9/9 ?????????????????????

Kedeman is venturing into semantic and philisophical debates. While i find his tenacity amusing, I see no need to try and show the light to the blind.

You can wax semantics all you want. And since I think the church gives meaning to peoples otherwise dreary lives, I'm not prepared to disprove the existance of god, so I don't care to engage anyone in a philisophical debate.

:)

'Course, I suppose that someone like yourself doesn't believe in a god, anyway, since you have the pride enough to believe man invented math and the reality around him. So I wouldn't be telling you anything new.

ignorance is a bliss right?

ok, these are two very fundemental axioms.

for any real number x where x>0 1/x is a real number and x>0.

for any real number y there exists a natrual number n such that n>y.

and it's not too difficult to show that there's also a natrual number n in the form 2^p * 5^q. (2^n would do.)

if you want to dispute one of these feel free to go along to the nearest university and take it up there, but I warn you they won't take too kindley to you asking them to re-write all the maths books because you've decided that they're wrong.

I sure hope so, Kedeman.

so you don't want to proove me wrong?

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Originally posted by Sam Finch
and it's not too difficult to show that there's also a natrual number n in the form 2^p * 5^q. (2^n would do.)

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I believe such numbers exist. But... what is 0.000....001?

I wrote it down, so it must be a number. And I can produce this number:

1 - 0.999...

So what is that number? I'm just trying to look out for that little number.

Now...

if your reality is true, then for y = 1/x, y is equal to 0 for very large x.

Hmmm... that bothers me. That bothers me a lot. And I know it bothers the academic sector. Y is not equal to 0, it approachs 0, but it is never equal to zero.

For the very same reason behind that, 0.999... is not equal to 1. It approaches one, but it can not equal.

Now, you can say they are equal, so long as you understand you are approximating.

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Originally posted by kedaman
so you don't want to proove me wrong?

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Well, just because I know that knowledge is not bliss, doesn't mean I can prove that ignorance is or isn't bliss.

that was a qwestion, i was refering to my earlier statement, there's no known reality to refer to. You define it, sometimes whether you want it or not.

still as serious as before.

*grumblegrumblegrumble*

Okay, Kedeman.

I have this many rocks on my desk...


* * * * * * *

I can't split that number of rocks into equal portions for any number of people (save # # # # # # # many people or # many).

Show me where I created the reality, where I affected it?

Sam: The part of your proof that I disagree with is the following.

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So if 0.9... < 1 there must exist a non recurring decimal c such that

0.9.... < c < 1

clearly this is not the case so 0.9... = 1.

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How can you assert without proof that there is no such number? It does not exist because you do not know what it is? It does not exist because I cannot specify what it is?

I gave a valid proof (see below) that recurring .9999 is less than one. That proof in conjunction with the first part of your proof proves that there is a nonrecurring number between recurring .9999 and one. To avoid the necessity of hunting through previous posts, an abbreviated version of my proof is as follows.

• DecimalSeries(N) = 9/10 + 9/100 + 9/1000 . . . + 9/10^N
• HexSeries(N) = 15/16 + 15/256 + 15/4096 . . . + 15/16^N
• DecimalSeries(N) < HexSeries(N) < 1 for all values of N.
• Hence there is a value between DecimalSeries(N) and one.
• Therefore DecimalSeries(N) not equal to one for any value of N.
• DecimalSeries(N) is a formal representation of recurring .9999
• Hence recurring .9999 < 1

It is true that the limit of both DecimalSeries(N) and HexSeries(N) is one. It is also true that DecimalSeries(N) is not equal to HexSeries(N) for any value of N, nor is it equal to one for any value of N.

One should not confuse a function and its limit. A proof involving a limit merely proves that some function gets arbitrarily close to its limit. Such proofs never prove nor assert that the function ever equals its limit.

In many contexts, it is valid to use a limit in place of the function. This is a matter of notational convenience and merely indicates that for practical purposes, the limit can be used instead of the function. Such usage does not imply that the limit is theoretically equal to the function.

Such usage of a limit in place of a function leads to posts such as this. Of course 9/9 = 1. To think that it equals recurring .9999 is theoretically absurd, but true for all practical purposes (If you ignore the wastage of printer’s ink and space on a page).

you are assuming so much again, i guess i could put up a list of millions of Why...? What...? How...? and you won't be able to answer none of them, if you try, you will find out there's a border you can't pass, that your knowledge is limited and that you can't distinguish reality.

for instance, "split" is a function we apply in several models, if you define one of them with reference to reality, you have prooven me wrong, go ahead.

Kedaman, who are you responding to?

Guv, very nice job. I don't think I said so earlier, but very pretty.

you of course :)

Guv, did you notice that

these:

DecimalSeries(N) = 9/10 + 9/100 + 9/1000 . . . + 9/10^N

Therefore DecimalSeries(N) not equal to one for any value of N.

doesn't match

0.99...

because if you put a value N you have a value with a specific amount of decimals, that is N

Quote:

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Originally posted by kedaman
you are assuming so much again

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What am I assuming? WHAT??? If you can't name it, then is it really so?

Quote:

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, i guess i could put up a list of millions of Why...? What...? How...? and you won't be able to answer none of them, if you try, you will find out there's a border you can't pass, that your knowledge is limited and that you can't distinguish reality.

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what the **** are you talking about?

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for instance, "split" is a function we apply in several models, if you define one of them with reference to reality, you have prooven me wrong, go ahead.

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Split? What the **** does split have to do with any of this? We are talking about numbers!!! wait.. I'll define sum() if you'd like!

Take a hand full of rocks and put them on the table.

Now take another hand full of rocks. Put them on the table.

You have just summed rocks.

NOW IF YOU WOULD KINDLY STOP WITH THIS RAMBLING CRAP AND EXPLAIN YOUR POINT!!! WHERE DID YOU AFFECT REALITY WHEN YOU SUMMED ROCKS???

I maintain that you didn't affect reality, at all. That the mathematics behind the summation of those two groups of rocks is purely a self-evident thing for which man has labeled... 1 + 1 = 2.

And I just proved it by putting rocks on the desk!

Quote:

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Originally posted by kedaman
Guv, did you notice that

these:

DecimalSeries(N) = 9/10 + 9/100 + 9/1000 . . . + 9/10^N

Therefore DecimalSeries(N) not equal to one for any value of N.

doesn't match

0.99...

because if you put a value N you have a value with a specific amount of decimals, that is N

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Damnit, Kedaman, you are pissing me off.

DecimalSeries(N) = 0.999... where N is equal to infinity.

What was so ****ing hard about that.

i picked "split" of two reasons, one because you mentioned, two because it's the best shortcut to the border i mentioned.

oh btw i saw that, infinity is not a number so you can't put it in N's place.

now if you calm down, we could talk more seriously.

Okay, I don't know where I mentioned split. And I don't care about shortcuts to borders ???

The point behind this whole arguement is wether or not infinity exists. So, you can put it in N's place. If you can't, then... why not say as N approaches infinity if that will make you feel better.

Now, for the last time.

Look over my examples. Where did I affect reality? According to you, I'm making this **** up. But every time I ask you to show me where, you simple say, "You can't understand it."

You know that is no way to prove anything.

Now, don't say another word until you show me where I affected reality.

Quote:

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Originally posted by CiberTHuG
The point behind this whole arguement is wether or not infinity exists. So, you can put it in N's place. If you can't, then... why not say as N approaches infinity if that will make you feel better.

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we weren't actually arguing about that, but i guess we could allow some "multithreading". If N approaches infinity, the series approaches 1.

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I can't split that number of rocks into equal portions for any number of people

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here's your statement. and you are assuming split as a function which we implicitely declare and hence risk ambigueties.

Quote:

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Okay, I don't know where I mentioned split. And I don't care about shortcuts to borders ???

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I would because this is already getting boring.

Quote:

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Look over my examples. Where did I affect reality? According to you, I'm making this **** up. But every time I ask you to show me where, you simple say, "You can't understand it."

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I'm trying to avoid things that happened between me and Gen-x. To spawn several issues at the same time doesn't lead anywhere.

Quote:

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You know that is no way to prove anything.

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i hope you know what you are saying.

Quote:

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Now, don't say another word until you show me where I affected reality.

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i am trying, now please give me a chance.

Perhaps there are answers to these questions, it's just possible (nay, probable) that the human brain cannot comprehend at this stage of evolution.

Maybe we haven't been exposed to the right stimuli so far to answer these questions.

Maybe we're at the point of the blind man who is trying to understand the colour green.

We can handle it, yet it's beyond us - if you see what I mean....


PS - Ciber,

I'd lay off Kedaman if I were you. He's one of the most respected people in these forums and it's for a reason.

yeah, we can handle most stuff, whatever stuff happens to be, if it actually is, or is it?
you might be right too arbiter :) and i certainly hope sometimes, but i think we still are too young

Quote:

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Originally posted by kedaman
here's your statement. and you are assuming split as a function which we implicitely declare and hence risk ambigueties.

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Okay... so what are the dangers? You aren't explaining anything.

Fox's statements still stand as a gross error. You have done nothing to help that.

Anyway, it is late on Thursday, so I'm heading home for the long weekend. See you on Monday.

As a species, I think we're too young for a lot of the things we're tying to do, genetic engineering for a start.

Hell, compared to the ridiculous amount of damage we could cause ourselves if we started dropping hydrogen bombs around and creating 'super plagues', I think we'd do well to carry on trying to calculate 9 / 9 !!!

eh, i'm not here to back up fox actually.
This is some idea that hit me long ago, and it surely stands.

If you want to have it said, you never prooved fox wrong either.

the danger of assuming, and you should know English is a very vague language, is getting missinterpreted.

i'll see you at Monday then

Sadly, Arbiter, you are too right on that :( If we can't even decide on this then we hardly have the right to go messing with even bigger things.

were so helpless it hurts to think how little we know :)

but then again, isn't it lovely being ignorant and stupid

Wahey, a new argument! Err.. not that I usually get involved in this kind of thing, I'm very shy really :rolleyes:

Personally, having read all of this, I think 0.999... is the same as 1. Okay I've stated my position.

This subject seems to be pretty linked to infinity - ie. 0.999 is conceptually one 'point' (a point being the smallest value possible) away from 1. That is my impression from what has been said about 'points' anyway.

I take issue with the idea that there is a smallest possible number or a largest possible number if you're not going to involve infinities, and I think infinity is a fairly unhelpful thing to involve if you're looking for a proof of something. Someone said that if you could take the smallest number and halve it, then it wouldn't be the smallest number. Isn't that statement in itself a disproof of there being a smallest number, by contradiction, though?

Of course, you could say that the number is infinitely small, and half of any infinitely small number is still an infinitely small number, but as far as I am aware that operation is not formally defined, just assumed.

Speaking of undefined operations, who decided that 1 - 0.999... = 0.000...1 ? I'm not disputing it necessarily, just wondering if it's true in formal mathematics. How would you formally write 0.000...1 ? 1*10^(- infinity) ? How?

I don't see Fox's statements as a gross error at all. Maths is just a language, it's a way of expressing things. What CiberTHuG is talking about is not maths, it's Physics. It's life, the universe and everything. Maths is the language of Physics, that doesn't mean that everything in Physics is representable using Maths, not necessarily anyway. Not yet even. Does that mean that the maths itself is wrong though? I don't think the 'points' that have been mentioned are existent in current maths, and I have a feeling they won't be unless some major issues with infinities get worked out.

Anyway, I'll reply to any flames as they turn up :)

Kedaman,

I must admit, I find your posts very amusing. I remember (not too long ago) having a debate on the subjectivity vs objectivity of reality. You were definitely arguing strongly for the objectivity of reality. You certainly have undergone some sort of transformation.

CiberTHuG,

Mathematics is about producing inventing models that enable us to simplify the complexity of reality. Don't make the mistake of thinking that they are reality.

What is real about complex numbers? The don't seem to be directly observable in the 'real' world but they allow us to construct helpful models that we can use to understand otherwise baffling problems.

so you think that i have changed my views.
I am totally aware on what i'm trying to achieve simon, and it's neither prooving the subjectivity nor objectivity of reality, and in this case i'm definitely not agruing for the subjectivity of it. I haven't even mentioned that reality exists yet, if you can comprehend what i'm trying to say, that is not taking any side in that issue.

I've only really scan read the posts ive missed overnight, but a couple of points come to mind.

The number or lack thereof between 0.9 recurring and 1 is neither infinity or infinitesimal, because it resides between 2 happily defined numbers.

Maths was deliberately invented by man. Complex calculations were devised when simply counting the stars was not enough. Think about it. If you were to ask this question of a member of any number of tribes, their response would be "many". If it were necessary to find a correct answer, said tribesman would sit down and count to it. And die of old age in the process.

The debate on whether mathematical models exist outside of our perception is key to the debate about the subjectivity or objectivity of reality.

Quote:

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Now you can't distinguish between what is real and what is not, in fact, you have to define what is "real" and sometimes you have to redefine, most of because the inaccuracy of your genetical or psycological functions.

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What you seem to getting at here is that there is know way of knowing how close our perception of the outside world is to the actual outside world.

If this is the case, is there any value in maintaining a belief in the absoluteness of the outside world if we are not (and will never be) capable of knowing it? :confused:

that's the case, and to you qwestion, everyone (or almost i think) could answer, except me. But as a human being, i am bound to be passive, my instincts tells me what to do, my brain tells me what to think and my decisions... who knows...

Very non committal.

Are you saying the rest of us aren't human? :rolleyes:

did i ever mention that you guys might not exists at all :p
my instincts tell me to believe but my mind tells me to think, i guess it's part of being human, so i can't say who you neither i am.

Kedaman, please tell me, what is the the number closest to 1, that is not 1 and is smaller than 1? :confused:

When .9 recurring ppl just round the number up to one but it's incorrect when looking at the number as a fraction.

9/9 <> 1....

Imagine it in terms of cutting a cake: You divide it into 9 perfectly equal parts - surely that can be done, no?

You take all 9 of them, and besides being a fat bastard, you have the whole 1 cake...

And since I've gotta go in a minute, I'll refute the first possible argument I can imagine, that being that you can't get 9 perfectly equal parts in the first place.

To that I say: if that is true, then there is a flaw in the logic that 1 can even be divided by 9. If it can't be done into 9 equal parts, then 1/9 <> 0.11... nor does 2/9 = 0.222, hence we would be assuming that 2/9 = 2 * 1/9, and since that apparently can't be done, 1 can't be divided into 9 equally, so we still get the conclusion 9/9 <> 0.9999, it = 1.

My maths is limited, as is my time, so that'll do ya for now.

:D

Quote:

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Originally posted by sail3005
Kedaman, please tell me, what is the the number closest to 1, that is not 1 and is smaller than 1? :confused:

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What level of accuracy do you want to represent? Integer accuracy? Then the answer is 0, (but of course zero implies nothing, but you see my point...)

There is no definite number. You define your own limits on the problem, and use that to model a solution.

Who the hell cares anyway ...

When was the last time that you needed a precision of a million figures after the comma in calculations or predictions?

It makes for a nice speculative philosophical debate but for all intents and purposes an approximation will do just fine in our daily maths.

Practically spoken, for me 9/9 equals 1 (and I don't care if it also could be 0.99999... (recurring), it's too impractical to work with)

This is just philosophical number crunching.

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Who the hell cares anyway ...

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I do.

Quote:

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This is just philosophical number crunching.

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It's great, isn't it?

Possibly

but it's far from practical ;)

Quote:

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Originally posted by Wally Pipp
Possibly

but it's far from practical ;)

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Thats the spirit ;)

I thought this would have been clear to everyone that this isn't a practical concern. It is an abstract, mathematical, philisophical concern for asthetic purposes only. Personally, those are the types of concerns that interest me most.

abstraction is worthless without application :p

Kedaman

Quote:

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abstraction is worthless without application

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I couldn't disagree with this more.

Many things have been studied purely in the abstract with no notion of practical application (like number theory) but have yielded practical applications in the future.

However, even if a topic has no practical application, it is only "worthless" from a practical perspective. It may have asthetical value.

Simon
application doesn't nessesary have to be physical, ex,

X=X=true
can be applied on
5=5
= true

Variations on this topic have been argued at VB Forums several times. One question which keeps coming up relates to the number less than one which is closest to one. There is no such number. Cantor, Dedekind, or somebody proved a more general statement than that, which is applicable.

Cantor might have made some statement about a real number being represented by an infinite number of decimal digits, but I doubt that he ever based any of his proofs on operations involving the decimal system of notation.

Most of this post has been posted to a similar thread.

This thread includes a proof by Sam Finch that recurring .9999 is equal to one. The proof is esoteric and difficult to follow. I am not sure that it is valid. At any rate it essentially claims that recurring .999 is equal to one because there is no non recurring value between .9999 and the limit. While his proof might be valid, I do not see that it proves that recurring .999 equals the limit. There are real numbers between recurring .999 and one. Perhaps there are no non recurring numbers, but the existence of any real number between recurring .999 and the limit should be sufficient to show that .999 is not equal to one.

It is just not valid to set a value equal to its limit, although is it is sometimes stated that doing so for computational purposes will not result in erroneous results.

Consider the following two limits.

Limit(1 + 1/n) is one as n grows without bound.

Limit[ (1 + 1/n)^n ] is e (approximately 2.7182818) as n grows without bound.

If the limit value for (1 + 1/n) is used in the latter expression, this implies that 1^n = e, or that the second limit is one rather than e. Neither of these two alternatives is correct.

Do you propose picking and choosing which values are equal to their limits and which are not? Do you claim that in some contexts it is OK, but in other contexts it is not? Sometimes you allow it, at other times you do not.

You can look up the formal definition of a the limit of a series with an unbounded number of terms in any good college text. It is a bit cumbersome, but essentially states that the difference between the sum and the limit becomes arbitrarily small. It does not define the difference as zero.

There is never a statement like: When n equals infinity, the sum of the series equals the limit. Such a statement is invalid. As far as I know, formal texts on calculus and analysis do not use the term infinity. They use statements like grows without bound.

The following proof would never be considered valid by a formal mathematician.

x = .9999. . .

10x = 9.9999. . .

10x - x = 9

9x = 9

hence x = 1

One or both of the bolded statements are not valid operations. One must define such operations using limits or other formal methods, rather than relying on intuition or common notions.

It can be shown that there are many numbers between .99999. . . and one. It should be noted that it is probably not valid in some contexts to use decimal notation as the basis for mathematical proofs. This is probably one such context.

Recurring decimal .9999 is 9/10 + 9/100 + 9/1000 . . .+ 9/10^n

Recurring Hex .FFFF is 15/16 + 15/256 + 15/4096 . . .+ 15/16^n


The above two series are not equivalent, although one is the limit value for both. Using decimal equivalents, the first few sums are the following.

.9000 — .9375
.9900 — .99609375
.9990 --- .99975586 . . .
.9999 — .99998474 . . .

The above clearly indicates that for any finite value of n, the sum of the Hex series is always greater than the sum of the decimal series and less than one. Hence, the sum of the Hex series is always between the sum of the decimal series and its limit.

There seem to be people who post at the VB Forums who believe that there is a place called infinity where magic takes over and mathematical logic is no longer valid. Nobody has ever traveled there to see what happens. There are some common notions like parallel lines meet at infinity or the sum of an infinite number of terms is equal to the limit, but mathematicians do not agree with these notions.

For every value of n that we can investigate, the hex series is greater than the decimal series and less that one. Do you really believe that when you get to the land of Oz (infinity), the hex series will no longer be greater than the decimal series? If they both become equal to one, what happens if you add one more term? Do they both exceed one? Does some monster keep you from adding one more term?

As for me (and many serious mathematicians), recurring decimal .99999 is equal to recurring decimal .99999, and recurring Hex .FFFF is equal to recurring Hex .FFFFF and both have one as the limit value. Neither is ever equal to one, and the Hex series is always between the limit and the decimal series. There is a huge (probably unbounded) number of series whose sums are between the sum of recurring nines and the limit value.

Just as parallel lines never intersect, asymptotes never meet their limit lines, and the sum of an infinite series never equals its limit. Follow the parallel lines or asymptotes for light years, billions of light years, et cetera, the parallel lines never intersect and the asymptotes do not touch their limit lines. Sum the series for millions, billions, trillions, et cetera of terms, there is always some infinitesimal difference between the sum and its limit

Why assume that the above situations somehow change at or on the way to some Land of Oz at a place called infinity? When you get there, is it impossible to go further? Do the parallel lines terminate there? At infinity does one of the parallel lines make a right angle turn to meet the other? Do both the parallel lines gradually bend toward each other? Do they start bending toward each other before you get to infinity or afterwards? At infinity can you no longer add even one more term to a series? What prevents you from adding one more term? If you can add one more term, do you exceed the limit?

There may be text books written for pre-college courses which are informal in some of their statements. No modern text written for serious mathematicians will claim that it is valid to set the sum of a series to its limit.

I think I have seen instances in which a text made some statement like: Erroneous values will not result from use of the limit for computational purposes. I think it is sometimes valid to define a function as equal to a limit for a particular value of the independent variable.

Guv
Are there any practical use of the notation with recurring decimals? Wouldn't it be better to type all rational numbers as rational numbers: 1/3=0.1₃ 15/16=0.F₁₆

Kedaman: Your idea sounds good to me. I worry a bit about what to do when somebody asks me to compute results like.

.1₃ + .F₁₆

or .1₇ * .EF₁₆

Maybe we should improve on the Babylonian approach. They used a base 60 number system because so many ordinary fractions did not recur. Perhaps a number system vased on 7! or 13! would help.

Once you design & memorize the 5040 or 6,227,020,800 symbols for digits, you do not have to bother with many recurring fractions.

Well, maybe its easier if we type them as fractions instead 1/3 and 15/16 and then let the computer do the messy stuff, similar goes with irrational numbers, let the computer do the dirty work but there's no need for recurring madness