

I have shown above that an infinite decimal cannot exist, therefore we can't have a largest or smallest decimal.
However if we take the open interval (0,1) then 0.999... is larger than every number in this interval as has been shown before in this thread.
We are not trying to say that 0.999... is the largest decimal below 1, it is 1. It always has been and always will be because that's what we have defined it to be with mathematics.

On this note I will bow out of this thread, we are still arguing about the same things as we were 500 posts ago and it seems futile to carry on.
Have fun those of you who choose to continue it.

 

[Keego]

As an engineer, I'd round up to 1.

 

[MadRat]

Originally posted by: Haircut
We are not trying to say that 0.999... is the largest decimal below 1, it is 1.

Since no decimal can exist higher than it then it is the highest decimal possible, correct? If you carry the 9's out to infinity then you have just used the same argument to disclude .000...1 as what discludes .999... from the real numbers set. To include the .999... in the Real numbers set for your reasoning would be a perversion or double-standard of the same meaning; this would then not be a logical explanation of .999...=1.

Originally posted by: Haircut
We are not trying to say that 0.999... is the largest decimal below 1, it is 1. It always has been and always will be because that's what we have defined it to be with mathematics.

When you say "we" then I understand that as your colleagues, correct me if I'm wrong. Then what you are saying is your colleagues believe this definition by doctrine, or what one would call dogma. Definitions by doctrine are not absolute rules but rather generally percieved notions. Unfortunately for your colleagues this notion is not a very popular one with the general populace.

Someone awhile back noted how people once believed the earth was flat. It wasn't because people believed it was flat, but rather it was a defintion by doctrine of the Church. Some brilliant scientists for their day could prove that the earth was not flat and were forcibly squelched. The holders of the dogma, the Church, could get away with censoring these early scientists simply because of their positions in society. Thank God we are long past this point in society, where people need deflect arguments by introducing their credentials, although I fear some people around here wouldn't mind forcing their opinions down all of our throats. An opinion counter to .999...=1 has already spurred some rather unpleasant and primitive reactions from a small group of simians out there. Again, thank God, these types of people are in the minority!

Haircut, I have to say your participation in this thread has been a big positive. Its nice to see someone that firmly believes in their idea of truth without resorting to unpleasantness. Cya around!

 

[TuxDave]

sooo... if 0.000...01 does exist.... are you claiming that the real number system is discretized with the increment of 0.000...001? So that way the next number after 0.000..0001 is 0.000...002? So what number comes after 0.000....009?

 

[RossGr]

You have pushed to new levels of trollhood with your current line of nonsence. Opinion polls mean nothing, as you are well aware. We could conduct an opinion poll on the winner of next weeks lottery, it will have no effect on the outcome. The poll merely shows that this is not a obvious result. It requires the rigor of mathematics to determine the meaning of .999...

To claim mathematics is opinion, or that we have simply defined .999... to be one is further nonsence. The Real numbers came about because of the exisitance of irrational numbers, along with the structure required to make sense of numbers like Pi and Square root 2, came the fact that there are multiple representations for all points on the number line.

Your denying the validity of mathematics is really shooting your intire philosophical system in the foot. Your are correct, Math is an off shoot of Philosophy. Where do they connect? At the root, each share the same logic system. So if the logic of mathematics is flawed so is the logic of philosophy. If the reasoning and detuctions which have lead to mathematical conclusions are false so are all of the conclusions of philosophy. Therefore all of your conclusions are meaningless. This by your own arguments.

Your really ought to be ashamed of yourself, your misuse and abuse of the system of logic makes a lie any claim you have made of being a philosopher. You are a charlatan pure and simple. You are the ignorant peasent living in a mud hut denying that the earth is a shpere because you can clearly see that it is not. You go further by closing your eyes and mind to any evidence to the contrary, after all in your many journeys to the outhouse and back, you have never witnessed any curvature. Therefore you can be sure that all others are lying.

You are free to live in your flat world, charlatan, but I resent your comming into public forums such as this to spread your lies and misinformation.

 

[ndee]

One of the worst thread EVAR.

 

[MadRat]

Originally posted by: RossGr
You are free to live in your flat world, charlatan, but I resent your comming into public forums such as this to spread your lies and misinformation.

LOL, a big word you didn't spell wrong - you MUST have dug out the Thesaurus!

 

[RossGr]

Originally posted by: MadRat

Originally posted by: RossGr
You are free to live in your flat world, charlatan, but I resent your comming into public forums such as this to spread your lies and misinformation.

LOL, a big word you didn't spell wrong - you MUST have dug out the Thesaurus!

Actually, the Franklin!

 

[TuxDave]

Originally posted by: ndee
One of the worst thread EVAR.

aww.. come on.. just a couple more posts before we let it die. So what about my argument on what comes after 0.000...09?

 

[ElFenix]

Originally posted by: MadRat

Unfortunately for your colleagues this notion is not a very popular one with the general populace.

the problem with that is that the general populace is made up of, as a gov't professor of mine once said, idiots.


the problem with this thread is that rossgr is arguing from the very strict, yet clear mathematics position, in which .999... = 1, and madrat is arguing from some other system, in which a decimal can end once it has reached "infinity", this being the infinith decimal position. which would make .999 != 1. the problem is, which of these systems was the original poster refering to? probably the math one, though i can't say for certain. rossgr has presented complete proofs for the math sytem, whereas madrat has done more philosophizing than anything else.

 

[MadRat]

Originally posted by: TuxDave

Originally posted by: ndee
One of the worst thread EVAR.

aww.. come on.. just a couple more posts before we let it die. So what about my argument on what comes after 0.000...09?

10^-(infinity-1)

Since one cannot assign infinity a number then the expression would be at its simplest form. Since infinity is an indeterminate value we cannot progress the equation into any finite form.

I know that earlier someone was trying to say "infinity + infinite = infinity", but those are bastardizations of the definition of infinity as a limit. Also don't confuse 'moving back towards zero' with 'moving further away from zero' beyond the limits of infinity. It wouldn't be possible to say 10^(infinity+x) if x>0 because infinity is itself the limit of the numbering system. Therefore going past infinity would be illogical. But subtracting from infinity is moving below infinity and it is logical that you can move from infinity back down to null if you are staying within the context of null and infinity as the definite limits of the numbers. In this respect it is logical to assume that "infinity - infinity = null'.

 

[MadRat]

Originally posted by: ElFenix

the problem with that is that the general populace is made up of, as a gov't professor of mine once said, idiots.

OT: I've never hung with a general populace that acted much like idiots. From my point of view most people are about average on the scale of "IQ" and it takes pretty complex judgement skills to attain that average score of "100". I'd like to see these people that this gov't professor called idiots. The idiots he is probably observing are probably alot more significant than he gives them credit. Then again more than a few professors out there are patently elitist and have generally poor perceptions as to what is "normalcy". Alot of smart people aren't necessarily all that well equipped to live in an environment that includes other people.

 

[TuxDave]

Originally posted by: TechDeck
Wow, this thread is crazy.

hahaha... perfect time to join in. Welcome to ATOT!

 

[RossGr]

Originally posted by: MadRat

Originally posted by: ElFenix

the problem with that is that the general populace is made up of, as a gov't professor of mine once said, idiots.

OT: I've never hung with a general populace that acted much like idiots. From my point of view most people are about average on the scale of "IQ" and it takes pretty complex judgement skills to attain that average score of "100". I'd like to see these people that this gov't professor called idiots. The idiots he is probably observing are probably alot more significant than he gives them credit. Then again more than a few professors out there are patently elitist and have generally poor perceptions as to what is "normalcy". Alot of smart people aren't necessarily all that well equipped to live in an environment that includes other people.

What has the world come to! I agree with Madrat.

To say that the average IQ of this board is, well, average may not be quite correct, I would think that it is, in fact above average. But then, what does IQ mean? I don't put much weight on any given measurement of it, seems like generally it is simply a mearsure of your ability to take IQ tests.

In their own way every individual has an element of genius. Einstein, in the stone age, would propbably have been the viliage idiot, each human has incredible abilities, but if the conditions to bring those abilities to the forefront do not exist, the genius may never be realized.

As I stated a few posts back this question is not obvious. I do not find proofs which simply push opertations further and further down the line such as 1/3 + 1/3 + 1/3, you know the rest! 0r x = .999.... '> 10x = 9.99.... very convincing since they imply mathematical operations which never complete. To internallize this proof in a completely convincing fashion requires mathematical sopistocation beyond a majority of the population in general, the fact that only 50% of the readers on this topic are not aware of the fact is not surprising. I only hope that a couple of them read enough of the posts to get the glimmer of an idea, that just maybe, .999... =1.