/sci/ - Science & Math

[Anonymous No.6546839]

Given the fact that the interval between 0 and 1 is so densely populated with transfinite numbers (infinitely unpredictably uncountable similar to pi) that it is virtually impossible to ever get from 0 to 1 if one were to take the time to count on the linear line 1, 2, 3, and so on...

What makes some think that 0.999... = 1 ?
I'm genuinely curious.

[Anonymous No.6546844]

>>6546839
>is so densely populated with transfinite numbers
No, it isn't you moron.

[Anonymous No.6546847]

>>6546844

Go read Cantor.

[Anonymous No.6546856]

>>6546839
> 0.999... = 1
because there is no number between them.

[Anonymous No.6546865]

>>6546847
> transfinite
> transcendental
Learn the difference: it might save you from looking like an uneducated idiot talking out of his ass.

[Anonymous No.6546869]

>>6546847
Do you actually know what a transfinite number is?
Do you really believe things like aleph-null lie in the interval [0,1]?

[Anonymous No.6546874]

>>6546865

he meant transfinite though...

>>6546869

>he's never even read Cantor or understood set theory

/sci/ = /shit/

[Anonymous No.6546876]

>>6546839
>Given the fact that the interval between 0 and 1 is so densely populated with transfinite numbers (infinitely unpredictably uncountable similar to pi) that it is virtually impossible to ever get from 0 to 1 if one were to take the time to count on the linear line 1, 2, 3, and so on...

Implying anyone here understands this sort of nuanced point. Have fun arguing semantics with autists OP.

[Anonymous No.6546881]

>>6546856
There are many numbers between them. They just can't be represented as decimals.

[Anonymous No.6546886]

>>6546874
> They seem to know what they're talking about, so I'll troll them by samefagging and pretending I'm not OP just to see what will happen.
Just go look up transfinite numbers, OP, and then don't post again until tomorrow. You've looked stupid enough for 1 day.

[Anonymous No.6546902]

>>6546881
There are no REAL numbers between them.

[Anonymous No.6546914]

>>6546902
Is there some kind of other number between them?
complex and quaternions don't have an order so between doesn't make sense right?

[Anonymous No.6546921]

>>6546876
OP, quite being a samefag

Saying transfinite vs transcendental is mere semantics is like saying 1 = 2 and calling "autism" when people say, actually 1 does not equal 2.

>nuanced
No, it is woolly bullshit.

[Anonymous No.6546922]

>>6546914
Well, you could try to argue that infinitesimals were between them, but yeah, there's no number between them in the end.

[Anonymous No.6546924]

>>6546921

>transfinite vs transcendental

No one mentioned transcendental. Do you even know what transfinite means in set theory? Please refer to Cantor for an education.

[Anonymous No.6546927]

>>6546922
>there's no number between them in the end.

What is this "end" you imagine? Do you know how infinity works, do you know what infinite repetition implies?

[Anonymous No.6546928]

>>6546924
So tell me how a transfinite number is a member of the set [0,1]

[Anonymous No.6546934]

>>6546839
>infinitely unpredictably uncountable similar to pi
you don't know what these words mean

[Anonymous No.6546937]

>>6546924
>Do you even know what transfinite means in set theory?
Yes I do, and transfinite numbers are not real numbers, yet every element in [0,1] is a real number, so how is [0,1] "populated" with non real numbers like transfinite numbers.

Also, why do you think pi is a transfinite number?

[Anonymous No.6546939]

>>6546839
>thinks pi is "uncountable"

[Anonymous No.6546946]

>>6546927
>What is this "end" you imagine?
It means that after you investigate said arguments you reach the conclusion that it is not the case.

[Anonymous No.6546947]

>>6546946

It's not the case that an infinite has an end.

[Anonymous No.6546953]

>>6546947
it's an idiom not a mathematical statement.
you read his post wrong.

[Anonymous No.6546955]

>>6546947
"in the end" is a figure of speech you 1/10 troll.

[Anonymous No.6546957]

>>6546839
Find a real number <span class="math">k[/spoiler] such that <span class="math">0.999... < k < 1[/spoiler].
What's that? You can't?
Then 0.999... is exactly equal to 1.

If you still have doubts, just go talk to an actual math professor.

[Anonymous No.6546959]

>>6546957
fucking constructivist

[Anonymous No.6546962]

0.333... = 1/3
Multiply by 3
0.999... = 1

QED

[Anonymous No.6546966]

>>6546959
find a real number <span class="math">k[/spoiler] that satisfies <span class="math">0.999... < k < 1[/spoiler]. Then you can insult me all you want.

[Anonymous No.6546974]

>>6546966
There is none of course. 0.999... = 1

But your "find" implies the only valid proof of the converse is by explicitly finding such a k.

There are other ways to prove things than by construction.

You are too retarded for /sci/, try >>>/mlp/

[Anonymous No.6546978]

>>6546974
>But your "find" implies the only valid proof of the converse is by explicitly finding such a k.
When the hell did I imply this?

I was just giving the proof of 0.999...=1 that seems most intuitive. Autists, please, just lay the gun on the table, that's right, just calm down...

[Anonymous No.6546986]

>>6546978
>When the hell did I imply this?
The first 3 lines of >>6546957

>whats that? you can't
>then 1 = 0.999...

Though your conclusion is correct (0.999... does equal 1) your reasoning is dumb-fuck tier.

>Autists
The complaint of the idiot. You can't do math without rigor. 0.999.. = 1 is a classic example of that.

[Anonymous No.6547003]

>>6546986
So you're saying that the fact that there is no real number <span class="math">k[/spoiler] such that <span class="math">0.999... < k < 1[/spoiler], does NOT imply that 0.999... = 1 ?

...are you trolling?

The real numbers form a complete metric space. Between any two distinct real numbers, there is an infinite number of other real numbers.
Therefore, if when given two reals <span class="math">a[/spoiler] and <span class="math">b[/spoiler] you cannot find any real numbers between them, then <span class="math">a = b[/spoiler].

[Anonymous No.6547007]

>>6547003
>So you're saying that the fact that there is no real number k such that 0999k1, does NOT imply that 0.999... = 1 ?
Nope. I'm saying that not being able to produce such a number is not proof.

Not being able to find an odd perfect number is not proof no such number exists.

[Anonymous No.6547011]

>>6547003
alternatively, you could also just say that the real numbers are a Hausdorff space, which is more general.

[Anonymous No.6547012]

You can't write zero point 10

I win

Everybody go home

[Anonymous No.6547018]

>>6547007
>Nope. I'm saying that not being able to produce such a number is not proof.
It is, though. The real numbers are a Hausdorff space; i.e., any two distinct real numbers have disjoint neighborhoods.
Have you even taken topology? You sound like an angry highschooler.

>Not being able to find an odd perfect number is not proof no such number exists.
Congratulations on finding a completely different problem that has absolutely nothing to do with 0.999 ≠ 1.

Just take a basic topology course, for fuck's sake.

[Anonymous No.6547019]

>>6547007
>try to find
>you can't

You can't construct one. The computable numbers lie dense in the real numbers. It isn't a proof exactly he offered, but a sketch to one and there are easier proofs for ops problem, but I think what he says makes perfect sense.

[Anonymous No.6547025]

>>6547018
Well it's a bit more complicated, since one could assume, that you can seperate them, but with numbers wich are not computable so you can't find any. However the computable numbers lie dense in the real numbers, so that's not a problem and the proof just works as you said.

However I think it's easier to directly proof that there is no number k with that property and not just that we are unable to find one.

[Anonymous No.6547038]

>>6547025
>I think it's easier to directly proof that there is no number k with that property and not just that we are unable to find one.
Yes, I know. I was just trying to show OP what I think is the most intuitive way of showing that 0.999... = 1 in this post: >>6546957. I just didn't expect that it would blow up into an argument.

[Anonymous No.6547039]

>>6547018
Sure, but I am not criticising sound proofs that 0.999. = 1. There are plenty. I am not criticising your conclusions either.

It is your logic that I am shitting on, because of how full of holes it is.

In particular am criticising your assertion that if I cannot produce an example of a number between 0.999 and 1, then the matter is proven, done, the end, and we can safely conclude that no such number exists.

>can't into analogy
If your reasoning was sound, it would form the sound basis for other proofs, I could say the fact that I cannot give you an odd perfect number is proof that none exists.

Get it now? Or is your second rate brain unable to actually think critically not just about the subject matter of mathematics, but the reasoning behind it.

[Anonymous No.6547043]

>>6547038
>intuitive
No. The best way is to say, suppose you CAN find a number between them, then show it leads to a contradiction.

[Anonymous No.6547050]

ok guys i've got a feeling that i'm breaking the spell right now but i absolutely love these threads. Does anybody else mainly browses /sci/ for these transfinite levels of trolls and shitposters and laughs their asses of?

I think this board consists of maybe 20 posters consistently given richer and richer variants on a couple of theme, to the point almost of perfection. I mean
>densely populated with transfinite numbers (infinitely unpredictably uncountable similar to pi)
If that's not a collective work of art then i don't know what is..

I really don't mean this sarcastic.

[The Only Good Communist Is A D(...) No.6547053]

>>6546839

http://en.wikipedia.org/wiki/0.999...#Analytic_proofs

An entire Wikipedia article dedicated to the subject. Or, if you're one of those "WIKIPEDIA IS TEH BAD SOAUCE LOLROFLSHIT!", talk to a math professor. Or learn Calculus up to infinite series.

[Anonymous No.6547056]

>>6547039
so have you taken topology or not?
The answer is no, obviously, so just stop trying.

>I could say the fact that I cannot give you an odd perfect number is proof that none exists.
That's different.
I was saying that if you can prove that there is NO real number <span class="math">k[/spoiler] such that <span class="math">0.999... < k < 1[/spoiler], then it follows (from the basic topological properties of the real numbers) that 0.999... and 1 are the same real number.

>>6547043
>The best way is to say, suppose you CAN find a number between them, then show it leads to a contradiction.
I guess this might be more intuitive, but either way it amounts to the same thing.

[Anonymous No.6547065]

>>6547056
>so have you taken topology or not?
Yes, two courses, analytic and algebraic. I am a maths graduate. But this is irrelevant. We are arguing the reasoning, not the subject matter. We both agree on the conclusion.

>I was saying that if you can prove that there is NO real number
No, you weren't. f you had I would have agreed.

>it amounts to the same thing.
No, it doesn't.

I seriously suggest you take a proper course in how proofs work, and maybe a course in foundations too.

[Anonymous No.6547077]

>>6547065
When did I say I was trying to do a formal proof? Show me.

In my first post, all I did was try to show something to OP in a way that I thought would be intuitive. There was absolutely no reason to treat it as a formal proof.
But then a bunch of autists got enraged for no good reason.
Is /sci/ always like this?

[Anonymous No.6547078]

>>6547039
I think you missunderstood his argument. Trying it a few times and failing is no proof, okay, but I think what he meant was, that there is no way to construct such a number. You can, and I think that's overkill, proof that there is no way to construct a computable number k with that property and you can build a rigorous proof on that.

[Anonymous No.6547092]

>>6547077
I think the only person who seems enraged is yourself.

As I started to attack your arguments on grounds of rigour, you defended them, bringing up topology, attacking my credentials, and all kinds of butt-hurt sophomorish shit. If you aren't claiming it as a sound proof, you should just say so instead of defending it as such.

>Autists
This is pretty much what people say when they can't into rigour. Ironically, all your "muh topology" depends on the rigour you claim is autism, when it suits you.

[Anonymous No.6547095]

>>6547077
>Is /sci/ always like this?

/sci/ is anti-intellectual.

If you want intellectual discussion go to /pol/ or /lit/

[Anonymous No.6547096]

>>6547092
He said that proving that it is impossible to find a number k such that 0.999... < k < 1 means that they are both equal. A simple and true statement. Then you started going on about constructivism and how it is a wrong argument, but then you said:

>I'm criticising your assertion that if I cannot produce an example of a number between 0.999 and 1, then the matter is proven, done, the end, and we can safely conclude that no such number exists

This is textbook strawman, because that is not remotely what he said. You misunderstood him, and it wasn't an understandable misunderstanding either, because he specifically said that you need to PROVE that no such number exists. Admit that you're wrong and let's move on.

[Anonymous No.6547097]

>>6547092
>As I started to attack your arguments on grounds of rigour
Why did you need to attack my arguments in the first place? I never claimed a formal proof.

>you defended them, bringing up topology
Because I thought you were completely misunderstanding what I was saying. I now understand that you were just butthurt about my lack of rigor, even though I never claimed rigor.

> >Autist
> This is pretty much what people say when they can't into rigour.
It's what people say when they don't claim to be rigorous and others attack them for not being rigorous.

[Anonymous No.6547098]

>>6547095
The anon you are replying to is anti intellectual.

[Anonymous No.6547105]

>>6547096
>>6547096
>He said that proving that it is impossible to find a number k such that 0.999... < k < 1 means that they are both equal.
No he didn't.

I would have had no problem with that.

>Strawman
It's what he actually said. Read the post here >>6546957


Here is the thing you are failing to understand

****Being unable to show something exists is not logically equivalent to showing it does not exist****

>he specifically said that you need to PROVE that no such number exists.
find me the post

[Anonymous No.6547108]

>>6547097
>I never claimed a formal proof.
Then use tentative language. Not emphatic statements of certainty.

[Anonymous No.6547116]

>>6546839
In all truth the 0.9999... just proves that 1 is infinitely far away from 0.9 and as soon as we start filtering all that space we start getting data that as far as 1 is from 0.9 that 9 is from 10 and so on and so forth

The the real question here is, is 1 a real number?

[Anonymous No.6547120]

>>6547105
I think you misinterpreted this post: >>6546957
> Find a number k such that 0.999... < k < 1
> What's that? You can't?
> Then 0.999... is exactly equal to 1.

Obviously, the statement rests on the fact that it is impossible to find such a k.

[Anonymous No.6547125]

>>6547108
>Then use tentative language. Not emphatic statements of certainty.
I'm not going to change the way I write just because some mathematicians get butthurt about it.

[Anonymous No.6547127]

>>6547120
>Obviously, the statement rests on the fact that it is impossible to find such a k.
So it rests on that which it is trying to prove?

That's even worse reasoning.

[Anonymous No.6547129]

>>6547127
Once you prove that there is no such k, there is another step required before you conclude that 0.999... = 1 (the Hausdorff property of the reals).

[Anonymous No.6547131]

>>6547125
Your call, but you will waste less of your time and others' time if you signpost when you are interested in arguing rigorously and when you are just shooting the breeze.

>butthurt
No, you don't get it, rigourous mathematical argument is what I enjoy.

[Anonymous No.6547133]

>>6547129
Okay, so it was circular reasoning with a missing, unstated step in the circle.

[Anonymous No.6547172]

Here is a simple proof using geometric series:)

[Anonymous No.6547194]

>>6547172

every proof of 0.999... = 1 will implicitly assume a rounding off of 0.999... = 1 at some point, in order to prove it. Just like the proof you posted.

For example, near the end it says 9(1/9/10) which is 9( 10/9) which is 9 * 1.11111, which is 9.99999999...

But what does this proof do? It assumes exactly what it wants to prove! And turns 9.9999... into 10, right at the last line.

Many mathematical proofs commit this sort of circular reasoning without even knowing it.

[Anonymous No.6547196]

>>6547172
That shit is retarded yo. Why would you not just take the 9 out of the sum and then evaluate that geometric series? You gain nothing from all dat complexity.

[Anonymous No.6547198]

>>6547194

[Anonymous No.6547241]

>>6546839
I have a better idea. If it's not written down, it doesn't exist.

[Anonymous No.6547262]

<span class="math">1/(1-0.999...) ?[/spoiler]

[Anonymous No.6547268]

>>6547194
I hope this is bait. On the off-chance you are being serious:

9(10/9)=9/9(10)=1(10)

[Anonymous No.6547289]

>>6547262
undefined.

[Anonymous No.6547310]

>>6547289
That's the point.

[Anonymous No.6547334]

>>6547268
>9(10/9)=9/9(10)=1(10)

There is always a phase where 10/9 = 1.111...
And then you multiply it by 9 to get 9.999...

You are just using simplified arithmetic to ignore this fact.

[Anonymous No.6547356]

>>6547334
Okay, I'll do it extra slow for you:

x*(y*(x^-1))=x*((x^-1)*y)=(x*(x^-1))*y=1*y=y

The first equality is commutativity, the second one is associativity.

Those equalities always hold true in the real numbers.

[Anonymous No.6547365]

>>6547334

hmm this is surprising

[Anonymous No.6547390]

>>6547365

Yeah, it's surprising he is that retarded.

[Anonymous No.6547394]

>>6547390

huh? but it's correct.

[Anonymous No.6547432]

>>6547394
There arises the term of 9*(10/9) and he evaluates it to 9*1.111... and then to 9.999... and then rounds up to 10 by using what is to be shown instead of calculating it by:

>>6547356

Then he accuses the proof of circular reasoning. I mean, he decides to calculate the term using what is to be shown instead of using the commutativity and associativity of the real numbers. He missinterpretes the proof in a really strange way, so strange I'm still unsure whether it's a troll or he just didn't understand it.

[Anonymous No.6547477]

>>6547432

i dunno what you don't get, he's right.

[Anonymous No.6547526]

>not knowing about cardinality and aleph numbers
Even a greasy physics major like me knows at least the basic concepts.

[Anonymous No.6547546]

>>6547526
That comic's explanation of why the cardinality of integers and reals is different is wrong. The same argument would hold for the rationals, but the rationals are clearly countable.

[Anonymous No.6547564]

>>6547477
There is nothing to get, dude. He says you need to use the equality of 0.999... and 1, but it's simply not true. You can use it and he does, but there is no need to do so. I mean, of course we could use his proof and of course it would be circular, but we simply don't because we don't need to and it makes no sense to do so. I was actually surprised anyone would think that way, just because it seems to be really absurd.

[Anonymous No.6547569]

>>6547526
>there are an infinite number of real numbers in a single integer
>"in"

what does that mean?

[Anonymous No.6547639]

>>6547116
No.

[Anonymous No.6547663]

>>6547569
How many reals are between 0 and 1?
There's your answer

[Anonymous No.6547690]

>>6547564

>we don't need to use the identity to prove it

except you use it even when you think you aren't, you just aren't thinking deeply enough

[Anonymous No.6547830]

>>6547003

OP should read Dedekind and stop trying to cum Cantor's sacred name on our faces. He obviously doesn't know jack about transcendence theory.

[Anonymous No.6547854]

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

QED

[Anonymous No.6547855]

>>6547854
Third line is supposed to be
10x-x=9.999...-0.999...
idk what happened

[Anonymous No.6547857]

>>6547854
why are you posting grade school level "proofs" on a site for adults discussing math and science?

[Anonymous No.6547880]

>>6547196

Isn't that exactly what he did?

[Anonymous No.6547896]

>>6546839
>Given the fact that the interval between 0 and 1 is so densely populated with transfinite numbers (infinitely unpredictably uncountable similar to pi)
The stupid burns. And I didn't even read the rest of his post.

[The Only Good Communist Is A D(...) No.6548632]

>>6547857

People are honestly arguing whether or not the 100%, inarguable fact that 0.9999…=1 is true. Trust me, there are no "adults" here.

[Anonymous No.6548660]

>>6547857
Troll post for troll thread.
Not sure what is wrong there

[Anonymous No.6550709]

x=0.999...
10x=9.999...
10x-x=9.999...-.999...
9x=9
x=1

[Anonymous No.6551841]

OTP

[Anonymous No.6551964]

>>6547896

>he hasn't read Cantor yet
>he doesn't understand what transfinite numbers are and where they exist

[Anonymous No.6551973]

>>6551964
>caring about the quality of posts on a 0.999...≠1 thread
>not saging a 0.999...≠1 thread

[Anonymous No.6551997]

>>6546839
>troll status achieved
>you have earned, "obviously has never taken calculus" along with your award!
>congratulations

leave /sci/ pseudo-intellectual faggot

[Anonymous No.6552019]

>>6551964
Transfinite numbers are obviously larger than any number between 0 and 1.

[Anonymous No.6553542]

>>6552019

high schooler detected

[Anonymous No.6553578]

.999999999.... can be represented as 9/10 + 9/100 + 9/1000......

This makes it a geometric series, with r = 1/10, and since abs(r)<1, it is convergent, and the convergence of a geometric series is S=a/(1-r), a being 9/10 so its (9/10)/(1-1/10) which is (9/10)/(9/10), or just 1.

Thus .999...=1

[Anonymous No.6553758]

>>6553578
>it is convergent

>assuming convergence in order to prove convergence

Hilarious! circular reasoning has done it again.

>assume the bible is true, well then these facts about Jesus must follow!!! Wow!

The problem with these proofs is the very idea of convergence of an infinite.

[Anonymous No.6553766]

1/9 = .1111111...
.111111111*9=1

.333333333... = 1/3
.333333333...*3 = .9999999...
.999999999... = 3/3
3/3 = 1
.99999999... = 1

it's that simple

[Anonymous No.6553769]

"Transfinite - You can't count to it, but you can count to it!" -Fuckheaded math enthusists, Cantor

[Anonymous No.6553794]

>>6553766
>1/9 = .1111111...
>.111111111*9 = 1
>.111111111*9 = 1

0.111111111*9 does not equal 1.
It's not even close to one.

0.111... * 9 can't be multiplied.
And even if we allow it, 0.999.... is still infinitely far away from 1.

[Anonymous No.6553805]

>>6553578
Fractions and decimals are not interchangable, especially in proofs of math.

It's like none of you have ever even tried to get your masters or phd.

[Anonymous No.6553836]

>>6553794
1/9 * 9 =/= 1
this is your argument?
please leave
we have enough trolls already

if you are genuinely this dumb
please type 1/9 into whatever bad calculator you rent from your high-school
then type * 9 and hit enter

compare this with the results of multiplying 9/9 by 1, or 1/3 by 3

[Anonymous No.6553886]

>>6553836

>calculators prove things now

oh boy

[Anonymous No.6553896]

>>6546839
> 101 posts
congratulations, you have successfully reached the troll limit.

Pls, no more posts after this one.

[Anonymous No.6554038]

>>6553896

thanks for the bump

[Anonymous No.6554042]

>>6546839
>densely infinitely unpredictably uncountably virtually impossibly
GTFO

[Anonymous No.6554046]

>>6553886
>1/9 * 9 isn't 1
u wot

[Anonymous No.6554053]

>>6554046
>1/9 * 9

representing 0.999... as a fraction doesn't help your case.

We all know what it actually means and why it doesn't converge to any particular digit.

[Anonymous No.6554073]

>>6554053
>it doesn't converge
Lrn2converge

[Anonymous No.6554077]

>>6554073

>this infinite series that never ends
>well it ends here

>it never ends
>ok but now it ends

Convergence!

[Anonymous No.6554081]

>>6554077

it never ends and it ends.
Why? Proof?

Well we need it that way!

[Anonymous No.6554423]

just visit r/trollmath on reddit

[Anonymous No.6554477]

>>6554077
An infinite series can converge. That doesn't mean it has a finite number of terms, dumbass.

[Anonymous No.6554496]

I will try one last time.
Cut an item into three pieces.

.33333... + .33333... +.33333... = 1

[Anonymous No.6554526]

>>6552019

>Can't tell if troll

>In a .999 thread

>2k+14

[Anonymous No.6554541]

.9999 =/= 1

however this pic proves otherwise...still, if it did =1 it would be 1, not .9999

what about .99999999999999999999998 does that = .999? what about .99999999999999999999999999999997 does that = .999998?

its a vicious cycle of numbers not meaning what they literally mean.

[Anonymous No.6554556]

>>6554541
2/10 I replied.

[Anonymous No.6554560]

>>6546839
You're so retarded, it hurts to read this crap.
Am I the only the only one thinks that whoever starts these threads should be banned?

[Anonymous No.6554563]

>>6554541
in case you're not trolling

0.99999999999999999999998 is not equal to 0.999,
0.99999999999999999999999999999997 is not equal to .999998,
and 0.9999 is not equal to 1.

However, 0.999... is equal to 1.

[Anonymous No.6554859]

>>6554563

0.999.... is an infinite series that never ends.
It is always infinitely far away from 1.

Infinity has no end you goofball. It never finishes approaching. It never terminates. It never magically becomes a whole number just because you want it.

As far as mathematicians are concerned convergence is just a short-hand and pragmatic tool for doing calculations, it has no proof and makes no logical sense.

[Anonymous No.6554866]

>>6554563
>However, 0.999... is equal to 1.

0.999... repeating, forever, is just as far from 1 as 0.999999999997 is

Namely, they are both infinitely farm from it.

The idea of convergence is a contrivance and can't be proven.

[Orlando Herrerra No.6554920]

Straight from Wiki. Best proof I've seen.

[Anonymous No.6554925]

>>6554920

>assumes convergence to prove a convergence

nice circular reasoning there chump. come back to us when you graduate middle-school.

[Anonymous No.6554938]

>>6554925
Explain to me how it "assumes" convergence, faggot. Protip: It doesn't

[Anonymous No.6554941]

>>6554925
The convergence of it is already known. The proof is specifically about which value it converges to.

back to middle school

[Anonymous No.6554961]

Bill nye the druggie guy

[Anonymous No.6554972]

>>6554938
>Explain to me how it "assumes" convergence, faggot. Protip: It doesn't

prove the possibility of convergence of an infinite series.

the problem isn't this particular case of 0.999... = 1, the problem is more general than that.

[Anonymous No.6554991]

Good number for today: 3.12353343423435642342342352352342342348888888888883231283126312756371274836473748236478623894623654237642375478237482364786378462378462783647823647823664156341263512693627362738917387183617668547812634781256371268589589496969672384738942378647236478263476246269

[Anonymous No.6555027]

>>6554991

implying pi "approaches" or "converges" to some finite whole number

This is about as stupid as thinking 0.999... converges to some particular whole number.

[Anonymous No.6555071]

>>6555027
>This is about as stupid as thinking 0.999... converges to some particular whole number.
infinite decimals are defined as the limit of a series. series are defined as the value to which their partial sums converge

[Anonymous No.6555080]

>>6555071

>god is defined as existing

well, I guess we can just define things without proving them now.

[Anonymous No.6555088]

>>6555080
I guess you aren't too familiar with math.

So long as you are consistent with that which is already defined, you may go ahead and define what you like.

Whether your definition is consistent, I don't know. Theology doesn't interest me.

[Anonymous No.6555089]

>>6555088
>Whether your definition is consistent, I don't know

convergence is not a consistent or logical definition in any possible sense.

[Anonymous No.6555090]

>>6555080
nj wetburger please go

[Anonymous No.6555093]

>>6555089
Well, you would have to argue that with all analysis since Cauchy, Weiestrass, et al.

[Anonymous No.6555104]

Strange. Shouldn't the result be 1 no matter how often it is multiplied by itself?

[Anonymous No.6555106]

>>6555104

rekt is right

[Anonymous No.6555118]

>>6555104
http://www.wolframalpha.com/input/?i=%28.999...%29%5E999999

[Anonymous No.6555172]

>>6555104
but 0.999 (not-repeating) is not equal to 0.999...(repeating)

[Anonymous No.6555180]

1, 2/2. 3/3, 4/4, 1.0 1.00, 1.000, 1.0000, etc

[Anonymous No.6555195]

>>6555172

0.999...(repeating)

Is only equal to 0.999... repeating, it never reaches 1. Ever.

[Anonymous No.6555198]

>>6555195

>it never reaches 1. Ever.

because it is 1 you dummy

[Anonymous No.6555204]

saying that 0.999... is not equal to 1 is like saying 3/3 is not equal to 1.

[Anonymous No.6555205]

1-0.999...=0.0....01

[Anonymous No.6555216]

>>6555198

>an infinite, non-ending series is a whole number

[Anonymous No.6555218]

We have 1.0 and 0.9

There is a difference between them of 0.1

1.0 - 0.9 = 0.1

If we add another nine, the difference decreases

1.0 - 0.99 = 0.01

If we add another nine, the difference decreases further

1.0 - 0.999 = 0.001

If we never stop adding nines, 0.999... the difference never stops decreasing, at a fixed rate of reduction of ten times smaller per nine added.

difference = 1/10^x, with x being the amount of nines.

0.990... has an infinite number of nines, meaning the difference between 0.999...and 1 is infinitely small, which by definition, means it does't exist.

If there is no difference between two real numbers, again by definition, it means they are the same.

Which takes us to the conclusion that 0.999... equals 1.


btw, as a teacher, I love these threads, lots of opportunities to learn new ways to present the same argument. It's really helpful in making everyone understand.

It's kinda like playing tennis against a wall, you're not going to win, but you can still get something useful out of it.

[Anonymous No.6555223]

>>6555216

1.000... is an infinite, non-ending series AND a whole number

[Anonymous No.6555243]

>>6555223
So is 0.999...

The same whole number as it happens

[newt !!TSnMjm2ZKHc No.6555250]

>>6546957
K = (0.999.. + 1) / 2

# rekt

how about you learn some basic maths before you start acting smart.

[Anonymous No.6555270]

>>6555218
>>6555243

>thinking 0 adds something.

The repetition of 0 is irrelevant. 0.999... requires a repetition of infinite nines, otherwise it becomes a terminal number namely 0.999999

[Anonymous No.6555274]

>tfw this thread isn't against the rules

[Anonymous No.6555299]

>>6555270

>0.999... requires a repetition of infinite nines

so?

What is the hold-up?

[Anonymous No.6555307]

>>6555299

an infinite series is not a whole number.

[Anonymous No.6555308]

>>6555307

how so?

[Sceak No.6555309]

>>6546839

There's casual rounding, where people are usually instructed to round, there's also an incapacity for some people to fathom infinity, and there is also some people breaking the ... sound barrier.

[Anonymous No.6555939]

>Infinite Series converges to 1

[Anonymous No.6555956]

>>6555308
How is a cat not the same as a dog?

[Anonymous No.6555962]

>>6555218
>infinitely small, which by definition, means it does't exist.
That's where your argument fails. Something that is infinitely small can be treated as non-existent, by it by definition exists. You're treating an approximation as exact.

I'm glad you're not teaching my kids.

[Anonymous No.6555963]

>>6555962
>by it by definition exists
*but by definition it still exists.

[Anonymous No.6556010]

All these arguments for 0.999...=1 have been disproved on
http://www.reddit.com/r/trollmath

[Anonymous No.6556104]

>>6547854
X=.999...
10x=9.9999...
10x-x=9.999...-.999...
9x=9
X=1

This proof denys the existence of extraneous solutions, and that an equation can produce a value that does not actually solve the equation.

[Anonymous No.6556314]

>>6556104
>10x-x=9.999...-.999...

multiplying and subtracting infinities
nice.
real neato

[Anonymous No.6556433]

OP this is just the Dichotomy paradox from Zeno's paradox
http://mathworld.wolfram.com/ZenosParadoxes.html

[Anonymous No.6556450]

>>6556314
>thinking a number less than 10 represents infinity
Back to riddet with you.

[Anonymous No.6556496]

>>6546957

0.9999

#rekt

[Anonymous No.6556525]

>>6555962

The majority of /sci/ will never accept that science is a philosophy of best guesses based on assumptions and circular reasoning, sadly.

[El No.6556551]

>>6556525
True dat.

[Anonymous No.6557404]

>>6556525
but where discussing maths not science here, and we're not dealing with approximations we're dealing with a convergent sequence

[Anonymous No.6557406]

>>6557404
*we're

[Anonymous No.6557435]

>>6557404

Alright, I'll bite. Let's play the semantics game.

Aristotle called mathematics a science. And by the purest and oldest definition of the word, it is, to some degree at least, because it is some form of knowledge.

If you insist it is an art, then math is entirely subjective, and we all know that is not the case.

Definitions aside, however, there is still the matter of an infinite series of numerals in 0.999... You can play number games all you like, but it is still approximation, because at the end of the day you are saying that an endless series of numerals has an end after all, and that is simply not the case. 0.999... is certainly not infinitely large, as it is still less than 1, but it is infinitely long. What you are saying is tantamount to saying that a ray has two ends.

It doesn't.

[Anonymous No.6557442]

>>6546839
> transfinite numbers like pi
> unpredictably uncountable
> take the time to count
> the linear line 1, 2, 3, and so on

kill the thread, kill it with fire

anyone who posts without sage deserves cancer

[Anonymous No.6557444]

>>6557442

>he's never read Cantor

[Anonymous No.6557449]

>>6557444
> 2bait4me

[Anonymous No.6557624]

>>6557449

>implying you've counted to the end of pi

[Anonymous No.6557658]

This topic makes me thank that Math is not democratic.
When will we be opened a thread about the Pythagorean Theorem?

[Anonymous No.6557671]

>>6557658

Me too, otherwise virtually everyone would believe 0.999... = 1

OH WAIT

(captcha: services xtryone)

[Anonymous No.6557836]

>>6557435
Maths isn't a science because the conditions for establishing a mathematical truth is different from the conditions for establishing scientific truth, I wont go into this any further because it is merely a question of semantics as you say.

As far as your second point, is concerned, you're obviously missing the point completely, so i'm gonna try and give my own proof (I'm only a first year engineering student, so excuse my lack of rigour)

Start by considering two cauchy sequences 1/10^n and 9/10^n (i'm not going to bother proving that these are cauchy sequences because it's obvious). Let S be the sum of the first sequence from 1 to infinity, it's easy to see that this equals 0.111... (again to obvious to bother with a proof) or 1/9, therefore 9S = 1. Let T be the sum of the second sequence from 1 to infinity, again we can see that T = 0.999... (also too obvious to bother with a proof), now observe that the second sequence is simply the first sequence multiplied by 9, therefore T = 9S and as we determined earlier, 9S = 1.

In simple terms, I'm saying that given 1/9 = 0.111... then 0.111... * 9 = 0.999... = 9 * 1/9 =1

Also, no one is saying that 0.999... has an end, what we're saying is that the interval between 0.999... and 1 does not exist, therefore one cannot sensibly call them distinct numbers. Essentially, 0.999... and 1 are two different representations of the same number in the same way that 1/9 and 0.111... are representations of the same number.

tl;dr go read a real analysis textbook

[Anonymous No.6557851]

>>6557836

>no one is saying that 0.999... has an end, what we're saying is that the interval between 0.999... and 1 does not exist, therefore one cannot sensibly call them distinct numbers

What you're saying is that what I will describe (for the sake of this argument's clarity) as 0.000...1 is so vanishingly small as to be inconsequential to the point of it being more practical to ignore it entirely. I don't mind it being done because at least there's a valid reason, but for god's sake don't pretend the difference between 0.999... and 1 literally does not exist at all. That's the kind of ignorant shit that pisses me off.

>particles below the planck threshold don't exist because they have no proper dimensions herpaderp

That's what it sounds like. If you're a physicist, I'm sure you can understand what I mean.

tl;dr do what is most efficient and sensible but don't magically will numerical constructs completely out of existence because they're untidy and awkward

[Anonymous No.6557856]

u mad bro?

[Anonymous No.6557890]

>>6557851
Obviously I'm not explaining myself well enough. Let's go back to our sequence from before. First we let the upper bound for 0.999... be 1 and the lower bound be 0.9, then the difference between the upper bound is 1/10, the first term of our sequence. Then we add 0.09 to our lower bound to get 0.99, now the difference between the lower bound and the upper bound is 1/10^2. We can generalize this by saying that for every term of the sequence 9/10^n we add to the lower bound, the difference between the two becomes 1/10^n, therefore the more terms you add to the lower bound, the smaller the difference between the upper bound and the lower bound of the sequence becomes. Therefore the difference between the lower bound and the upper bound after an infinite number of terms is zero, therefore the 0.999... and 1 must by necessity be equal, either or calculus must be fundamentally flawed, because both ideas work on the same principle.

>this is like saying that particles below the planck threshold don't exist.

No this is nothing like that. Real numbers are abstract platonic objects that can only be approximated by physical reality so the comparison is not meaningful.

[Anonymous No.6557902]

>>6557890

>the difference between the lower bound and the upper bound after an infinite number of terms is zero

At what point does the difference magically change from 1/x to 0? That's right, it doesn't.

>if I keep making it smaller and smaller eventually there won't be anything left right guise

I suppose if you keep multiplying 1/2 by itself it will eventually become zero as well.

>we can generalise
>must by necessity be equal

No, that's not how generalisations work. As I said before, I can see it having practical applications but that doesn't mean that there literally is no difference whatsoever between 0.999... and 1. If there were, you wouldn't call them two entirely different things. It'd either be just 1, or just 0.999...

[Anonymous No.6557906]

>>6557890

Oh, and also

>or calculus must be fundamentally flawed
>implying it is perfect and complete as is

[Anonymous No.6557916]

>>6553542
Imbecile detected.

[Anonymous No.6557917]

>>6557902
>'The present President of the US' and 'Barack Obama' are two entirely different names, so they can not be the same person

This is how dumb you sound now

[Anonymous No.6557920]

>>6557906
>implying you have found an improvement to calculus

[Anonymous No.6557930]

>>6557917

jesus h christ no

There is a clear and distinct difference between a TITLE of a POSITION, and a NAME of an INDIVIDUAL. This is how ignorant you sound now.

>>6557920

>what we have works fine if we make some stupid-ass assumptions and generalisations and adjustments, obviously what we have is perfect

Fuck off with your shitty nonlogic.

[Anonymous No.6557943]

>>6557902
fuck it, you're either a troll or monumentally retarded, in any case, it seems as if summer's come early. Anyway i've beaten my head against the wall enough for tonight.

[Anonymous No.6557946]

>>6557930
Yes I know faget, but my point was that you are perfectly fine with using two names for one thing in this case, so why would you consider 0.999... and 1 different for this reason?

>stupid-ass assumptions and generalisations and adjustments
please give an example

[Anonymous No.6558075]

>>6557930
Both things still refer to the same person dipshit.

[Anonymous No.6558215]

>>6557851
>for god's sake don't pretend the difference between 0.999... and 1 literally does not exist at all. That's the kind of ignorant shit that pisses me off.

they don't get it bro, they are too dense

[Anonymous No.6558226]

Using the logic that 1/10^n is zero.
multiplying 0.9 by itself results in smaller numbers each time, reaching zero at infinity
multiplying 0.99 by itself results in smaller numbers each time, reaching zero at infinity
multiplying 0.999 by itself results in smaller numbers each time, reaching zero at infinity
multiplying 0.999... by itself results in smaller numbers each time, reaching zero at infinity
multiplying 1 by 1 by itself results in 1,regardless of number of multiplications.
#rekt

[Anonymous No.6558292]

>>6558226
You silly man, did you even look into the definition of a limit? It's all very clearly defined, but you just sound like someone who is only used to straight lines and rambles on about how a circle never can exist, because it is nowhere straight.

Please point at an error in a reasoning that uses limits in the way they are defined.

[Anonymous No.6558699]

>>6547096
> find a whole number between 1 and 2
>you can't, therefore 1==2

[Anonymous No.6558719]

>>6558699
Whole numbers are different from real numbers.

[Anonymous No.6558995]

>>6557943

>summer
>implying

>>6557946
>>6558075

Wrong. One refers to a bureaucratic position, not the individual holding it. The other refers to an individual, not the bureaucratic position held by them. You're comparing apples to oranges. When you say Obama 'is' the president, that's shorthand for 'Obama holds the position of president'. And I'M the dipshit? Riiiiight...

>please give an example

>>6557890

You gave it yourself.

>>6558215

I know, but goddamn, if no one ever corrects them they will propogate their flawed logic. Call it thought eugenics, but ignorance and stupidity is one of the worst blights on the planet and one of the most dangerous threats to humankind we have ever faced. Thanks for the support, though.

[Anonymous No.6559100]

>>6555205

No. 0.0....1 must have an integer number of 0s before the 1, because you can't have 'an infinite number of 0s, THEN a 1'. Yhis means your 0.999...has an integer number of 9s and is not an infinite sequence of 9s, which it was supposed to represent.

[Anonymous No.6559113]

>>6559100

>what is ℕ
>what is aleph-null

really?

[Anonymous No.6559116]

>>6557917

In 10 years time, who will be considered 'The present President of the US'?

[Anonymous No.6559121]

>>6558699

All you actually prove is that there are no integers between 1 and 2, and that therefore 1 and 2 are consecutive numbers. Which they are.

[Anonymous No.6559135]

>>6559113

Doesn't aleph-null apply to finite strings, which 0.999... is not?

[Anonymous No.6559139]

>>6559135

"Mathematicians use N or \mathbb{N} (an N in blackboard bold, displayed as ℕ in Unicode) to refer to the set of all natural numbers. This set is countably infinite: it is infinite but countable by definition. This is also expressed by saying that the cardinal number of the set is aleph-null (\aleph_0)."

from http://en.wikipedia.org/wiki/Natural_number#Notation

[Anonymous No.6559141]

>>6559116
Hu Chunhua?

[Anonymous No.6559144]

>>6546839

The problem with 0.999... = 1 is that the ellipsis means you're not specifying any particular number.

[Anonymous No.6559691]

>>6559144
>The problem with 0.333... is that you're specifying any number

In general you're reasoning sounds like
>Look I understand i can have a set of 2 integers, and of 3 integers, and of any finite number, but you can't magically go to a set with infinite elements like N, right guise?
>I never understood limits

[Anonymous No.6559692]

>>6559691
NOT specifying (dammit)

[Anonymous No.6559763]

>>6558995
on the off chance that you're not just a troll, I recommend reading Spivak's Calculus. If you can get through it and still hold your moronic beliefs, then at least you might be informed enough to try and provide a more coherent refutation of the past 200 years of mathematical thought than the OP.

[Anonymous No.6559809]

>>6559763

My 'moronic beliefs', as you so quaintly put it, are merely that infinitely small != nonexistent. Not that 0.999... = 1 is invalid in mathematical equations. I'll read the book, though.

[Anonymous No.6559846]

>>6559809
Except that on the real numbers, the number "infinitely small" DOES NOT EXIST.
That's like saying a 1/3 is an integer number, it isn't, that number does not exist on the integers.

[Anonymous No.6559886]

>>6559809
You also claimed the standard way of doing calculus was flawed, you however failed to give examples of such.

[Anonymous No.6559895]

>>6559846

But you and I both know that fractions and decimals are not integers, you even said so yourself just now... so why even bring this up? There's not even a bijection between the reals and the integers - the former is an uncountable set, the latter is a countable set.

So what does "there is no 'infinitely small' token" have to do with this discussion? You're basically saying "Apples are not yellow, that's like saying bananas are red" when you're trying to convince me that pears are fluorescent blue.

[Anonymous No.6559901]

>>6559895
I'm saying that the number you are referring to doesn't exist.

[Anonymous No.6559905]

>>6559886

>claimed

I said you're assuming that calculus is perfect. I also said that you're generalising by saying that the difference between the two values is so small that it is literally, LITERALLY nonexistent. It isn't.

I can't immediately prove that calculus is flawed. What I *can* do is tell you that a set of infinite tokens is not *literally* the exact same thing as the sum of those tokens. The set itself is a thing in and of itself, separate and independent of the tokens. Otherwise there would be no such thing as a null set.

[Anonymous No.6559937]

>>6559905
Well if you can't give me any reason to doubt the validity of calculus, I think the only reasonable thing to do is to accept it.

One of the theorems is: if the difference between two real numbers is smaller than every positive real number, then these two must be equal.

And that's all there is to it. If you claim this is not the case, I really want to see where the proof of this theorem breaks.

[Anonymous No.6559962]

>>6559809
If you choose to work with hyperreals (reals + sum infinite and infinitesimal numbers), still 0.999... = 1.

Okay, maybe not in all constructions of hyperreals, I haven't read nonstandard analysis, but professor who spent 90's doing nonstandard analysis and lectured my intro real analysis mentioned it during one lecture.

[Anonymous No.6559967]

>>6559937
>>6559962

By this logic 0 = 1 because the sum of the tokens in a null set equals 0. That is incorrect because sets themselves have separate values from the sum of the tokens they contain. QED.

It may certainly be true that 0.999... = 1 in practise and application due to generalisations or special cases. It is certainly *not* true that they are *actually* equal. There is a definable difference between the two values.

[Anonymous No.6559973]

>>6559967
That does not even make sense. No-one talks about sum of tokens in sets in calculus, it is something you invented, and you have not defined it in this thread, so we cannot speak about it.

Please tell me what this difference is.

[Anonymous No.6559977]

>>6559973

>something you invented

Do you even set theory.

[Anonymous No.6560029]

>>6559967
if there is a finite definable difference then maybe you could tell us what it is, or give an expression or it, go ahead and try.

Also your 'proof' by contradiction is monumentally flawed. First of all, you've failed to define what you're tokens actually are. If they're real numbers, then the proof is false because there is a finite interval between the two values, similar reasoning holds for the integers and other such sets.

Secondly, your claim that the sum of tokens in a null set is meaningless. How can you possibly define addition in the null set, there are no elements for an addition operator to operate on, moreover, even if you could define 'addition' in the null set, the sum of all the 'tokens'(elements) in the set cannot be zero because zero is not contained in the null set.

Lastly, how exactly do you define a value for a set? What, for example, is the value of N or {1,6,23,2}. Moreover, how exactly does 1 and zero come into your proof? It seems to me that you must be assuming that the null set is equivalent to zero (it's not) and that 1 comes into it because the cardinality of the set is 1, also illogical.

Also

[Anonymous No.6560045]

>>6560029

>the sum of all the 'tokens'(elements) in the set cannot be zero because zero is not contained in the null set
>legitimately being this ignorant

You're looking at it wrong. It's not that '0' is in the set. It's that there are zero TOKENS in the set. The null set is, by definition, empty. EMPTY As in CONTAINING ZERO TOKENS. Yet it still exists as a set, a thing. A. As in ONE.

I'm too tired to deal with the rest of your half-educated shitposting, assuming you're the same guy who's been giving me all this bollocks from the beginning. I'm going to bed to get some much-needed rest. I'll be back to address your questions tomorrow.

[Anonymous No.6560047]

>>6559905
We're not generalising, we're simply observing that there is no finite real number besides zero which is the difference between 0.999... and 1.

Also, since i'm fairly sure at this point that you're a philosopher(or philosophy student more likely), I can't help but wonder how exactly you plan to resolve Zeno's paradox without analytic continuity.

[Anonymous No.6560075]

>>6560045
What does this even have to with set theory, we're talking about 1 as in the axiomatically defined multiplicative identity in R, not the cardinality of the null set, they're two different things, so how about you stop trying to shift the goal posts just because you obviously don't know the first thing about real analysis.

>
You're looking at it wrong. It's not that '0' is in the set. It's that there are zero TOKENS in the set. The null set is, by definition, empty. EMPTY As in CONTAINING ZERO TOKENS. Yet it still exists as a set, a thing. A. As in ONE.

I'm well aware of this, obviously you've confused sums and cardinalties

[Anonymous No.6560082]

>>6560045
>An empty bag can never contain zero things, because it's A bag, A as in one.

This is how dumb you sound

[Anonymous No.6560995]

>>6560029

>definable difference

The definable difference is that one is 0.999... and the other is 1. You are using two separate numerical constructs, they are different types. By definition they are two separate things and have two separate values.

>>6560047

Whether there is a finite real number besides zero between the two is incidental. There is still a definable difference between the two, and they are not the same thing. Even if it's some sort of imaginary or irrational number, there is a value difference between 0.999... and 1.

>Zeno's paradox

Which one, and what does that have to do with this? (And yes, I have studied philosophy for over a decade.)

>>6560075

>axiomatically defined
>claiming 0.999... is objectively equal to 1

Every natural number is a set, genius. How am I confusing sums and cardinalities.

>>6560082

what the fuck are you on about

[Anonymous No.6561041]

ITT: People who don't understand how infinity works/ Trolls

[Anonymous No.6561047]

>>6555962

>It's kinda like playing tennis against a wall, you're not going to win, but you can still get something useful out of it.

[Anonymous No.6561065]

>>6560995
i finally figured out your particular method of trolling. you're deliberately confusing the natural number 1 with the real number 1

[Anonymous No.6561079]

>>6561041
ITT people forget geometric series exist and argue the same shit they have been arguing for years.

[Anonymous No.6561136]

>>6561079

convergence is an unproveable concept

[Anonymous No.6561143]

>>6561065

>accusing people of trolling when they are the one deliberately ignoring logic and reason

[Anonymous No.6561179]

>>6561143 here. I am actually discussing this with a maths major at the moment, and they are on the fence about it.

They blame the alcohol.

[Anonymous No.6561263]

>>6560995
>two separate numerical constructs
By that logic, couldn't one say that 3/2 is not equal to 1.5, or more appropriately, that 1/3 is not equal to 0.333...(this effectively equivalent to saying that 0.999.. = 1). Maybe in whatever area of philosophy you deal with those things aren't considered equal and that's fine, but in REAL ANALYSIS, that is the analysis of the real numbers 2/3 and 0.333... are equivalent and so are 0.99... and 1.

>There is still a definable difference, even if it can be rigorously proven that there isn't one

>Even it's some sort of imaginary or irrational number
>Imaginary numbers in R
Why has no one given you a fields medal yet?

>Which Zeno paradox
The one that gets introduced in most uni level calculus texts. The intuitive idea is that if you take a runner covering a certain distance, you observe that he covers half the distance over a given time interval, then you observe that he covers half of that distance in another time interval(the time interval doesn't matter as long as it's finite) and so on so that the distance between the runner and the finish line after n such steps is 1/2^n. Therefore, the runner cannot reach the finish line in a finite number of steps. Despite this, the runner can reach the finish line in a finite time interval because the distance becomes zero(the runner reaches the finish line) after an infinite number of steps. Essentially the same reasoning applies to our problem except that in our case we look at 1/10^n as the difference between the sum 9/10 + 9/100 +... + 9/10^n and one.

>Every natural number is a set.
I'm well aware of that, it doesn't change the fact that your 'proof' by contradiction takes far too many liberties with logic to be taken seriously.

>>6560082
Think of it this way, the bag contains Nothing.

[Anonymous No.6561268]

How i think of it is:
1/9 = 0.11111 for eternity
9 * 1/9 = 0.99999 for eternity

But 9/9 simplifies to 1

[Anonymous No.6561290]

>>6561268

only honest post on /sci/ ^

[Anonymous No.6561293]

>>6561268
>But 9/9 simplifies to 1

as long as you guys admit its just a convenient short-hand rounding thing....

it doesn't actually simplify to 1 if we are being strict intellectuals.

[Anonymous No.6561328]

>>6561293

This. This has been exactly my point all along.

[Anonymous No.6561331]

>>6561263

>Imaginary numbers in R

Alright, wise guy. What is 5/0? Or any n divided by 0 such that n ∈ ℕ ?

Where's my medal.

[Anonymous No.6561342]

You don't even know what's the definition of field

>Keep insisting that 0.999... is different from 1


I think I smell engineers...

[Anonymous No.6561343]

You are proving that you don't even know the definition of Field. Chapeu.

>Keep insisting that 0.999... is different from 1


I think I smell engineers...

[Anonymous No.6561368]

http://en.wikipedia.org/wiki/Infinitesimal

[Anonymous No.6561469]

>>6560047
> that there is no finite real number besides zero which is the difference between 0.999... and 1.

[Anonymous No.6561488]

>>6561293
>>6561328
What I love about this discussion is that the one and only argument is: this thing has two different descriptions, so it cannot be the same thing.

Please elaborate on your definition of infinite series, I am really interested.

Do you also believe that 1+2 and 2+1 are not the same number, if we are *strict* intellectuals?

[Anonymous No.6561539]

>>6561488

That's a matter of order, not type. Besides which, it's like a two-sided coin that is infinitely thin. The coin still exists between the sides.

http://en.wikipedia.org/wiki/0.999%E2%80%A6#Infinitesimals

"All such interpretations of "0.999..." are infinitely close to 1. Ian Stewart characterizes this interpretation as an "entirely reasonable" way to rigorously justify the intuition that "there's a little bit missing" from 1 in 0.999....[51] Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about 0.999... < 1 are erroneous intuitions about the real numbers, interpreting them rather as nonstandard intuitions that could be valuable in the learning of calculus.[52][53] Jose Benardete in his book Infinity: An essay in metaphysics argues that some natural pre-mathematical intuitions cannot be expressed if one is limited to an overly restrictive number system:"

[Anonymous No.6561556]

>>6561539
Yeah you can do such things in non-standard analysis, but the point is that people here are claiming there is a contradiction in standard calculus, a claim for which noone provided evidence.

[Anonymous No.6561570]

Question to everyone who thinks 0.999...=/=1
What do you need to add to 0.999... to get 1?

[Anonymous No.6561575]

>>6561570
1/(-1/12)

[Anonymous No.6561583]

>>6546839
>>6546876
>>6546921
>>6546921
Transcendental is correct, for a quick background a transcendental number is a non algebraic number. If you have never worked with them before a good example is e or pi.

Its any number that cannot be represented using algebraic conventions. And as OP suggests, there are far more transcendental numbers than algebraic numbers.

Then why havent I seen them? Well your probably not dealing with that echelon of mathematics yet. This only comes into play in the more advanced mathematics, so its completely understandable to not be aware of their existence.

Also note that while .999 REPEATING is equal to 1, . 999999999999997, is definitely not equal to one, and may be a transcendental number.

[Anonymous No.6561586]

>>6561575
Numberphile fucks around with the sums of infinite series, hes a dipshit.

[Anonymous No.6561591]

>>6561586
>What is the Riemann Zeta Function?

[Anonymous No.6561593]

>>6561583
>this rational number may be a transcendental number
top kek

[Anonymous No.6561604]

>>6561593
What are you talking about?
.999999999999997 is obviously not algebraic,
just like 1+1=/=2 and .999...=/=1

[Anonymous No.6561605]

>>6561586
>not recognizing epin sci trelling

[Anonymous No.6561610]

>>6561593
> rational number
You say that but I dont think you really know what it means.. This is a real number..
>>6561604
thanks anon, good to see some people are on top of their shit

[Anonymous No.6561612]

>>6561605
>that fucker got me.

[Anonymous No.6561620]

>>6561604
>>6561610
.999999999999997 = 999999999999997/100000000000000
REKT

[Anonymous No.6561636]

>>6561620
"No, you're stupid, there can only be one unique way to write a number"
~49.999...% of this thread

[Anonymous No.6561667]

>>6561556

The point here is that the calcheads are claiming they're the same value and same number. We're saying that they aren't because there is a nonzero value separating the two. Clearly the nonzero value is 1/10^H.

[Anonymous No.6561684]

>>6561667
Yeah, but H is as non-existent in the standard real numbers as 1/2 in the integers, so this value between the two is zero, so they are equal. If you cannot see why H doesn't exist I think you should read a book about analysis.

[Anonymous No.6561734]

>>6561684

so
H ∈ R ≈ 1/2 ∈ a
⇒
H ∈ R
∵
1/2 ∈ a
∎

If you cannot see why it does exist I think you should read a book about logic.

[Anonymous No.6561737]

>>6561734

Z, not a... not sure how that happened

[Anonymous No.6561768]

>>6561734
I quit. Nobody here has any idea of what is and what isn't allowed in standard calculus. You just *don't* have infinitely small things in the real numbers. It's just as wrong as talking about the largest prime number in N.

I now see why philosophy is such a messed up discipline. Nobody has any idea what they are talking about, you change definitions while you're talking about it or you just don't talk about them at all.

[Anonymous No.6561825]

>>6561768

Hey, you're the one who said "H is as non-existent in the standard real numbers as 1/2 in the integers", or H ∈ R ≈ 1/2 ∈ Z, and if 9/9 isn't a fraction then I don't know what is. It's the only way your silly 1/9 = 0.111... then 0.111... * 9 = 0.999... = 9 * 1/9 =1 proof works. So you've proved that it's a generalisation with your own logic. 0.999... = 1 only in practical application.

You should look into philosophy, you might actually learn something.

[Anonymous No.6563087]

>>6561825
he doesn't realize when you divide 1/9 the remainder is 1/10^n for each 1's position in 0.1111.... so 1/9 > 0.111... because 0.111... represent an incomplete division process(which is approximated by a limit, which is the upper bound and not the result(something all these retards conveniently forget at every opportunity ))

[Anonymous No.6563088]

>>6563087

It's like arguing with a rock, isn't it? Except of course that the rock doesn't argue back.

[Anonymous No.6563128]

10x-x = 9x
10*0.999... - 0.999... = ?

[Anonymous No.6563141]

>>6563087
But it doesn't, you retard. Numbers don't represent incomplete processes. Please do not pretend you understand anything about analysis because you finished babby set theory.

Let me explain it for the last time:
infinite series (like 0.9999...) have a value because we DEFINED it to have a value. So the only value it has is the value that we get when we apply the definition. If you apply this definition you will see that the partial sums of the infinite series 0.999... (0.9, 0.99, 0.999, ...) are arbitrarily close to 1. So it IS 1. This is not a subject of discussion (muh infinitesimals). You have to take another definition, another number system or you are forced to accept this.

>imcomplete process hurr durr
is simply not relevant here.

>result
The result IS the upper bound in this case.

>I don't know the definition of a series.
Please go read a good book on calc.

[Anonymous No.6563170]

>>6563141

>arbitrarily close
>is

It's literally only one or the other. Even by definition, they're only the same because you say they are, and that is purely subjective. Please go read a good book on logic.

[Anonymous No.6563171]

This entire thread

[Anonymous No.6563188]

>>6563170
An infinite series is just a symbol. It has no value until we define such a value. You talk like .999... already is a value an sich, but that notion is wrong.

We already have defined finite addition in R and if we want to talk about an *infinite* addition, we have to define it. That's how the game works.
The partial sums (.9, .99, .999, ...) are arbitrarily close to 1, and therefore, and only because of this, we defined the infinite series to be 1.

Let me give an example: Suppose I want to use the word flarp. Flarp doesn't mean anything, until I give a definition for it. If I define flarp to be a table, it IS a table. It is not 'for practical reasons only' a table. It is not 'if we are strictly intellectual' different from a table. It just IS the same. Flarp is not a flair or a flap because the name is similar. It is also not different from a table because it has a different name. It just is a table.

This has nothing to do with logic, it's just that you fail to understand that .999... is defined to be something, and not something that follows from only finite addition.

tl;dr the infinite series is a meaningless symbol until I define a method of finding a value for it.

>captcha: ilogyst pardonable

[Anonymous No.6563196]

On a scale of 1 to 10, what is four times 3 and why?

[Anonymous No.6563202]

>>6546856
>>6546957

>ITT: Kids playing with infinity..


Yes there is:

1 - 0.9 = 0.1
1 - 0.99 = 0.01
1 - 0.999 = 0.001
1 - 0.9999 = 0.0001
(...)

So the number you are looking for is:
0.0(...)1

For every decimal place you add I can also add one and there's still a number between them.


In other words:

If you got the half-bounded interval "[0, 1[", then
1 the supremum (but not the maximum).

[Anonymous No.6563204]

>>6563196
π, because the question's irrational

[Anonymous No.6563206]

>>6563170
In some cases, you would be correct. If I have numbers arbitrarily large, but that doesn't mean I have infinity.
However, in the case of real numbers, this is true. The reason is that real numbers are defined as equivalence classes of Cauchy sequences of rationals. Because the sequences (0.9, 0.99, 0.999...) and (1,1,1,...) converge to the same limit, they represent the same number.

[Anonymous No.6563207]

>>6563188
>I define 0.999... to be 1 therefore proving conclusively that 0.999...=1

[Anonymous No.6563212]

>>6563206
1/1,1/2,1/3,1/4
and
1/2,1/4,1/6,1/8
bot converges to the same limit.
Doesn't mean they're equal at any step.

[Anonymous No.6563265]

>>6561331
there cannot be a number x such that x/0 = y, this would imply that there is a product x*y = 0 for x and y != 0. This contradicts the axioms of the real numbers and is therefore undefined.

[Anonymous No.6563285]

>>6563212
(1/n)=(1/(2*n)) as n increases to infinity
or even more ridiculous
(1/n)=(1/(n^n^n^n)) as n increases to infinity
since both sides (like 0.999... and 1.0) have the same limit they're both equal.

[Anonymous No.6563297]

>>6563212
No but they are in the same equivalence class and therefore defined to be equal in R

>>6563207
Congratulations, you have grasped the basic idea of a definition.

>>6563202
>ITT: philosophy students raping definitions.

As you ought to know, the Least Upper Bound property holds in R:
Every set S in R that as an upper bound has a least upper bound.

Suppose we have infinitesimals. We look at the positive infinitesimals. (I take as definition a number h>0 such that for all natural numbers n we have h < 1/n).
Look at the set K of positive infinitesimals, which is not empty. It is obvious that 1 is an upper bound of K. So K has a least upper bound, say c. So h =< c for all positive infinitesimals h. We assumed there were infinitesimals, so there is an h such that 0< h=<c, so 0<c. So c<2c, and 2c cannot be infinitesimal, it is larger than an upper bound of K.

If 2c is not infinitesimal, there must be an n such that 2c >= 1/n. But then c/2 >= 1/4n. So there don't exist infinitesimals larger than c/2, because those numbers are larger than or equal to a number of the form 1/m for natural m. So c/2 is also an upper bound of K, but is is strictly smaller than c itself. But c was already the least upper bound. Contradiction. Our only assumption about K was that it was not empty. Therefore K must be empty, there are not positive infinitesimals, 0.0(...)1 is not a number, and .999... and 1 are equal.

>inb4 I reject the Least Upper Bound property
In that case we are not even talking about real numbers anymore.

>inb4 I don't understand this proof.
Please take a solid Analysis course.

[Anonymous No.6563311]

>>6563188

>just a symbol
>no value until we define such a value

So you're saying that math is a human construct, and does not exist independently in nature?

Let's go with your word problem regarding 'flarp'. We already have a defined name/label for a table. It's 'table'. A 'flarp', then, is not a table, but rather an arbitrary name you decided to give a table. One is not perfectly divisible by three, nor by nine. Not in the reals. We make do with the clearly imperfect system we have of understanding things and groups of things by allowing for this infinitesimally small margin of error - "0.999... = 1", even though it isn't perfectly equal. See >>6563207

>>6563265

I've honestly never understood the problem of 'undefined error' for x/0 = y. If we consider that any n for n != 0, and 0 - n = -n and 0 - (-n) = n, then it follows that 0 = R. This allows for dividing by zero with no issues, does it not? Further, it allows for 0.999... != 1. What is the problem with defining 0 as R, exactly? Because as I see it, accepting a system where dividing by a number results in (undefined) is not optimal. There is obviously a flaw in such a system, especially if it causes crashes and malfunctions in certain systems when attempted. Better to rehash the system in such a way that everything actually works properly, even if it makes some people uncomfortable because you're changing things.

Change isn't bad. What's bad is clinging to outdated, imperfect modes and models based solely on the fact that "well, it's worked well enough for us so far, it must be correct".

Did you know that it hasn't even been a full century since the scientific community finally agreed that the universe consisted of more than the Milky Way? And yet here we are, still arguing about how infallible calculus is and falling back on arbitrary approximations to make one equation work, while completely excluding the possibility of another because "we don't like getting (undefined) or infinity for an answer".

[Anonymous No.6563345]

>>6563311
No I'm saying that our symbols do not have a meaning independent of the reader.

>A 'flarp', then, is not a table, but rather an arbitrary name you decided to give a table.
A 'table' is also an arbitrary name you gave to a flarp.

>One is not perfectly divisible by three, nor by nine. Not in the reals.
You cannot be serious. Do you have any idea how mathematics works? It is almost by definition that we can invert 3 in the reals. The only reason we cannot present it in a finite number of decimals is because we usually use base 10, and 3 doesn't divide 10. Or do you think 1 is perfectly divisible by 2 because we can represent it with 0.5?

>I've honestly never understood the problem of 'undefined error' for x/0 = y.
This is because you never understood algebra in the first place. If we define x/0 the real numbers will go down faster than the Titanic.

Suppose we follow your unholy idea of defining such a value: suppose x/0 = y for certain x and y. Then x = y*0, because that is how we want division to work in the reals. But hey, we also *defined* 0 to be the element such that z*0 = 0 = 0*z for all z. So x=y*0 = 0. So if we tried to define x/0 for any x != 0 we reached a contradiction.

0/0 is also undefined, because suppose it is defined. Because 1*0 = 0 we have 0/0 = 1, because that is how division works, and 0/0 was defined so there is no reason why we cannot do this. But also 2*0 = 0, so 0/0 = 2. But wait, now we have 1 = 0/0 = 2. Contradiction.

As you probably see, wannabe Gauss, your idea leads to contradiction in no time. It is not that we are afraid of change, it is that any attempt to define x/0 leads to contradictions and total failure. Go on, define values for x/0. Your system will break.

[Anonymous No.6563357]

>>6563345

>flarp nonsense

Why the hell would you arbitrarily name something that has already been named? It's redundant and pointless.

>we can invert 3
>3 doesn't divide 10

I'm the one who can't be serious?

>x/0 stuff

1 = 0/0, and 2 = 0/0, but 1 != 2. This wouldn't be a contradiction since 0 literally would be defined as equalling every and any other real number. It would essentially be a variable. How difficult of a concept is that to grasp?

It's like you're all children, I swear.

[Anonymous No.6563360]

>>6563357
>since 0 literally would be defined as equalling every and any other real number
I feel a bit ashamed I fell for this. Good troll 7/10

[Anonymous No.6563389]

>>6563360

I appreciate the compliment, but how is it trolling? 0 is equal to the sum of literally any positive fraction, decimal, or integer and its negative counterpart (the reals). Any positive or negative fraction, decimal, or integer (the reals) divided by or multiplied by 0 equals 0. It follows that 0 = R.

[Anonymous No.6563558]

>>6563202

this is brilliant.

[Anonymous No.6563637]

>In mathematics, the repeating decimal 0.999... (sometimes written with more or fewer 9s before the final ellipsis, or as 0.9, 0.(9), or 0.9 with dot over the 9) denotes a real number that can be shown to be the number one. In other words, the symbols "0.999..." and "1" represent the same number. Proofs of this equality have been formulated with varying degrees of mathematical rigor, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

[Anonymous No.6563661]

So if according to you, 0.999..=1, then
1.1999.. = 1.2 ?
From where i come from = means EQUAL or SAME.
I dont see 0.999.. being same to 1. You nerds need to get back to reality.

[Anonymous No.6564329]

>>6563661
Yes 1.19999... = 1.2

[Anonymous No.6564349]

>>6546839
Not this shit again
Didn't any of y'all take calc 2

[Anonymous No.6564355]

1/9 = .111… (9)(0.111...) = .999...

[Anonymous No.6564362]

>>6564355

Thank you for repeating what every other calculus-worshipping troll has already regurgitated eleventy billion times. That really helps clear things up. /sarcasm

[Anonymous No.6564378]

>Begin with a general Geometric Series with first term a, common ratio r, and n terms
<span class="math">S_n=a+ar+ar^2+ar^3+....ar^{n-2}+ar^{n-1}[/spoiler]

>Multiply the series by r
<span class="math">r S_n=ar+ar^2+ar^3+ar^4+...ar^{n-1}+ar^n[/spoiler]

>Subtract the second equation from the first: Note that the first and second series both have the exact same terms, except for the first term of the first series and the last term of the second series.
<span class="math">S_n-rS_n=a-ar^n[/spoiler]

>Factor both sides and solve for S_n
<span class="math">S_n=\frac{a(1-r^n)}{1-r} [/spoiler]

>Apply formula to infinite Geometric Series by taking the limit as n approaches infinity. For notation, drop the “n” subscript on S_n
<span class="math">S=lim_{n\rightarrow \infty} \frac{a(1-r^n)}{1-r} [/spoiler]

>The limit depends only on the r^n term. Restricting |r|<1, the limit resolves to
<span class="math">lim_{n\rightarrow \infty} r^n=0 [/spoiler]

>The formula then simplifies to
<span class="math">S=\frac{a}{1-r}[/spoiler]

>Noting that
<span class="math">.999...=.9+.09+.009+...[/spoiler]

>We see that .999... is an infinite geometric series where a=.9 and r=.1. Our formula then tells us that
<span class="math">.999...=\frac{.9}{1-.1}=\frac{.9}{.9}=1[/spoiler]

>Q.E.Fucking.D.

[Anonymous No.6564384]

>>6564362
>implying that a proof does not prove things
...wat

[Anonymous No.6564391]

>>6564378

You are wrong form your very first equation. The formula you give for the summation is finite (it stops at the n-1 term).

Therefore, it does not apply to 0.999... which has no final term.

[Anonymous No.6564394]

>>6564391
>The formula you give for the summation is finite (it stops at the n-1 term).
And then he takes the formal limit as n approaches infinity. Literally high-school level math.

come back after you've finished middle school.

[Anonymous No.6564397]

>>6564391

>Didn't bother to read far enough to see where I took a limit to apply this series to an infinite series.

Nice reading comprehension.

[Anonymous No.6564403]

>>6564394
>>6564397

The limit of a summation and a summation itself are not the same thing.

Just as you can only speak of the limit as x approaches 1/x instead of the value at 1/x, you cannot talk about the limit as n approaches infinity versus the value when n is infinity.

[Anonymous No.6564404]

Jesus, is this thread really still at the top?

[Anonymous No.6564406]

>>6564403
>The limit of a summation and a summation itself are not the same thing.
Wrong. For an infinite sum, they ARE the same thing.

>limit as x approaches 1/x
...what? I hope that's a typo.

[Anonymous No.6564409]

>>6546839
because there is no number between .999... and 1.

Also because 1/9=.111...
2/9=.222...
3/9=.333...
4/9=.444...
5/9=.555...
6/9=.666...
7/9=.777...
8/9=.888...
9/9=.999...
9/9=1 as well

[Anonymous No.6564428]

>>6564403

The limit of a finite sum as the number of terms approaches infinity is the same as an infinite sum. In fact, ALL infinite sums are finite sums where the limit is taken, we just write the upper bound of summation as infinity as a notational short hand. The proper expression for an infinite sum is

<span class="math">lim_{n\rightarrow \inf} \sum_{i=i_0}^n a_i[/spoiler]

[Anonymous No.6564429]

>>6561469

>show me a number between 0.999.... and 1

0.999...
there are infinite 9's between it and 1.

Why do people think infinity ends somewhere?

[Anonymous No.6564441]

>>6564429
There are not infinite 9's between them, there are infinite 0's between them

[Anonymous No.6564448]

>>6564429

>mind blown

[Anonymous No.6564449]

>>6564441

there are infinite 9's by definition.

0.999... is just as far away from 1 as 0.999

namely, infinitely far away from 1.

[Anonymous No.6564451]

>>6564449

lmao

.999....-,999=.000999....
1-.999...=.000....1

Even if you think a number like .000...1 exists (Under the real number system, it doesn't), surely you'd agree that .000999.... is a larger number than .000...1 would be?

[Anonymous No.6564468]

>>6564428

>notational short hand

You guys keep doing this - spouting your nonsense about how calculus is infallible and definitive, and then saying things like "generalise" and "approximate" and "notational short hand". It's like you don't even know what the fuck you're saying. You just spew the same contradictory shit over and over and never listen to what you're actually saying.

Tell you what. Why don't you actually READ the entire thread instead of just replying to a single post and thinking you're clever. Save us all the time and trouble of a rehash.

[Anonymous No.6564546]

>>6564449
....

You're retarded, the difference between them is not infinity, nor is it an infinitesimally small amount, it is 0. That's why 0.999... Is 1

[Anonymous No.6564623]

>>6564468

Well, I was responding to the guy arguing that the proof I posted was invalid. Was I supposed to talk to someone else while I did that?

My point to him also was not me espousing some calculus cock sucking bull shit either. When you right "infinity" on top of a sum to indicate an infinite summation, that IS a notational short hand. The true way of expressing that is to write it as a summation to some arbitrary n and take the limit as n approaches infinity. It gets repetitive as fuck to write that every time you want to write an infinite sum, so we drop the limit bit and just write infinity, with the understanding that it is in fact a limit. Since the person I was responding to made the claim that taking the limit as the number of terms in a sum approaches infinity is different from an actual infinite sum, I feel my response was rather appropriate.

Since you seem to think yourself the king of this thread though, please instruct me which post I should look at more closely? Cause it seemed to me like all of these posts were the typical

>1=3/3=3(1/3)=3(.333....)=.999.....

and

>you can't find a number between .999.... and 1 so they're the same!

and I had thought posting and discussing the actual proof of the equality would be a good direction for the thread, but apparently I was wrong!

[Anonymous No.6564635]

>>6564546
>That's why 0.999... Is 1

0.999... is 0.999...
1 is 1

0.999... doesn't end and doesn't reach any thing, it's always infinitely far away from 1. Separated by an infinite amount of 9's.

It's like an asymptote, and 1 is a point it never, ever touches. So how can it "be" 1?

[Anonymous No.6564646]

>>6564623

I'm literally 95% of the replies arguing that 0.999... != 1, if that answers your question. I legitimately can't tell who's arguing what anymore, after three hundred-odd posts it's all become one hazy blur of "muh calculus" and "lol infinity". I'm just arguing simple logic and 3 does not divide 10 (also the bit about 0 = R).

Sorry for snapping at you if you weren't arguing for 0.999... = 1. What with all the repetition, the shit is literally so thick in here that I can't see past my nose anymore.

[Anonymous No.6564654]

>>6564468
if you know of any flaws in the major theorems of calculus and real analysis then please don't hesitate to enlighten us.

[Anonymous No.6564660]

>>6564654

Literally everything I have to say is in this thread, friend. All you need to do is read.

[Anonymous No.6564672]

0.999... never reaches 1 in the same way 1,2,3,4... never reaches some definite end.

There is no end to an infinity. Converge is a conspiracy theory.

[Anonymous No.6564730]

>>6564623
> and I had thought posting and discussing the actual proof of the equality would be a good direction for the thread, but apparently I was wrong!

>and I had thought posting and discussing the actual proof of the equality would be a good direction for the thread, but apparently I was wrong!

refer to >>6564378

>>6564660
when i say flaw, i mean a rigorously examined counter-example that can be derived from the axioms for the real numbers.

It seems to me that you're main line of reasoning rests some mistaken assumptions about how the real numbers are defined. The reals are defined as an ordered field that satisfy the least upper bound property. Consider the following case: It is known that there is no positive rational number x such that x^2 = 2, however, the rationals are split up into two sets, the set of x such that x^2 < 2 which we will call A, and the set of x such that x^2 > 2. Clearly A and B are both infinite, that is, for every x, there is a y that is larger/smaller than x such that y is closer to being the square root of two than x is. Therefore we can establish that the number z such that z^2 = 2 is both the least upper bound of A and the greatest lower bound of B. This is more or less how the real numbers are defined, 0.999... and 1 are defined in the same way, specifically 1 and 0.999... are both least upper bounds for the set of rationals r such that r = 9/10^n, 0.999.. is simply a decimal expansion of one. I am of course aware that the above demonstration does not really account for the existence of the transcendental numbers, but it's only an intuitive outline of Dedekind's method for constructing the reals, not the full proof. Moreover the essence of >>6563297 is that if we assumed that there was an x such that 0.999... + x = 1, but it was infinitely small, it would lead to a contradiction of the axioms for the real numbers, therefore such an x cannot exist in the real numbers.

[Anonymous No.6565109]

>>6564646

Well, we're back to where we started then. I am in fact arguing that 1=.999.... just I'm not using (correct, but not proof) arguments like 3/3=.999.... and asking for a number between .999.... and 1.

[Anonymous No.6565151]

>>6565109

There is a definable nonzero difference between the two. I don't know how to explain it any more clearly. 1/3 and 1/9 don't translate perfectly, and

>what is http://en.wikipedia.org/wiki/Recurring_decimal#Every_repeating_or_terminating_decimal_is_a_rational_number
>what is http://en.wikipedia.org/wiki/Recurring_decimal#A_shortcut
>what is http://en.wikipedia.org/wiki/Recurring_decimal#Repeating_decimals_as_an_infinite_series
>what is http://en.wikipedia.org/wiki/Approximation

Every single argument I have heard has stated that the equation of 0.999... and 1 is arbitrary and/or an approximation. Ergo,

0.999... != 1
0.999... ≃ 1

[Anonymous No.6565195]

>>6565151

No, there is NOT a definable nonzero difference between the two. In alternative number systems like the hyper-reals, you can get a difference between them, but in the real number system, there is no difference.

And again, I never argued based on the 3(1/3) argument, I gave the proof based on geometric series in >>6564378

[Anonymous No.6565515]

>>6565151
Do you understand the least upper bound property?

Can you point out a flaw in >>6563297 ?

[Anonymous No.6565587]

Whenever I begin to lose the motivation to study for my real analysis exam coming up, I read this thread.

[Anonymous No.6566420]

>>6565195
>>6565515

Why is it so hard to accept that 0.999... = 1 is only an applied definition. If it is untrue in even a single mathematical approach, then it is untrue. Approximations and arbitrary subjective definitions are not ontological truths.

[Anonymous No.6566432]

>>6566420
Is it really so hard to understand, if we permit 0.999... != 1, it leads to a contradiction, therefore there cannot be a number such that 1-0.999...!= 0.

>If it is untrue in even a single mathematical approach, then it is untrue.

Again no, if we're dealing with the real numbers (which we are), and such an approach shows that 0.999...= 1, then that approach cannot be valid because it leads to a contradiction in the axioms of the real numbers. This is the essence of what we're saying, the real numbers are defined in terms of a certain number of axioms, if a number violates one of those axioms, then it is not part of the real numbers, trying to say it is is like saying that 1.5 is in N. As >>6563297 has pointed out, infinitely small numbers cannot be defined in the reals therefore there cannot be a difference between 0.999.. 1, it really is that simple, it has nothing to do with approximations or anything like that, it is a simple question of deriving results from axioms.

[Anonymous No.6566451]

>>6566432

>deriving results from axioms
>not taking into account axioms are assumptions in the first place

http://en.wikipedia.org/wiki/Real_number
"The real numbers include all the rational numbers"

Literally the second sentence. See >>6565151

Every repeating or terminating decimal is a rational. Every rational is in the reals. Therefore both 0.999... and 1 are rationals, and as individual rationals cannot equal each other. Otherwise the nonzero value difference between the two, or the poorly represented "0.000...1", is zero, as you say, which means there is zero distance between any and all decimal values in N, which in turn means that 0 = [1,2,3,...] and you have broken it far more than I have. So your maths are wrong.

[Anonymous No.6566480]

>>6566451
First of all, all of mathematics is built from axioms, all of it. That's how we define number systems, we define axioms and build a system of numbers that all obey these axioms, this is how mathematics works.

>Both 0.999.. and 1 are rationals, and individual rationals cannot equal each other

You haven't proven that they are individual rationals, they have different representations but they are
PROVABLY(see>>6564378,>>6563297 and >>6557836) the same number, in exactly the same way that 1/3 and 0.333... are the same number.

>if 0.999.. = 1, then every rational number must be equal (for some reason...) therefore 0 = N

Give up. You're embarrassing yourself, you're obviously just not cut out for maths, this is just totally incoherent reasoning.

[Anonymous No.6566493]

>>6566480

It's perfectly coherent. If there is zero distance between 0.999... and 1, then every digit of 0.999... is zero distance from every other, and thus 1=2,2=3,3=4,...

Put simply, if 0.999... = 1, then 0.999...8 = 0.999..., 0.999...7 = 0.999...8, etc. Therefore all numbers are the same because there is zero distance between them. This applies to the reals since every repeating or terminating decimal is a real.