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/sci/ - Science & Math

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6859226 No.6859226 [Reply] [Original]

How advance can maths get?

>> No.6859228

>>6859226
a little past calculus

>> No.6859238

>>6859226
about tree fiddee

>> No.6859260

>>6859228
Would you tell me what calculus is please, does it have any real world benefits

>> No.6859280

>>6859226
Until it goes full-circle and you're writing down trivial statements via random arrangements of letters and symbols while crying about how your academic adviser lied to you about the 300k starting.

Also, there really isn't a "most advanced" field mathematics is a tree of topics with no particular order once you get past a certain point.

>> No.6859281

>>6859260
nah, it's mostly theoretical. something about the proving 0.999... = 1 LOL talk about stupid.

>> No.6859283

>>6859260
the study of how things change.
do things change in the real world?

>> No.6859285

>>6859283
I don't know?

If there is a chair and I break it I guess that's a change, but does that count?

>> No.6859294

>>6859285
Yes. Calculus is mostly used to explain how chairs break.

>> No.6859300

>>6859281

I learned how to prove this in Calc 2. If this is advanced stuff for you, then you're in trouble :(

>> No.6859301

>>6859294
Haha I'm pretty sure its not but the guy or you, said it was to do with things that change

>> No.6859315

>>6859301
Differential calculus deals with the rate of change of the function (think of the slope of the function at any point), whereas integral calculus deals with finding the area under curves. There's much more to it, and I'm certain someone else can elaborate upon what I'm saying, but that's the most basic summary I can give you.

>> No.6859317

>>6859300
So that dude was serious calculus os about 0.9999999 becomes one?

Well yeah if you round up? I thought calculus was hard

>> No.6859319

>>6859315
Thanks

>> No.6859338

>>6859317

x=.999…
10x=9.999…
9x=9
x=1=.999…

There, now even your feeble brain can wrap around the FACT that .999…=1

>> No.6859413

>>6859338
I know that it is an epic meme and all, but is is mathematically speaking wrong to say that .9999=1? I know that the sum of natural numbers isn't -1/12 because you are doing algebra with infinities, but the .9999 has a series convergance proof I think of you take it as a continued fraction.

>> No.6859437

>>6859226
true fact we still havent been able to prove that 1 million isnt the biggest number in the world... some wonder if it will ever be proven...

>> No.6859444

>>6859338
No, 1> .9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999

To says his isn't true is to say fuck logic.

>> No.6859445

>>6859437
I've been working on this proof for a while dude, could you peer-review it for me?

>100001

>> No.6859446

>>6859445
annals of math here we come

>> No.6859450

Hmmmm I guess there's Riemann geometry, Topological quantum field theory (that'd be the highest maths I guess the today theoretical physicists would use), and then there's Heegaard Floer homology.

Also those are research fields my maths/physics professors are interested in. What's funny is that the phyiscist at my uni wants to research non-integral and maslov indices in wavepackets (which should be the pure/applied mathematicians job) and then you a pure mathematician wanting to go into research for quantum field theory (which is for the physicists, lel).

>> No.6859451

>>6859280
>2014
>doing STEM
>attempting to take any job in academia
>not going into industry/gov't
>not raking in the shekels

>>6859317
0.9999999 ≠ 0.99.......

>> No.6859463

>>6859450
the guy who said something about math being a tree of pre-requisites is closest to the truth IMO

who's to really say whether riemannian vs symplectic geometry is "more advanced"? from a logical point of view they're just different structures you can place on a manifold.

TQFTs may not be hard if you think categorically as a default state, but something like the theory of (\infty,1)-categories and work of lurie, baez, hopkins, etc. might be a candidate for current "edge of knowledge" level math

>> No.6859552

>>6859300
holy sh9t, calc II, you must be a god among men

>> No.6859562

saying 0.999=1 is like saying 999=1000. there is a tendency to round up because 1 is easier to deal with than 0.999 but it is still wrong to say they are equal.

>> No.6859568

>>6859260
Calculus has to do with rates of change. Derivatives are rates of change and allow you to find the tangent line at a point on a function graph. The tangent line is the slope of the line at that point.

Integrals are basically derivatives in the other direction. They allow you to find the area under a curve from one point to another.

It's the most useful, versatile subcategory of mathematics there is, and it's a shame that many people don't learn it. Everyone, from businessmen to engineers, can find it useful.

>> No.6859733

>>6859445
After careful analysis I have concluded that one hundred thousand and 1 is less than one million. I am sorry if I made a mistake, I had to solve a triple integral to get this result

>> No.6859770
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6859770

>>6859226
Okay, I'll try to tackle this in a serious manner...
>inb4 OP was trolling anyway
One can look into the mathematics of almost every known structure or theory and how "advanced" it gets depends solely on the approach an mathematical prowess of the beholder. Several hard problems of elementary number theory (especially prime number theory) are ridiculously simple to state but as of yet undecided. Or think Fermat's problem: even a primary school pupil (second thought, let's make this grammar school) could in principle grasp the concept of Fermat's conjecture, but Andrew Wiles needed 7 years, a shitton of advanced maths, 150 pages in print (and a coworker) to actually prove it.

To give you an impression one field that is considered to be incredibly obscure and advanced by many mathematicians, consider the theory of algebraic stacks. The Stacks Project (http://stacks.math.columbia.edu/)) has assembled an "introduction" to the subject that is at 4520 pages in PDF atm, all graduate and post-graduate mathematics.

Hope I could help.

>> No.6859782

>>6859260
Elementary Calculus is a joke to real mathematicians and even physicists.

And Yes Calculus is applicable. Applied Mathematicians, and Physicist, and atmospheric scientists use it heavily.

Abstract Algebra, Geometry, Topology, Analysis, and Number theory is what real mathematicians play with, and I guess you could fluid dynamics (where most applied mathematicians go into research...).

>> No.6859814

>>6859463
It's true that math is a tree and lots of results are incomparably advanced, but some results are definitely more advanced than others.

In theory math can get infinitely advanced, but that's not really true because math is done by humans. You only have one lifetime to study math so any one person can only climb so far up the tree.

We're already kind of reaching that limit now. The vast vast majority of math was developed after 1900. An average PhD student knows almost everything discovered before then, but only a tiny fraction of 20th century math. Because of that I think there's been more and more horizontal branching in the last 50 years, and it will only get more pronounced.

>> No.6859883

>>6859782
>Geometry, Topology,
Not up on your topological quantum fluids?

>> No.6859918

OP here just letting you guys know I appreciate the responses, but my head hurts and even something physically has been done to my eyes.

>> No.6859946

>>6859444
Complexity <span class="math"> f(x) \to \infty [/spoiler] advanced math.

>> No.6859996

>>6859338

This is exactly how NOT to calculate with infinite numbers.

The "trick" is here:
10x=9.999…
9x=9

By that you suggest you can just move the comma and "cut out" a part of an infinite number. But this is wrong, because you are cheating on the "last digit" - in other words: since it has not last digit, that kind of calculation is not allowed.

Imagine "0.999.." rather as a rule, not as a number.


>>6859413
>>6859946

No, 1 isn't 0.999..

1 = 0.9 + 0.1
1 = 0.99 + 0.01
1 = 0.999 + 0.001
1 = 0.9999 + 0.0001

And so on.

Therefore:
1 = (0.9999..) + (0.00..1)

Since (0.00..1) > 0, it's not the same.

>> No.6860005

>>6859450
>Topological quantum field theory (that'd be the highest maths I guess the today theoretical physicists would use)
Nah TQFTs are used more or less exclusively by maths guys. As far as we know, spacetime does give a shit about the metric.

>> No.6860011

>>6859996
>By that you suggest you can just move the comma and "cut out" a part of an infinite number.
Exactly, which is why this doesn't work either:

x = 0.333...
10x = 3.33...
9x = 3
x = 1/3

Oh wait, it does work you retard.

>But this is wrong, because you are cheating on the "last digit" - in other words: since it has not last digit, that kind of calculation is not allowed.
There is no last digit so you can't cheat on it. It is allowed.

>Therefore:
>1 = (0.9999..) + (0.00..1)
0.00...1 does not exist. You can't have a last digit when "..." means there is no last digit.

What you are actually extrapolating is that

1 = 0.999... + 0.000...

Which means 1 = 0.999...

>> No.6860057 [DELETED] 

>>6860011

>0.00...1 does not exist.

What is "Lim (1 / 10^x)" ..?

Exactly!
The difference between 1 and 0.999...

>Exactly, which is why this doesn't work either:

No, because you weren't "cutting out" a piece like you did before.

>10x = 9.99...
>9x = 9

OK, I'll show you why this is bullshit:

x = (0.999..) = [SUM(9/10^x), with x >= 1]

<-> 10x = 10 * [SUM(9/10^x), with x >= 1]

<-> 9x = 9 * [SUM(9/10^x), with x >= 1] = [SUM(81/10^x), with x >= 1] = (8.1 + 0.81 + 0.081 + .. ) = 8.999...

<-> x = [SUM(9/10^x), with x >= 1] = 0.9999


This is the only valid way, you just won't get "1 = 0.999.." without cheating.

>> No.6860067

Math can get assumably infinitely complex, but through increments of finite boundaries.

What I mean by that is that mathematics either tries to give statements about the world around us, ablut itself, and ultimately, both.

Therefore, the amount of knowledge you put into this function is how complex math becomes. If you want to throw in abstract vector spaces, an infinite amount of variables, or an infinite amount of dimensions (which is actually basic analysis, i guess, but still), you can, theoretically.

In fact, maybe math is simple, and we're just dumb.


It's probably a little bit of both, OP.

>> No.6860074

>>6860057
That limit you've referenced is zero. That summation converges to 1.

>> No.6860076

>>6860057

>What is "Lim (1 / 10^x)" ..?

FUCK!
I made a mistake here.. :)


OK, let's clarify this:

"Lim (1 - 10^(-x))" is indeed 1.

But 0.9999.. is just a number with an infinite amount of nines.

It's not the same:

The first version is like "what's 1 minus an infinte small number?"

The second version is like "what's a number that has 9 on every digit?"

>> No.6860077

>>6860074

Yeah, you confused me..
Look here:

>>6860076

The second part (about how to calculate with sums) still remains valid. That's why 9x = 8.999.. and NOT 9x = 9.

>> No.6860083

>>6860057
>What is "Lim (1 / 10^x)" ..?
As x goes to infinity? 0. QED

>The difference between 1 and 0.999...
is 0.

>No, because you weren't "cutting out" a piece like you did before.
Yes I did. All I did was switch 9 for 3.

>This is the only valid way, you just won't get "1 = 0.999.." without cheating.
You just did the exact same thing except you wrote the digits as an infinite sum. Do you understand what "=" means?

>> No.6860086

>>6860005
there are some physically-realistic situations where TQFTs are useful, like in condensed matter. It's utility might be more in the field of quantum computing where you have a lot of research into quantum systems where tons of things are invariant under homotopy (because you dont want little variations to mess with your calculations)

>> No.6860089

>>6860076
Your clarification is totally devoid of any sort of rigor. This argument is like using hearsay in mathematics.

If .999... and 1 were two distinct points on the real line, there must be a number between them. However the series "[SUM(9/10^x), with x >= 1]" converges to one, so there cannot be a d > 0 such that d(.99... , 1) > d.

>> No.6860094

Can you guys be more considerate to the ones of us who aren't fucking mathmaticians, what are you talking about.

>> No.6860095

>>6860076
>But 0.9999.. is just a number with an infinite amount of nines.
>It's not the same:
You just proved they're the same.

>The first version is like "what's 1 minus an infinte small number?"
1

>The second version is like "what's a number that has 9 on every digit?"
1

>> No.6860098

>>6859814
Yeah, you're right in some ways IMO, especially in that things will get more "horizontal" now since fewer individuals can keep all of it in their heads. On the other hand I think there are a few counter-balances that keep driving "vertical" advances in theory:

1) automated proof systems will be able to verify everything before us once they get fast and efficient enough;

2) better results in the *pedagogy* of mathematics will allow students to retain more information and "compactify" it into a theory a bit better (for example remember how people had to deal with quadatic forms instead of matrices? a change of notation simplified linear algebra for everyone learning it for the first time);

3) the existence of the internet will foster communication (as it has been doing so far) and the existence of large-scale "programmes" will keep everyone's eye on the prize. There's a good chance that Langlands functoriality will be around after Langlands himself is dead; don't discount the benefits of having a long-living mathematical "meme" like that. The Weil conjectures could have lasted 100 years if not for Deligne and Grothendieck

>> No.6860102

Surely there is a natural upper limit to the level of complex math humans can learn? Simply by the virtue of human lifespan, unless learning techniques are vastly increased then humans are severely hindered by the time it takes to learn mathematics (or anything).

I mean it takes the vast majority of people until age 21/22 to get their undergraduate. Sure there will be savants getting their PhDs age 20 and under but they aren't the norm. Even if you spent the remainder of your lifetime consistently building on your existing knowledge base, there is still only a finite amount of time for learning.

Then the new generations start entirely from scratch. And they have to learn everything all over again. Is it not inevitable that we'll reach a saturation point where people will need to spend nearly their entire life learning the prerequisites required in order to tackle [X problem]?

I know that point is far off since realistically most mathematicians today could 'break into' any field of mathematics they like if they dropped their current field and dedicated all their time to learning a new area. But that doesn't undo the fact that eventually there must be a point where there are 'too many' prerequisite definitions and relations for a human being to adequately understand and make use of given the time that humans have to work with.

>> No.6860106

What you're all not taking into considerations is that math is an abstract logic.

A baby can do it in the womb.

>> No.6860112

>>6860076
0.999... = 9(1/10)+9(1/10)^2+...

This is a geometric series with constant ratio 1/10

Because 1/10 < 1, the series converges by the convergence theorem to

ar/(1-r) = 9(1/10)/(1-1/10) = (9/10)/(9/10) = 1

QED

>> No.6860115

>>6860089


>The second version is like "what's a number that has 9 on every digit?
>9 on every digit
>9

Yeah, it's totally not violating this rule if the first digit is not a 9.


>This argument is like using hearsay in mathematics.

Exactly! The symbol "0.999.." has no exact definition. I've never seen let alone used it in my study.

If you are playing with infinity you must be very careful how you put things down.

>> No.6860124

>>6860115
>The symbol "0.999.." has no exact definition.
Wrong. Here's a question for you: Do you also believe 1/3 is not defined as 0.333...

>> No.6860136

>>6860089
>>6860115

Sorry, I meant to quote this guy:
>>6860095

As to you:
My point was to show that the "proof" of
>x=0.99..
>10x=9.999..
>9x=9

..was bullshit, because you get 9x=8.999..
It's circular reasoning.

Now to your point:
[SUM(9/10^x), with x >= 1] indeed converges to one.

I think I got a problem with that way of putting it like "0.999...", because I think it's mathematically accurate to put it like this.

OK, fine, have you fucking "0.999".

>> No.6860137

>>6860095
>>6860098
>>6860102
>>6860112
>>6860115
>>6860124
see >>6860106

>> No.6860142

>>6860124
>Do you also believe 1/3 is not defined as 0.333...

Well, I studied serveral years of math without this shit symbol of "0.999..".

But if you insist to use it, so be it.

>> No.6860145

>>6860136
>..was bullshit, because you get 9x=8.999..
How does a true statement make the proof bullshit? All you've done is get a trivial result, you haven't shown how the proof is contradictory or flawed.

>It's circular reasoning.
x = 1
2x = 2
x = 1
CIRCULAR REASONING!!!1

>> No.6860152

>>6860145

>How does a true statement make the proof bullshit?

Beacuse it's circular reasoning.
You want to prove "1 = 0.999.." by "9 = 8.999..".

>CIRCULAR REASONING!!!1
You add no new information:

Hypothesis: Pi = 3

Let's proove it:
Pi = 3
<-> 2*Pi = 6
<-> Pi = 3

You see..?

>> No.6860161

>>6860136

How is that circular reasoning? 1=0.999... isn't given at the beginning of the argument.

9.999... - 0.999... = 8.999... Assumes that 0.999 = 1, but that's not used in the original argument.

>> No.6860166

>>6860161

x = 0.999..

leads to:

10x = 9.999..

leads to:

9x = 8.999...

leads to:

x= 0.999..

You ended right where you started.

>> No.6860172

>>6860166

What? No. No it doesn't.

x = 0.999...

leads to

10x = 9.999...

leads to

10x - x = 9.999... - 0.999...

thus

9x = 9.999... - 0.999... = 9
9x = 9

Thus

x = 9/9
x = 1

And so it follows:

x = 0.999... = 1

>> No.6860176

>>6860172
dont get trolled too hard, pls

>> No.6860181

holy fuck will this every fail to generate a shitstorm?

>> No.6860184

>>6860172

>9x = 9.999... - 0.999... = 9

No, it's:

9x = SUM(81/10^x) = 8.9999..

The proof you are using is not within your calculations, it's rather an assumtion.

>> No.6860189

>>6860086
Yeah, Google is big on topological quantum computing, right?
It should really be called homotopical quantum computation, if what you say is true (don't know much about the topic), which sounds much cooler.
I've heard that condensed matter physics makes use of TQFTs, but never seen it in action, if you'll excuse the pun.

>> No.6860195

>>6860176

I planned on stopping now anyway as I figured that person was either a complete moron or a troll, but thanks anyway.

>> No.6860198

>>6860181

Say thanks to the stupid bastards that jerk off to an inaccurate way of putting things.

Actually ever number has an unique notation, only this strange notion 0.9999.. as Limes as an exception.

It's just a shit symbol.

>> No.6860222

>>6860115
>Exactly! The symbol "0.999.." has no exact definition. I've never seen let alone used it in my study.

To say 1 is not equal to .999 is to say that they are distinct points on the real number line. This will work for any definition of .999 as long as it's a real number. I proved that they are the same point, indistinct from each other.

>> No.6860256

>>6860222

I don't like that "0.999.." notion, shove your Archimedean property up your ass.

I therefore define "0.00..1".

>> No.6860275

>>6860256
Fair enough. Can you prove that it is not equal to zero?

>> No.6860281

>>6860275
Not that guy, but wouldn't they need a definition first before they could prove anything about it?

>> No.6860284

>>6860152
>Beacuse it's circular reasoning.
You're an idiot. Equations are not circular reasoning. All valid manipulations of an equation are equivalent to the original statement. All you've done is give a trivial manipulation.

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

All of these statements are true. It's not circular reasoning to say that x = x. That is axiomatically true, you fuckhead.

>You add no new information:
SIMPLY MANIPULATING AN EQUALITY CAN'T ADD NEW INFORMATION YOU PIECE OF SHIT.

>Let's proove it:
>Pi = 3
><-> 2*Pi = 6
><-> Pi = 3
>You see..?
That is a valid but unsound argument because the introductory premise that pi is equal to 3 is false. The reason it's wrong has nothing to do with circular reasoning. Where did I make a faulty assumption? My argument is both valid and sound.

>> No.6860289

>>6860281
He said
>I therefore define "0.00...1."

I was interested in how he would use his definition--whatever it is--to prove that 0.00...1 wasn't equal to zero.

>> No.6860297

>>6860289
Ah, crap. I missed the "I" and got confused. Thanks

>> No.6860536

>>6860011
You can augment the reals to allow uncountably infinite decimal expansions, but I'm sure the person who posted that stuff would have no fucking idea what was going on if you started doing that.

>> No.6860550

>>6860102
Assuming that I have an unlimited amount of paper/storage media I can do arbitrarily complex math with my current brain. Anyone who isn't so retarded as to lack general intelligence can learn as much as they need, write down and verify their results, then forget the earlier stuff and learn new things to keep moving forward, knowing they already verified the previous work. The only thing stopping us from doing this type of thing and really hitting the limits of what we're capable of is mortality.

And people say we'd run out of things to do after 100000 years. We'd probably just be getting started.

>> No.6860556

>>6860124
Try constructing the rationals from scratch and you'll see why that definition can't work. It's fucking circular. Decimal expansions are defined in terms of rational numbers, which you them propose to define with decimal expansions.

>> No.6860557

>>6860550
correction: you can do arbitrarily complex *computable* math (including symbolic manipulation, approximations of continuous operations, etc.) with your current brain

>> No.6860567

>>6860557
Well just let me know when we determine that a noncomputable process is occuring in the universe somewhere, and I'll be sure to use it to modify my brain accordingly.

>> No.6860570

>>6860567
that's not the point. the point is that uncomputable processes exist in mathematics.

you said you can do "arbitrarily complex math". that's just not true

>> No.6860576

Why is everyone struggling with .999.... is === 1?
This is a PROVEN FACT about the number system using unarguable logic that some of you may learn if you ever take discrete mathematics.

There is LOGIC and FACTS to back it up! There are also different sizes of infinity. Learn how to do rigorous proofs instead of taking your own intuition. It lies to you.

>> No.6860578

>>6860570
What if I'm allowed infinite amounts of paper and time, is it possible then? I could write down every real number given an uncountably infinite amount of time and memory.

>> No.6860581

>>6860578
No, you can not. You can write out every rational number but NOT every real number.

>> No.6860583

>>6860578
the existence of uncomputable and uncountable objects in nature go hand in hand. if no uncomputable processes exist in nature, then very likely nature is countable (indeed, most likely finite) as well

>> No.6860592

>>6860576
It's a meme, you fuckin sperg.

>> No.6860595

>>6860578
Here's a (simplified) proof.

You can "List" all the rational numbers in a grid

1/1 1/2 1/3 1/4 1/5 1/6...
2/1 2/2 2/3 2/4 2/5 2/6...
3/1 3/2 3/3 3/4 3/5 3/6...
4/1 4/2 4/3 4/4 4/5 4/6...
5/1 5/2 5/3 5/4 5/5 5/6...
... ... ... ... ... ...

given an infinite time, you will be able to "list" every possible rational number.

If you try to list all the irrational numbers, I can provide any number that is NOT in the list.

lets take for example:

1.123...
2.345...
3.456...
4.567...
...

Now if I take the rows and index them, then take the row at that index and build a new number with the indexes added, then add a stipulation, every time I see a 1, switch it to a 2. Every other time, switch it to a 1. Following this rule, you will always create a unique number. It is impossible to "List" every real number.

Discrete Math.

>> No.6860602

>>6860592
I know it's a meme. It's a fucking stupid meme. You can't argue logic.

>> No.6860607

>>6860581
I didn't specify the order of uncountable infinity. Anything above aleph_2 makes it possible.

>>6860583
I guess this is a matter of mathematical philosophy or metaphysics. What does it mean for a mathematical structure to "exist"? What is the difference between arbitrarily accurate mathematical descriptions of the universe and the "real universe"? Is there one?

>> No.6860612

>>6860595
That's only if I'm given countable time and memory. In the continuous case it can be done.

Analysis.

>> No.6860619

>>6860607
to do what you're saying, you'd pretty much have to be an omnipotent deity that could change the laws of physics

even if you could write a single digit in base whateverthefuck on each cube with side = planck length, you still wouldnt be able to write down all the real numbers.

and if you think you would end up becoming an omnipotent deity that can bend the laws of physics, then you should get the fuck off this board and attend a local church

>> No.6860620

>>6860612
>>6860607
Jumped the gun and read "What if I'm allowed infinite amounts of paper and time". Missed the "uncountably infinite" at the end.

>> No.6860629

>>6860619
Engineers vs Math majors.
Kudos though, I agree.

>> No.6860650

>>6859338
Or simpler.
1/3 = 0.3...
3 · 0.3... = 0.9...

>> No.6860708

>>6860595
You are proving that .9999... cannot equal 1.
A critical part of Cantor's diagonal argument hinges
on generating a diagonal number that is "not on the list".
If real numbers are allowed to have
multiple representations:

.9999... = 1
8.999... = 9
.3333... = 1/3

the entire proof goes down the drain.

>> No.6860714

>>6860708
It would only go down the drain if you could prove that the number you created already existed and it was a different representation of that, which you can't because it gives a contradiction.

>> No.6860747

>>6860714
The burden of proof is on those claiming the cantor number is not on the list.
Since there are an infinite number of numbers, actually an uncountably infinite number
of numbers, that have potential multiple representations that would be a tricky proof.

>> No.6860775

>>6860094
theyre talking about how 3/3 isnt equal to 1, but 0.99... instead

>> No.6861195

>>6860556
you can define the reals by decimal expansions, I'm 99.999...% sure

>> No.6861198

>calculus is a study of change
>Walter Bad told me that chemistry is the study of change

does calculus chemistry??

>> No.6861205

>>6860708
it doesn't matter if there are duplicates

>> No.6861413

>>6859226
>How advance can maths get?
Pretty fucking advanced MATE!

>> No.6861714

>>6860708

>>6860714
> It would only go down the drain if you could prove that the number you
> created already existed and it was a different representation of that,
> which you can't because it gives a contradiction.

>>6861205
> it doesn't matter if there are duplicates

These are claims by anons without any explanation.

The Cantor proof, as is usually demonstrated,
has a major problem with duplicate representations.

>> No.6861716

indefinitely advanced

>> No.6862184

>>6861714
Check out Robert Gray's "Georg Cantor and Transcendental Numbers." It's a fun read.

>> No.6862606

>>6859315
isn't that like high school calculos? lol

>> No.6862850
File: 295 KB, 667x967, rgray01b.jpg [View same] [iqdb] [saucenao] [google]
6862850

>>6862184
pic related

This paper is discussing topics related to this thread.

The author is attempting to prove properties of the diagonal method with an arbitrary enumeration.
For example, that a transcendental number can be generated from an ordered list of the algebraic reals.
It is not disputed that the diagonal method can at times produce new real numbers.
It is not clear that this particular work satisfies the burden of proof specifically mentioned.

>> No.6862929

>>6862850
>with arbitrary enumeration
The enumeration is not arbitrary. It is an ordered enumeration of polynomials with roots in an interval.

>> No.6862944

>>6862929
> The enumeration is not arbitrary. It is an ordered enumeration of polynomials with roots in an interval.

> ar·bi·trar·y
> based on random choice or personal whim, rather than any reason or system.

mmmmmm. Well it is on "personal whim" - that is he chose it for his own proof needs.
Maybe better is "specially chosen"

In the context of this discussion the "reason or system" is not relevant, or so I am questioning at least.

Could have phrased it better maybe.

>> No.6862952

>>6862944
it was the same kind of thing Cantor did, only he ordered the polynomials differently. But this is of no consequence.

>> No.6862964

>>6859568
your understanding of calculus is very limited

>> No.6862967

>>6862944
I'm just saying I personally don't see relevancy.
I'm not disputing his discussion of producing a transcendental.
He is trying to prove some specific yet limited aspects of the mathematics.
But someone please show the proof that more generally addresses the complaint of an infinite number of
multiple representations not causing a problem. This paper shows that a "specially chosen" set that
contains a complete countable subset of the real numbers will (diagonally) produce a number not in
the set. That is very different than saying a complete list of all flavors of real numbers will always
produce a real not on the list, given the multiple representation problem.

>> No.6863005

>>6860708
Anyhow, back to the .999... and 1 discussion

infinite set 1000000000...
not equal to
infinite set 9999999999...

.999... not equal to 1

QED

>> No.6863015

>>6862967
are you disputing the paper I linked or the proof of uncountability?

>> No.6863033

>>6859770

Thanks for that stacks link.

>> No.6863249

0.1 = 10^-1
0.01 = 10^-2
0.001 = 10^-3

0.000...(n-1 zeroes)...01 = 10^-n

0.000...(∞ zeroes)...0001 = lim(n->∞) (10^-n) = 0.

>> No.6863250

>>6863249
>>6859996

Forgot to link.

>> No.6863258

>>6863005
That's not how sets work. If you mEan "the set containing one and then an infinite number of zeros" then sets don't have multiples of the same elemet. Either 0 is in the set or it isnt, it's not in there multiple times. If you mean "a one followed by infinite zeros" well that doesn't have a meating for integers, just decimals.

So instead of a proof, you just have 2 things you've said that don't make sense.

And honestly, if you can't even be bothered to read the Wikipedia article why are you here?
http://en.m.wikipedia.org/wiki/0.999......

>> No.6863304
File: 42 KB, 625x626, 891.png [View same] [iqdb] [saucenao] [google]
6863304

>>6863249
1=!0.9999... because 0.99999...+d=1
It's literally the definition of an infinitesimal. Yes it's small but it still exists and serves to separate .9999... and 1.
Otherwise you're saying that d/dx is a division by zero and all of known calculus falls apart. Congrats you made me respond.

>> No.6863307

>>6860136
Are you sure you understand what circular reasoning is?

>> No.6863335

>>6863015

Not necessarily either, which is probably causing some confusion.
I am the author of these posts
>> 6860708
>> 6860714
I'm after the proof or explanation that addresses these posts.

>> No.6863337

>>6863335
ahhhh thats not right

these posts
>>6860708
>>6860747

>> No.6863440

>>6863258
> And honestly, if you can't even be bothered to read the Wikipedia article why are you here?

Some excerpts from the Wiki page:

>"In particular, the real number 1 is the set of all rational numbers that are less than 1."

That settles it then.


<span class="math"> \lim_ {n \to \inf} \frac{1}{10n}=0 [/spoiler]
> This limit is plain if one understands the definition of limit.

Proof by "plain" ???

>> No.6863486

>>6863258
> And honestly, if you can't even be bothered to read the Wikipedia article why are you here?

Wiki is known for constantly violating their own policies of no original research.
This is because they have all these college people volunteers that can cite whatever
work they choose and want to put on Wiki and then defend it with moderator powers,
and also because they wish to seem relevant as a contemporary source.

They also have a demonstrable history of attacking people attempting
to post true facts that are beyond dispute as to truthfullness if it does not fit
some vague and ambiguous political agenda.

Use Wiki at your own risk. There is a lot of usefull information, but it's not the ultimate
source of fact, rather a playground for a special group of elite Wikipedians.

>> No.6863551

>>6863335
OK, I see. Please be careful in understanding the difference between a duplicate NUMBER and a duplicate REPRESENATION. Not that either actually matter, but this is a real distinction.

The existence of duplicate elements is problematic not due to any generation problem, but because in order for the list to be "countable" you have to have a BIJECTION. But if n maps to both <span class="math">0.1_2[/spoiler] and <span class="math">0.0\overline{1}_2[/spoiler] then you no longer have a bijection. Then how could Cantor prove what he wanted?

It's not a problem with "diagonalization." Diagonalization will generate a representation not in your list of representations, duplicates representations or not. But duplication impacts what you can say about countability/uncountability due to the definition of "countable" which is "a bijection between the elements and the naturals."

I hope that helps clear things up.

>> No.6863725

>>6863551
Not sure I totally understand you but I'll do my best.

To me a duplicate NUMBER in the context of the current discussion would be like
9, 9, 9.0, (8+1), (90/10), 9 since these do not involve an infinite series
a duplicate REPRESENTATION of a number would be
.99999 and 1, 1/3 and .333333, 0 and (1 -.9999...) since these involve an infinite series

> But if n maps to both <span class="math"> \0.1_2 and [\math] <span class="math"> \0.01_2 [\math] then you no longer have a bijection.

Well, you can semantically define a numeric equivalency mapping and then still satisfy the one to
one bijection of infinite set comparison. This would be analogous to putting the naturals in
correspondence with positive and negative integers.

> Then how could Cantor prove what he wanted?
That would depend on whether the number of allowed/accepted alternate numeric representations
is finite or infinite.

> It's not a problem with "diagonalization."
It may actually not be a problem, but there is no proof in this thread yet as to the question
of the consistency of the diagonal argument if we accept .999... = 1.

> I hope that helps clear things up.
A simple proof would do better.
The matter of
>>6860708
is something any high school student may come up with as an objection to Cantor's
diagonal method. What is the response ?[/spoiler][/spoiler]

>> No.6863727

>>6863725
> But if n maps to both <span class="math"> 0.1_2 and [/spoiler] <span class="math"> 0.01_2 [/spoiler] then you no longer have a bijection.

Well, you can semantically define a numeric equivalency mapping and then still satisfy the one to
one bijection of infinite set comparison. This would be analogous to putting the naturals in
correspondence with positive and negative integers.

> Then how could Cantor prove what he wanted?
That would depend on whether the number of allowed/accepted alternate numeric representations
is finite or infinite.

> It's not a problem with "diagonalization."
It may actually not be a problem, but there is no proof in this thread yet as to the question
of the consistency of the diagonal argument if we accept .999... = 1.

> I hope that helps clear things up.
A simple proof would do better.
The matter of
>>6860708
is something any high school student may come up with as an objection to Cantor's
diagonal method. What is the response ?

>> No.6863730

>>6863727
I don't see how 0.999... = 1 has anything to do with anything, frankly.

>choose your bijection
>voila, an element not in your sequence
Simple.

It only gets slightly more complicated if you wish to show that there are irrationals which are not algebraic. But that has nothing to do with 0.999... = 1.

>> No.6863765

>>6863730
If it's so "Simple" then prove the conjecture
>>6860708
is false please.
Or at least give a detailed intuitive argument.

>> No.6863780

>>6859260
All of calculus deals with working of this newfangled mathematical invention known as the infinitesimal. It's halfway between being an imaginary concept, like infinity and a real number. If you want to find the area of something, the best way to do it is to break apart the area into little bits and count them up. The more bits you break it into the more accurate the area is. And if you break it up into infinite bits BAM, you're working with an infinitesimal. That's nutshell calculus

>> No.6863795

>>6863765
what conjecture

>> No.6863836

>>6863795
Prove the following conjecture

Background premises

1) Cantor's diagonal process defines a real number from an infinite list of real decimal numbers.

2) It can be shown that the new number will have a different sequence of digits than any of those
those already present on the infinite list with an appropriate algorithmic change of digits
along the diagonal.

Actual conjecture:

If real numbers with different digit sequences are allowed to be called the same real number
then Cantor's diagonal method alone does not prove the uncountability of the reals, further
proofs are required, since the diagonal process might just produce "clones" of a real number
already represented on the list.

>> No.6863846

>>6863836
The proof can account for some forms of clones, in particular the 999... recurring situation is easily tackled.

>> No.6863881

>>6863846
what proof ?
cantor or the other paper

and don't ask what other paper
it's here
>>6862850

BTW I should have said
"Prove or disprove the following conjecture"

>> No.6863892

>>6863881
I mean any retard can toss out a proof that accounts for recurring 9s. It's bottom feeder level math. You know this and should be ashamed for shitposting.

>> No.6863903

>>6863836
>If real numbers with different digit sequences are allowed to be called the same real number then Cantor's diagonal method alone does not prove the uncountability of the reals, further proofs are required, since the diagonal process might just produce "clones" of a real number already represented on the list.
just include both representations of these troublesome rationals in the sequence.

>> No.6863921

>>6859338
x= .999...
10x= 9.99...0
9x= 10x - x= 8.99...1
x=8.99...1 / 9 = .999...

>> No.6863932

It's rather easy. The size of the power set of {\R} restricted to [0,1) is 2^{\N}. Further, we can represent [0,1] as an uncountable sequence of binary sequences, which represent each unique binary expansion for each element of [0,1].

>> No.6863951

>>6859782
Fluid mechanics are still a big mystery, as far as I've heard.

>> No.6863955

>>6863932
>size of powerset of {\N} is 2^{\N}, which can be shown to be the same size as {\R} given how we build the completion of the rationals via Cauchy subsequences (modding out by {xn, yn : xn-yn -> 0})

>> No.6863958

.>>6859562
>being this oblivious to the problem

>> No.6863960

>>6863958
>there is a problem

>> No.6863972

>>6863892
>any retard can toss out a proof that accounts for recurring 9s
I guess I'm not a retard.

> It's bottom feeder level math.
It's 4chan. The obvious cannot be overstated.

>be ashamed for shitposting
Give an example. Not all of the posts on this topic are mine.
My first post was this one
>> 6860708
and some of ensuing chaos stemming from that post.

>> No.6863985

>>6863892
And what you just posted isn't an example of shitposting because...?

>> No.6864018

>>6863985
Because it was true.

>> No.6864034

>>6864018
The subject of what was true was rather superficial, and simply made you come across as an ass. Such useless posturing on an anonymous imageboard is shitposting.

>> No.6864066

>>6863985
Arguably most of this thread is "off topic".
Does discussing advanced math count as shedding light on "How advance can maths get ?"

>> No.6864093

>>6863836
Cantor's diagonal proof works if you only use transcendentals, which eliminates the possibility of both a terminatino decimal and it's "infinite nines" corresponding notation. (In fact it eliminates the possibility of either)

>> No.6864105

>>6864093
Does this discussion need it's own thread ?
People seem to be complaining the main thread question has been hijacked by specific topics.

>> No.6864109

>>6859226
Arbitrarily

>> No.6864131

>>6859226
Infinitely

>> No.6864356

>>6859260
optimization is probably the most useful part of first year calculus. it lets you minimize/maximize costs, supplies used, and such.

>> No.6864558

>>6860184
We don't need to calculate the value of 9x. We can calculate the value of 10x, and then subtract the value of x from that. There is no reason for use to try and find out what 9x is by any other method.

Though if we get 9x=8.99999.....

8.99999......=8+0.99999.....
8.99999......=9x=8+x
x=.99999.....=9x-8
9x-8=x
9x-x=8
8x=8
x=1

>> No.6864586

>>6863304
People have actually made that claim about calculus before, and in fact if you can't factor your infinitesimal out then you cannot take the derivative.

Essentially what we are doing in calculus is taking the rate of change at a point. Well a function doesn't actually change at a point does it? Really we need an average rate of change. Then we can say how much our function really changes over an interval. Because functions change over intervals not over points. The derivative at a point is always an approximation. The actual rate of change at a point is not definable.

>> No.6864612

>>6863304
even in the hyperreals, where there are infinitesimals, .999... still denotes a real number which is equal to 1

>> No.6864616

>>6864612
(nonzero infinitesimals, I should say)

>> No.6864620

>>6864586
no. all you said is bullshit.

>> No.6864633

>>6864620
Then how much does y=x change at 1,1. Not before or after. How much does it change at that point. How much does it increase or decrease at that point? It doesn't. If it did we'd get an undefined rate of change like we do when we take the derivative of a cubic at 0. In any other case a function does not change values at a point. It has a set value at a point. Rates of change only make sense if we move. And when we do we can only approximate by using the derivative.

>> No.6864651

>>6864633
Actually my bad. It doesn't even change in cubics. It changes at an X value, but it does not change at a point. There exists no rate of change at a point. Only the distance connecting points.