1=.999999...?

Omega124 said:
OK

Therefore, 3=.01/3=.00333...

No. No it can't.

1/3=.333...

.33*.0033...=.0011

.0011!=.333.....

1/3 is still .3333... You can't change values of pre-existing numbers without changing the base one uses (AKA, 6*9 actually does equal 42 when you use base 13).

EDIT: Furthermore, even if .333 DID equal 1/3, that still wouldn't change that 1/3=.333...

After all, (1/2)*(1/3)=1/6, but 1/6 still equals .1666... (Because, in fact, (1/2)*(1/3) ALSO equals .1666...)
Click to expand...
El_Machinae said:
1/3 = 0.33 + 3-bar = 0.333...
1/3 x 3 = 0.99 + 0.01 = 1
Click to expand...

But 0.3333... =/= 1/3, as has been demonstrated several times.

J
 
@NinjaCow64
Since the combination of sky-high arrogance and lack of comprehension capabilities you - from my point of you - showcase with such glee, I feel unable to continue this discussion with you, since for one it appears to be an extremely and unnecessarily laborious task to debate with you and for another the only joy I could get out of it at this point would be to mock you.
 
The finitists:
Oddly, I haven't seen a living person who would call him/herself a finitist. If you're going to adopt that position, you should be consistent with it though. That excludes differential and integral calculus for example, at least the way we know it.

Also, if you're a finitist, that's a thing you should inform for the very start. If you're saying just that 0.9999... is not 1, it is assumed that you speak of the standard maths and not some wacky cultist things.

Lastly, even the finitists would agree that in standard maths 0.999... = 1.

onejayhawk said:
Please read my post above. It answers this.
Click to expand...
Which post? I don't remember seeing you giving definition of it.

onejayhawk said:
1) See above. A series is the sum of a sequence.
2) A difference which is not defined. It can be described, but there (currently) is no definition.
3) Less than and not equal.
4) Conceded. What was your point?
5) That is the question you should have asked from the start. The answer is that the 0.9+0.09+0.009+... =/= lim (0.9+0.09+0.009+...)
6) Sure they do. Why would they not? There is even a standard notation for it:
lim Σ f (xn
n=1 to infinite
7) If you are defining 0.999... = 1, just say so and stop bothering with so called proofs. That was not a given, see #5. I decline to accept the definition. A series is the sum of all elements of a sequence, not the limit of the sum. The sum and the limit are typically not equal, e.g. 0.999 =/= 1.
Click to expand...

1. So series is not a sequence like I said.

2. How does that fit with the field axioms of the real numbers which say that for every x and y the difference x-y is defined?

3. I know what "strictly less" means. I was asking what is the thing you are saying is strictly less than some other.

4. That even if the partial sums 9/10+9/100 etc. are strictly less than 1, their limit can still be exactly 1.

5. Yes it is, even with your typo (or a thought mistake) corected. A sum of a series is defined to be the limit of the partial sums. That's in every textbook there is. If you have an alternative definition for it, please present it here.

6. That is series, not a limit of series.

7. That's how it is defined, whether you accept it or not. I can decline accepting that
"hawk" means the bird and insist it's a word for a person that has incestuous relationships, but if I understand the word so, it's my problem, not everybody else's. Also, what do you think is the difference between sum of all the elements of the sequence and a limit of the partial sums? Why would one be different from the other. Furthermore: the sum is originally operation of two numbers. You can expand it for more numbers in the usual way: a+b+c means (a+b)+c, a+b+c+d means ((a+b) +c) +d. Since the commutativity the parentheses are not needed. However, it can thus be defined for only for finite addition. You never get an infinite sum by expanding this definition. That's why the "infinite sum" is defined to be the limit of the partial sums. (They teach you this at the first year in the uni).


onejayhawk said:
Instead we find ways to use algebraic numbers as substitutions. Since one set is dense on the other, this works. However, note that the definition of "dense" involves a limit.
Click to expand...

It's astounding that you seem to know what is a dense set, and you might even know what an algebraic number is, but still don't know the basics of calculus.

I haven't seen you give any reason to think that 0.999... would not be algebraic. You haven't even given any reason to think it would not rational. That should be fun. :)
 
^It isn't that astounding. We are in the information age, and any intelligent person can pick up just about any math knowledge they want without having a uni degree on math.

Yet you still almost mystically speak of 'basics of calculus' without having once given a specific reason why 0.999....=1 was set as something other than an arbitrary convention for (supposed) 'practical purpose'.

I think i will just go read some math-site articles on 0.999.... and 1, cause we are going in circles here. At least then i can base specific questions on uni math department sites, in the hope we can bypass the 'it is astounding' or 'you know next to nothing' tropes ;)
 
Okay, decided to do more responding
Atticus said:
Maths doesn't claim that it reflects physical reality. It's claim is "if these axioms hold, then these theorems are also true".
Click to expand...
Strictly speaking, yes, I agree.
"Sigh"... this leads us to the old debate of weather math is "real" or just a tool. A debate which has never been entirely settled, as far as I know. But in spite of that, I find the matter rather easy to comprehend.
Math is foremost a tool and hence not a reflection of physical reality. I can go along with that. However, I'd maintain that quantities are a physical reality. But as with anything, we need a language to describe this physical reality. And this is was math is for, fundamentally. Isn't that painfully evident?
It is true that it is normal that not any single mathematical tool has a direct correspondence in physical reality - such as negative numbers, to use a simple example. However, in such cases, the assumption still seems to be that if not a physical status, then the numbers still reflect a physical relation of physical statuses, or put differently a relative physical status. For instance, a negative number represents the lack of a certain quantity. So it indirectly still fully represents a real physical thing. Just that the assumption of it being "missing" is included.
The same goes for all assertions about the relation of quantities math engages in, for all I can see. So I really don't see the god-damn problem in judging math not by its own rules but the physical reality of quantities. Though with an exception: infinity.
Atticus said:
Why would you think that is true? What is the difference between a number and a "series of numbers"?
Is 123 a series of numbers or a number? Is 0.22 a number or a series of numbers? What is a number in your opinion? How does a series of (supposedly) inanimate objects "try" to get close somethin? What it means to be infintely close to something?

If you first answer these questions satisfactorily, we can continue with the rest of your post.
Click to expand...
My assumption is that every single digit stands for an exact quantity and that a "proper" number stands for an exact total sum of those quantities (whereas "proper" refers to it being tied to physical reality by representing an exact quantity).
1 stands for something being there one time.
0.1 stands for the 10th part of something being there.
1.1 stands for something being there one time and additionally the 10th part of the same thing being there one time. Etcetera. That is the most simplest case of what I already said - math reflecting physical reality in terms of quantities. Which, I repeat, is all math does, even if at times interwoven with assumptions about the relative status of a quantity (with the exception of infinity).

In the case of infinite digits, there is no exact total sum of the quantities the number describes, since the parts of the sum go on forever. As a consequence, 0.999... does not represent an actual amount of something. But just a function for a number to "behave" as it user moves along the digits.

0.999... "tries" in so far to get close to 1 as it represents a series of quantities which move in the sum every closer to 1 as the number goes on without ever reaching it, and as in so far as the people who thought of it tried to find a way to express something they can't express and as a consequence had to settle with creating a system which steadily comes closer without ever getting there. So it represents the mathematicians "trying" to get as close to 1 as possible.
And that it how being infinitely close looks like.
Atticus said:
Also, the misthought seems to be that 0.999.... somehow would "be the same as"
0.9, 0.99, 0.999, 0.9999,...
Why would that be?
Is 1.5 similar to
1, 1.5?
Is it similar to
1, 1.5, 1.5, 1.5,... ?
Why not
1, 1.5, 1.6, 1.5, 1.5, 1.5.... ?
Click to expand...
They are all similar in so far as they describe exact quantities. 0.999... violates that.
Atticus said:
If you reject the definitions that are used in maths, can you give some coherent account on what are numbers, and how they should be interpreted?
Click to expand...
I hope to have already done so now. Maybe not in sufficient formality, but at least I hope to have conveyed the idea.
 
Atticus said:
The finitists:
Oddly, I haven't seen a living person who would call him/herself a finitist. If you're going to adopt that position, you should be consistent with it though. That excludes differential and integral calculus for example, at least the way we know it.
Also, if you're a finitist, that's a thing you should inform for the very start. If you're saying just that 0.9999... is not 1, it is assumed that you speak of the standard maths and not some wacky cultist things.
Click to expand...
Don't really know what a "finitist" is. But it is true that so far I wasn't aware of how important it was to refer back to the concept of infinity as such, since this is what it seems to come down to. I was just narrowly focused on the particular problem so far.
I don't have any problem with how math makes use of the concept of infinity. All I care about is that math is rooted in non-abstract real quantities and that in terms of those quantities 1 /=/ 0.999... Likewise differentials et all may also not be fully correct, but as said that is fine, since I agree that math is a tool. I just think that the "math is only a tool, only a tautology"-angel falls short of the truth, and that rather obviously so. And I thought that if we discussed whether 0.999... = 1 the real quantity-angle to it would be highly implicit and didn't need to get spelled out so that one isn't buried in such know-it-all dickedry of the "Math pros" :p
 
Terxpahseyton said:
the real quantity-angle to it would be highly implicit and didn't need to get spelled out so that one isn't buried in such know-it-all dickedry of the "Math pros" :p
Click to expand...

I don't know. Even worse than know-it-all dickery of professionals is that of the amateurs. Do you really think that a person who has studied a thing for years, done it as his job, and is familiar to the tradition as long as the written history should be humble and servile, while some random person who's knowledge about subject doesn't even touch the first undergraduate course dismisses his arguments with things like:
onejayhawk said:
It is difficult to be more wrong.
Click to expand...

So, you think that arrogant dickery is warranted for people who don't know anything about subject, but people who do know are to blamed when they inform the ignorant of their ignorance that the latter may not know of? (This relates also to what Kyriakos said. In the information age people can't just pick any maths knowledge from the internets. They have to study it, like it's always been. Just reading about algebraic numbers or dense sets from the wikipedia doesn't mean that you've understood what you've read. (The word "you" here is a passive, doesn't mean you specifically)).

As for the other things you say in your post, whether the maths models the existing world well is of course a good question. If it didn't that would be a terrible thing. However, your objections against it aren't very good.

First of all, in the reality there are minimum lengths of objects and there are maximum number of them. Should that limit the maths? It would be stupid. What would be the maximal natural number? What would happen if you added 1 to it? Would you be willing to take that possibility into account in all your calculations? What if we wanted to do calculations of a different kind of reality? On the minimum length, are the objects limited to a single grid where they can change their position from one slot to another, but not move in between them or be displaced slightly off from the grid? How is that grid positioned? How do you know? What if you wanted to study a thing in real life that hasn't been studied with maths before, like stock prices or computer operating once were?

Then, a whole another objection to your idea is the fact that the maths is based on the reality and real quantities. People just have spend couple of millennia to think it through and make it consistent. The system is such that you pretty much have no other way to understand the number 0.999... If you want to have decimal numbers such that 0.5 means half etc., you pretty much have no other option than to understand them as they are understood in the contemporary maths. Other ways would almost certainly be self contradictory.

Then there's the thing that you keep claiming things like:
In the case of infinite digits, there is no exact total sum of the quantities the number describes, since the parts of the sum go on forever. As a consequence, 0.999 does not represent an actual amount of something.
Click to expand...
despite the fact that you've been presented multiple times how a decimal going on infinitely can be well understood.

The number isn't defined as you think it is. The number is defined to be
lim_{n\to\infty} \sum_{k=1}^n 9/10^k.

It happens that that limit is 1.

That number has a corresponding quantity in the reality, since that number is 1.

EDIT: One more thing on the dickery charge, pretty much all the mathematicians on this forum do answer patiently to people who want to know stuff, me included. Even when those people present objections, they get fair treatment. What's wrong is that some people in this thread don't use a fleeting moment to consider what the maths people have written. That's dickery. Some of them deny that words mean what they mean, even when presented with sources. That's dickery. Instead of giving sources or anything to back up their claims they think their word for it is good, because sure, of course people who hasn't studied maths is more valid source of information than a mathematician or the numerous books on the subject he can refer to. That's dickery.
 
You've gone back to limits again. You cannot disprove a notion based upon the rejection of the validity of an incinitrly recursive number simply by saying that another infinitely recursive number proves it.

This conversation seems to have gone as far as it can. You don't have a proof that does not require assumptions that are a'priori rejected by the assumptions of your interlocutors. At thus stage all you can do is roll your eyes and say 'i've never met a living finitist' and 'look I have a degree in this'. If would be nice if you could simply admit that you are arguing against a perfectly valid but unusual position.
 
Atticus said:
It's astounding that you seem to know what is a dense set, and you might even know what an algebraic number is, but still don't know the basics of calculus.

I haven't seen you give any reason to think that 0.999... would not be algebraic. You haven't even given any reason to think it would not rational. That should be fun. :)
Click to expand...

That's my degree emphasis. The grad work was in stats.

The reasoning is simple. It's less than, but not equal to one, but the difference is not defined. Since 1 is rational, the other cannot be. If you could define two distinct but connected points, these would be candidates. I have said that the difference was undefined. Perhaps not measureable would be better. I will have to consider.

This grew out of contemplating dense sets. I realized that limits cannot be unique, or even countable. For any limit, there must be an uncountable set of points that fit the definition. Otherwise continuity fails. Of this uncountable set, we can define only one point. The set is open and sort of bounded, but the definition of bounded does not work. The real numbers are a measure space. How is that supposed to work?

This reminds me of non Euclidian geometry, relativity and quantum mechanics. The generalizations are there, but how often do you get used?

J
 
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