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/sci/ - Science & Math

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8773303 No.8773303 [Reply] [Original]

0.999...=1
prove me wrong
protip: you can't

>> No.8773372

>>8773303
Pic related is not entirely anatomically correct

>> No.8773394

>>8773303
DORON PLS OWN THIS FEG

>> No.8773552

>>8773303

It's approximately one, not one.

>> No.8773575

>>8773552
I't's e'xactlé one, thoust brainault.

>> No.8773595
File: 490 KB, 449x401, how_embarrassing.png [View same] [iqdb] [saucenao] [google]
8773595

>>8773552
No. It's exactly one.

>> No.8773614
File: 78 KB, 444x460, 1489452113574.jpg [View same] [iqdb] [saucenao] [google]
8773614

>>8773552
fuck off brainlet

>> No.8773617

>>8773595

It's exactly 0.999 and it is approximately 1.

>> No.8773618

>>8773303
Proved math is not a real science,
because outcome depends on personal choice

>> No.8773619

>>8773303
except it doesn't

>> No.8773673
File: 23 KB, 400x400, 1484783069773.jpg [View same] [iqdb] [saucenao] [google]
8773673

>>8773617
haha, no
that "..." is important

>> No.8773677

>>8773617
nigga you better be joking

>> No.8773852

[math]-1 = \sqrt1=1[/math]
[math]-1 = 1[/math]
prove me wrong fagtards

>> No.8773863

Niggas be postin' in a troll thread.

>> No.8773875

>>8773303
1-.999999 = 0.0000...1
myth busted

>> No.8773921

>>8773875
0.0000...1=0.0000...0999....

>> No.8773946
File: 151 KB, 611x427, oh.jpg [View same] [iqdb] [saucenao] [google]
8773946

>>8773852
-1 doesn't equal sqrt(1)

>> No.8773950
File: 256 KB, 640x360, look here, faggot.png [View same] [iqdb] [saucenao] [google]
8773950

>>8773875
0.000...1 is not an infinite decimal

You can't have an element in an infinite ordered set without a successor

>> No.8774026

>>8773946
wow you must be cs major
[math]-1(-1)=1[/math]

>> No.8774077

Hey ill give you 0.999 cents for every dollar you give me if you feel that you statement is true you should suffer no loss.

>> No.8774102

>>8773303
You could've written 0.(9) and made your statement much clearer.

>> No.8774115

[math]benis - feminine benis = no benis DD:[/math]

>> No.8774314

>>8774115
How do I thubs up :DD

>> No.8774325
File: 270 KB, 812x883, benin.jpg [View same] [iqdb] [saucenao] [google]
8774325

>>8774314
just give me the benin :DDD

>> No.8774330

>>8774102
or 0.(((9)))
the jews are behind these mathematical lies
keepin' the white man down.

>> No.8774332

>>8773946
>>8773950

baited

>> No.8774341

>>8774077
0.999... where 9 repeats infinitely

>> No.8774342

>>8773303
>>8773595
>>8773614
[math]\text{You guys ought to know better than to call this man a brainlet. But I guess brainlets tend to see brainletness everywhere. If you knew anything about infinitesimal numbers, which clearly you don't, you'd know }1-0.999\dots\text{ is exactly equal to }0.\ \ldots 01\text{. This is a very real quantity that we use all of the time to calculate derivatives. So next time, keep your mouth shut before lecturing.}[/math]

>> No.8774351

>>8774077
Hey fucker, I know [math]\sum_n n=\frac{-1}{12}[/math] so you're just trying to trick me into taking 1/12 of a dollar from you.

>> No.8774358

>>8774332
I know it's bait but it might as well have been a genuine [argument]

>> No.8774364

>>8773950
1 is successed by the 0 that precedes it, brainlet

>> No.8774369 [DELETED] 

[math]\hologo{LaTeX}[/math]

>> No.8774374

>>8774364
the position of that 1 is undefined

>> No.8774375

[math]\LaTeX[/math]

>> No.8774377

>>8774351
actually he is taking 1 dollar and giving one back
1-1+1-1+1....= s
1-(1-1+1-1...)=1-s
1-1+1-1+1...=1-s
s=1-s
2s=1
s=0.5

he is tryin to give you 50 cents

>> No.8774381

>>8774374
No it's not, it's defined to be in the same position as the last [math]9[/math] in [math]0.999\ldots[/math] such that they add to [math]1[/math].

>> No.8774391
File: 108 KB, 728x546, 1489983555459.jpg [View same] [iqdb] [saucenao] [google]
8774391

>>8774381
this is bait but still
>last 9

>> No.8774393

>>8774375
latex more like [math]G^AyTeX[/math]

>> No.8774399

>>8774391
>can't come up with an argument
>fails to see the logic
>"i know, just pretend it's bait"

>> No.8774407

>>8774399
there can't be an element without a successor in an infinite totally ordered set

>> No.8774413

>>8774399
>last digit of an infinitely repeating series of digits
What comes after it? If it were truly the last 9, then it would be followed by an infinite string of zeros. But clearly that isn't the case, because the number is defined as only 9s, repeating forever.

There is no "last 9", there will always be another 9 after it. Unless you think there's a highest number too

>> No.8774430

>>8774407
Sure there can, [math]0.999...[/math] is a coproduct of two infinitely ordered sets of digits, [math](a_i,b_n)=0.a_1a_2a_3\dots\times b_1b_2b_3\ldots[/math]. [math]a_n[/math] succeeds [math]a_{n-1}[/math], [math]b_n[/math] succeeds [math]b_{n+1}[/math] and [math]b_\infty[/math] succeeds [math]a_\infty[/math].

Shit's infinite and totally ordered, and [math]b_1[/math] doesn't have a successor. [math]QED[/math]

>> No.8774438

>>8774413
>If it were truly the last 9, then it would be followed by an infinite string of zeros
No, it's preceded by an infinite string of nines.

>But clearly that isn't the case, because the number is defined as only 9s, repeating forever.
Only when you look at the infinite part, not when you look at the number in reverse.

>There is no "last 9", there will always be another 9 after it.
*before it

>Unless you think there's a highest number too
It's possible to define infinite ordered number systems with largest elements.

>> No.8774454

>>8774430
How are you supposed to do arithmetic with that?

>> No.8774471

>>8774454
Do arithmetic on each of the sides, it's pretty obvious m8

>> No.8774477

WHO THE FUCK CARES

>> No.8774479

>>8774477
did you forget what board you're on?

>> No.8774496

>>8774471
>>8774430
You can't know the position of any of the elements in the b part since the position of an element is dependent on the position of its predecessor.

A decimal(less than 1) would look like

(10^-1)d + (10^-2)d + ... (10^n)d

with your definition, it would look like

(10^-1)d + (10^-2)d + ... (10^n)d + (10^m)

the value of m can't be derived from the other elements therefore it can't be a number

>> No.8774498

... (10^-n)d + (10^-m)d

>> No.8774508

>>8774496
Yes it can. That's like saying [math]a+bi[/math] can't be a number because you can't derive the position of a digit in [math]a[/math] from the position of a digit in [math]b[/math].

Complex numbers aren't ordered but I basically constructed them with an ordering.

You gotta be quicker about these things if you wanna keep up

>> No.8774525

>>8774508
The point is that it's not a real number. You can't do real number arithmetic with it, you'd haveto make up new axioms for it to make it work and even then it won't work for actual real numbers. the + operator is defined only for real numbers

0.999... + 0.000...1 is not a valid operation since one of them isn't a real number

>> No.8774529

>>8773852
[math]\sqrt{1}= \pm 1

-1\neq 1[/math]

>> No.8774541

>>8774525
real numbers are a subset of the superreals, [math]\mathbb{S}[/math]. [math]\mathbb{R}[/math] is actually [math]\mathbb{S}/\langle\mathbb{T}\rangle[/math], where we abandon tiny numbers [math]\mathbb{T}[/math]. This is really advanced stuff they only teach you in PhD programs, so it's understandable you don't know it.

>> No.8774554

>>8774541
operations defined for set does not necessarily imply that they would work for a set that contains that set. For example, there are arithmetic operations defined for subsets of R that does not necessarily work for all elements of R

>> No.8774561

>>8774554
In [math]\mathbb{R}[/math] they handwave away the details and make up bullshits about limit points to get around the fact that [math]\mathbb{R}[/math] only makes sense for certain irrationals and finite/repeating decimals. Seriously go ask your professor, any good professor will tell you this is true.

>> No.8774573

>>8774561
>Seriously go ask your professor, any good professor will tell you this is true.
Ask them what? Those retarded pseudo mathematical ramblings?

>> No.8774575

>>8773303
>0.999...=
the last valid number
0.9... <> 1; 0.9... <> 0
0.9... < 1, if 1 existed
I'd be an total idiot if I wasted my time trying to communicate intelligently with you. You know it all, when you aren't deliberately stirring up problems.
Prove how 1 unit came to be from the 0 unit dimensions of a singularity.
I couldn't care less if you babble I'm wrong. I keep an open-mind of a questing general scientist, so I can't be wrong. Maybe you are "better than a god" as you believe.

>> No.8774580
File: 428 KB, 1920x1080, 1490121979006.jpg [View same] [iqdb] [saucenao] [google]
8774580

>>8773303
0.999=OP is a bitch
prove me wrong
protip: you can't

>> No.8774587

>>8773372
>>8773552
>>8773617
>>8773619
>>8774077
>>8774342
>>8774438
>
>>8774430

[math]x=0.999...[/math]
[math]10x=9.999...[/math]
[math]10x=9+0.999...[/math]
[math]10x=9+x[/math]
[math]9x=9[/math]
[math]x=1[/math]
[math]0 . 9 9 9 . . . = 1 [/math]

Now shut the fuck up and go argue about something else you braindead mongoloid autists

>> No.8774589

>>8773673

It's not.

>> No.8774592

>>8774573
>brainlet gets btfo with deeper mathematical theories than he can process
>begins slinging insults with no refutations

fucking

lol

>> No.8774603 [DELETED] 

>>8774587

10x = 9.999...
x = 10/9.999...
x = 0.099...

>> No.8774610

>>8774587
[math]10x=9.999\ldots[/math]
[math]10x=9+0.999\ldots[/math]
is not a valid step,
[math]10x=9.999\ldots[/math]
[math]10x=9-8\times0.00\ldots 001+0.999\ldots[/math]
is correct, which gives you in the end that
[math]0.999\dots=1-0.00\dots 001[/math]
which is true

>> No.8774612
File: 297 KB, 300x456, 1437552316975.gif [View same] [iqdb] [saucenao] [google]
8774612

>>8774587
0.111... = 1

>> No.8774618

>>8774610
oops, that 8 should be a 9

>> No.8774620
File: 89 KB, 1936x1808, I was only pretending.png [View same] [iqdb] [saucenao] [google]
8774620

>>8774592
>is shown to be a fucking retard
>acts like he was only pretending to be retarded to not seem like a fucking retard on an anonymous kalmyk horse archery tip-sharing forum
>makes himself look even more retarded

>> No.8774631

>>8774620
you didn't show anything, though

>> No.8774641

>>8773303
in grade school they teach you bullshit like "there is a 1-1 correspondance between real numbers and decimal expansions", but they neglect to tell you that it's a lie, since their mapping, which ignores tiny numbers, maps 1 to both 1 and .999...

therefore, there's actually more (measure-wise) decimals than real numbers, and they handwave away the algebraic inconsistency to push an agenda

>> No.8774646

>>8774610
>not a valid step
Are you implying that [math]9.999 /ne 9 + .999[/math]
By that logic [math]9.5 /ne 9 + .5[/math]

>>8774612
kek

>> No.8774651

1/3 = 0.33333....
1/3 + 1/3 + 1/3 = 1.
1/3 x 3 = 1.

It's simple.

>> No.8774803

>>8773303
>prove me wrong

Son you havent proven yourself RIGHT to necessitate being proven wrong.

>> No.8774813

>>8774646
No, what you're saying doesn't even make sense. 10 times .999... = 9.999...9990 so 10-9.999...9990=0.000...0010

>> No.8775298

>>8774813
Pretty sure this is b8 at this point.

The nines don't ever end like you show.

>> No.8775307

>>8774651
the mappings between decimal expansions and real numbers are wrong

>> No.8775311

>>8775298
There are infinite nines INBETWEEN the ends, you doofus. I would expect a lowly undergraduate to get this, but I guess there is just no helping some people.

>> No.8775369
File: 26 KB, 300x354, headache.jpg [View same] [iqdb] [saucenao] [google]
8775369

>mfw people on reddit-lite are dumb enough to believe the sequence 1,0,0,0,0,... is the same as 0,9,9,9,9,...

>> No.8775566

>>8775369
Real numbers aren't nessisarily described by a single sequence of decimal digits.

>> No.8775577

>>8773950
>You can't have an element in an infinite ordered set without a successor
Do you mean predecessor? Are you denying the existence of sets with order type [math]\omega+1[/math]? Do you not believe [math]\{1-\frac{1}{2^n} : n \in \mathbb{N}\} \cup \{1\}[/math] is an infinite ordered set? Oh right, you don't know what ordinals are because you're just a popsci retard.

>> No.8776662
File: 10 KB, 236x236, bff5d074d399bdfec6071e9168398406.jpg [View same] [iqdb] [saucenao] [google]
8776662

0.(1) = 1/9
0.(9) = 9 * 1/9 = 1

Easy cheesy, lemon squeeze.

>> No.8777006

>>8773303
1/3 + 1/3 + 1/3 = 1
0.333... + 0.333... + 0.333... = 1
0.999... = 1

Why is this so hard for brainlets to understand?

>> No.8777123

As far as I know, it depends on the chosen metric wheter [math] 0.999... = 1 [/math]. You look on [math] 0.999... [/math] as a cauchy-sequence and if it convergeces at all and what its limit is depends on the metric.

>> No.8777135

>>8773303
0.999...=/=1
there
waddya gonna do about it?

>> No.8777155

Scientific notation is off

>> No.8777415

>>8777006
Because decimal expansions and fractions arent the same thing.

>> No.8777453

1 - 0.(9) = 0.(0)

0.(0) =/= 0

>> No.8777478

>>8777415

Yes they are. I think I can begin to understand why you can't get your head round this, it's because you're a simpleton.

>> No.8777496
File: 1.90 MB, 490x478, 1486519021187.gif [View same] [iqdb] [saucenao] [google]
8777496

>>8777415
wrong

>> No.8777654

.999 ≈ 1
.999 =! 1
there you go friendo

>> No.8777696

>it's a brainlets don't understand limits nor convergence episode

>> No.8777820

No, 1=1 and that's it. If 1 can equal anything else than 1 or an equation that also equals 1, then mathematics is broken as a whole and should be reworked to not be retarded

1 =/= ~1

>> No.8777847

>>8773303
are you saying that an irrational number can equal a rational number?

>> No.8777848

>>8773303
k

>> No.8777861

>>8777847
prove 0.999... is irrational

>> No.8777864

>>8777847

.999 recurring isn't irrational you dunce.

>> No.8777962

X ≥ 0
X+Y > 0

Solve for Y:

Y = .000...1

Otherwise this simple question has no answer.

>> No.8777988

>>8777962
how about Y=5?

>> No.8778000

>>8777988

Oops, I wrote the equation wrong, sorry.

X < 0
X+Y ≤ 0

Solve for Y:

Y = .000...1

>> No.8778012

>>8778000
so with X=-5 my Y could still be 5...
As long as Y >= X < 0 it should work in general.
i nearly get what you are trying to say, but i can't put it in an equation.

>> No.8778017

>>8778000
>.000...1

This isn't a number. The ... means "repeat forever".

>> No.8778022

>>8773303
infinitesimals

>> No.8778029

>>8774612
in base 2 it is

>> No.8778066

if 0.(9) = 1, then 1 + 0.(9) = 2
but it clearly doesn't. it equals 1.(9)
where is my Field's medal?

>> No.8778150

>>8777478
There's a very tiny number inbetween 0.99.. and 1. This is basic non-standard analysis. I know youre a pleb and have only seen calculus, and think youre some math genius, but youre wrong.

>> No.8778155

>>8777696
0.999... is infinitely close to 1, but its not 1 because there's an infinitely small number inbetween.

>> No.8778173

>>8778017
Yes it is a number.

>> No.8778174

>>8773303
I'll introduce the Axiom Of 0.999 Is Not Equal To One

>> No.8778179

>>8777820
>or an equation that also equals 1
You realize what .999... is, right? So you'd permit that 1/1 is an equation that equals 1, right? Would you permit that 9/10 is an equation that equals .9? And 9/10 + 9/10 equals .99? I'm sure you can see where this is going.

>> No.8778187

>mfw /sci/ is polluted with engineering sophomores who think they know everything about numbers because their calc professor simplified numbers to make their baby-tier limits calculations simpler

>> No.8778203

>>8778174
999/1000 is not 1000/1000

good job anon, not sure we need a new axiom for this though 2bh

>> No.8778207

>>8778173
>HURR

Oh you're retarded, carry on then.

>> No.8778210

>>8778203
It's fine, I'll just make another for 0.999... too. It's not like anyone can stop me, or prove me wrong.

>> No.8778213

>>8778203
Yup, 9/10=/=1,99/100=/=1, and so forth, .999... =/=1

>> No.8778219

an infinitely small number =/= 0

>> No.8778220

>>8778219
This.

>> No.8778222

>>8773575
>>8773595
>>8773614
>>8773619
>>8773677
>>8773950
>>8774374
>>8774580
>>8774587
>>8774651
>>8777696

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

>>8778012
>As long as Y >= X < 0 it should work in general.

That is an incomplete solution because it does not work for any X. You can't just throw extra restrictions into an equation to make your solution work. You need a Y that works for ANY X.

>> No.8778227

>>8778207
>mfw college freshman act like jerks because they can repeat shit they heard in a baby tier calc lecture

If you're such a math genius, why don't you prove infinitely small numbers don't exist, or that 0.999..=1. Without decimal expansion wankery, give an actual proof. But you can't, I'd bet money you're too stupid, that's why you just mock people who clearly know more than you.

>> No.8778251

>>8778213
please can you tell me what 0.999... is?
or at least put is as a fraction (numerator and denominator are finite integers, denominator non-zero)

>>8778210
interesting axiom

>> No.8778258

Lets try this a different way:

Line A = [0,1]
Line B = [0,1)

Note the ) instead of ] for line B. This means that line A includes 1, but line B excludes it.

So clearly Line A ≠ Line B.

How much longer is Line A then Line B?

>> No.8778262
File: 2.83 MB, 696x796, 1490476892141.webm [View same] [iqdb] [saucenao] [google]
8778262

>>8773303
>prove me wrong

Burden of proof.

>> No.8778264

>>8778258
>hurr durr 0.00...0001 longer

>> No.8778266

>>8778251
>interesting axiom
Thanks. I worked hard.

>> No.8778272

>>8778251
>please can you tell me what 0.999... is?
>or at least put is as a fraction (numerator and denominator are finite integers, denominator non-zero)

You can't do that with Pi or the imaginary unit, so why should you expect this of the infinitesimal unit? If you want a notational system, try this:

0@1 = .000...1
1@-1 = .999...
1_-1/2@0 = 1 + 2 + 3 + 4 + ⋯
1/0 = 1_0@0

I call this Base Infinity.

>> No.8778274

>>8778264

Is that your final answer? Is that all it takes to satisfy your curiosity?

>> No.8778289

>>8778272
your(?) argument only worked for fractions, so there was some confusion about the notation 0.999...

so there is no need to mention irrational numbers, unless a different argument is provided

>> No.8778312

>>8778289

We are supposedly talking about the Real set, which includes irrationals. You may be mixing my argument up with another, so I started using a post name.

>> No.8778337

>>8778312
the argument in >>8778213
the rationals are closed under finite addition and multiplication operations (it is a field)
it appeared that the argument used the formula
F(a/b) = (10a+9)/10b to obtain the next rational in the sequence to show for example 999/1000 =/=1000/1000=1

but assuming infinitely many applications leads to the same conclusion would be very silly

for example 1 is finite
1+1=2 is finite
and so on
1+1+1+...+1 infinitely many times is finite
so infinity is finite

>> No.8778424

>>8778337

That's not me. But the problem there is that there are contradictions in how you define infinite. You don't really have any explanation as to why an infinite series of inequalities would end in an equality. You have simply declared that a string of numbers can be infinite, and then declared that nothing can come after that. Why? Why not?

If there is no final digit to .999... then at what point to the string of inequalities become an equality, and why do they stop there?

>> No.8778445

>>8778424
>infinite series of inequalities would end
it would not end

the string of inequalities is in fact a string of infinite inequalities, there is no "end"

>> No.8778449

if you cannot find a number between a and b then

a = b

>> No.8778450

>>8778445
>it would not end
>the string of inequalities is in fact a string of infinite inequalities, there is no "end"

Then we are back where (the other poster) started:

.9 ≠ 1
.99 ≠ 1
.999 ≠ 1
.
.
.
.999...≠1

If there is no end, then what comes next? And why would the ≠ change to a = at this or any other point?

>> No.8778456

>>8778449
>if you cannot find a number between a and b then
>a = b

Okay then, in >>8778258 how much longer is Line A then Line B?

>>8778258

>> No.8778458

>>8778258

removing a measure zero set doesn't do anything lad you know this

>> No.8778459 [DELETED] 

>>8778450
please can you fill in the gap in that sequence?

>>8778258
they have the same cardinality (there is a bijection between them)

>> No.8778461

>>8778450
>>8778459
you have assumed an end point

>> No.8778462

>>8778456

in analysis you say

Line A = Line B (up to a set of measure zero)

>> No.8778487

>>8778458
>removing a measure zero set doesn't do anything lad you know this

So the lines are equal despite one having an element that the other does not? Or are they unequal with a difference of zero? When your theory tells you that 1=0, you have a problem.

>>8778461
>you have assumed an end point

No you have. My position is that nothing special happens at .999... To make the equality occur, you require the end that you insist doesn't exist.

>> No.8778502

>>8773303
1/3 is meant to represent one third of one.
So, 1/3 + 1/3 + 1/3 = 1.

The only way we can numerically represent 1/3 in decimal is 0.333...

0.333... + 0.333... + 0.333... necessarily has to equal 1; otherwise 1/3 is not the same as 0.333...

So either you must concede that it is impossible to represent 1/3 in decimal,
or you must concede that 0.999... is the same as 0.333... + 0.333... + 0.333... and the same as 1/3 + 1/3 + 1/3 = 1.

Both are equally correct, and neither of them allow 0.999... to represent anything other than 1.

>> No.8778503

>>8778487
the sequence can be shown to be fractions of the form 99...9/100...0, with finite 9s and 0s

the claim is that 0.999... is in this sequence, however it is not

we can get very close to this point, but it is not the same as being in the sequence

>> No.8778509

>>8773303
Yes sir you right
>>8773617
>>8778450
Faggots
[math]0.9=9\left\{ \left[\sum_{i=0}^{1}\left(\cfrac{1}{10}\right)^{i}-1\right]\right\} [/math]

[math]0.9=9\left\{ \left[\sum_{i=0}^{1}\left(\cfrac{1}{10}\right)^{i}-1\right]\right\}[/math]


[math]0.99=9\left\{ \left[\sum_{i=0}^{2}\left(\cfrac{1}{10}\right)^{i}-1\right]\right\}[/math]


[math]0.999=9\left\{ \left[\sum_{i=0}^{3}\left(\cfrac{1}{10}\right)^{i}-1\right]\right\}[/math]


[math]\downarrow[/math]


[math]0.999....9=9\left\{ \left[\sum_{i=0}^{\infty}\left(\cfrac{1}{10}\right)^{i}-1\right]\right\} [/math]


[math]\sum_{i=0}^{\infty}a^{i}=\cfrac{1}{1-a}\leftrightarrow a<1\rightarrow a=0.1[/math]


[math]\sum_{i=0}^{\infty}\left(\cfrac{1}{10}\right)^{i}=\cfrac{10}{9}[/math]

[math]0.999....9=9\left\{ \left[\cfrac{10}{9}-1\right]\right\} =1[/math]

>> No.8778511 [DELETED] 

>>8773303
>>8773617

Technically real numbers don't exist in the real world. They are just a concept. There's only fractions/decimals.

>> No.8778520

Technically whole numbers don't exist in the real world. They are just a concept. There's only fractions/decimals.

>> No.8778561

>>8778503
>we can get very close to this point, but it is not the same as being in the sequence

.999... is defined as being the end of that sequence. And yet it is also correct to evaluate it as you have. Thus .999... ≠1.

>> No.8778565

>>8778509

You are just burying the same false assumption in deeper jargon.

>> No.8778578

>>8778561
>the end of that sequence

please see >>8778445
and >>8778461

>> No.8778582

>>8778578

Please see that I have already responded to both those posts.

>> No.8778596

>>8778227
>infinitely small numbers don't exist,
It's an elementary consequence of the definition of the real numbers.
0.99...=1 assumes that "0.99... is a real number" is part of your hypothesis/definition. If you allow it to be a hyperreal, you can easily just define .99...=1-h=/=1 where h is the hyperreal element.
If you define it to be real, though, then it's clearly contained in [0,1] and an upper bound to every set [0,a) with 0<a<1. Meaning it's contained in every set of the form [a,1] whose intersection is precisely the single element set containing 1. But 0.99... is contained in this single element set by hypothesis. Hence 0.99...=1.
Remember were not dealing with real, actually existing things, but rather definitions and their consequences. Pure argument. What "0.99..." equals essentially reduces to your choice of definition, and why you've made it.

>> No.8778605

>>8778596

Then people should stop saying ".999... = 1" without specifying what set they are talking about. It's like saying "1/2 = 0" without bothering to mention that you are talking in the set of whole numbers.

".999... = 1" is just trolling unless you specify that you are excluding important number sets where this is not true.

>> No.8778610

>>8778565
>false assumption
>no arguments
>deeper jargon
>jargon
go back to elemetary school

>> No.8778612

>>8778582
personally, at no point ITT have i referred to the existence of the "end point" of an infinite sequence, or even the existence of 0.999...

>>8778605
>is just trolling

even in a place as maths-oriented as /sci/ it is accepted that we work in sets where this is true, and on the rare occasion someone wants to talk about say, hyperreals then they will state so

>> No.8778622

1/3 = .33333...

(.99999....)/3 = .33333....

>> No.8778624

come on people...

>> No.8778641

>>8778612
>personally, at no point ITT have i referred to the existence of the "end point" of an infinite sequence, or even the existence of 0.999...

But you are posting anonymously, so I can't avoid mixing your statements with everyone else who argues on the other side.

>even in a place as maths-oriented as /sci/ it is accepted that we work in sets where this is true

If you want to make a statement, you can't ask others to figure out under what context that statement is true. You have to state your full position. Otherwise I could say "Bicycles are faster than cars." and accuse you of being foolish for not realizing that this is true while riding through forest trails.

>>8778622

>1/3 = .33333...
This is what I meant by hiding your assumptions and circular logic. .33333... can only approach 1/3, just as .999... can only approach nine. I can't see how anyone questioning one equality would ever accept the other.

>> No.8778642 [DELETED] 

>>8778624
>come on people...
Smile on your brother
Everybody get together
Try to love one another
Right now

>> No.8778657

>>8778641

i thought it would be easier if i explained it that way, but let me break it down even more.

the number .33333... has infinitely many digits, not just arbitrarily many.

if it was not the case that
1/3 = (.9999....)

then there should be a decimal place where they differ, but you cannot find one. both numbers are three repeating.

>> No.8778664

>>8778657
* i meant 1/3 = (.99999...)/3

>> No.8778672

>>8778657
>then there should be a decimal place where they differ

I can find one, you just don't like it when I show it to you.

>> No.8778678

>>8778672

you're confusing me with someone else. link me the reply, but i'll tell you now, you're wrong.

>> No.8778697

>>8778641
>You have to state your full position

>>8778672
here would be a good example of a place to state your full position

also this was the initial problem, as i didnt get an answer how 0.999... was defined by >>8778213

>> No.8778735
File: 113 KB, 634x815, 3DC458D600000578-4264002-image-a-10_1488207420797.jpg [View same] [iqdb] [saucenao] [google]
8778735

It all boils down to this:

The set of Real numbers is built upon a collection of assumptions or Axioms. One of these axioms is that there are no non-zero infinitesimals. Other sets, like the Integers, make different assumptions.

As soon as someone questions .999...=1, they are questioning the nature of the Real set. The only logical response to this should be to bring up other sets such as the Hyperreals that are capable of presenting meaningful answers to the question.

Instead, many people respond to the question of .999... with something resembling religious fervor. Answers like "1/3x3" or "10x=9.99.." are just insulting because they assume the very axiom of the Real set that is being questioned. It is like using Whole number analysis to prove that 1/2 does not exist! This makes all of higher mathematics look like a scam. The questioner SHOULD distrust people who don't see the problem with those answers - they may be more educated, but they aren't very smart.

It doesn't make any difference whether or not the questioner, or even the answerer has the education to understand Hyperreals. What matters is that the questioner has uncovered something important and (sometimes) useful in mathematics. Such knowledge, even if incomplete, is not to be feared.

>> No.8779069

>>8774587
x=0
10x=0
10x=0+x
9x=0
x=0
not x=0.000000...01

>> No.8779184

>>8778735

>>8778312
>We are supposedly talking about the Real set


really makes you think

>> No.8779355

>>8778735
You can look at the reals from different points of view. Not only as a set, defined by axiom, but also as a field or vector space or ...

>>8777123
Assume euclidean metric

>>8778150
and others:
Assume standart analysis.

>>8778227
>>8778262
Analysis 101, first semester style proof:

We say a sequence [math](a_n)_{n \in \mathbb{N}}[/math] converges to [math]a[/math] if:

[math]
\forall \varepsilon > 0 : \exists N \in \mathbb{N} : \forall n>N, n\in \mathbb{N}: |a_n - a| < \varepsilon
[/math]

right? So claim:

[math]
\forall \varepsilon > 0 : \exists N \in \mathbb{N} : \forall n>N, n\in \mathbb{N}: |\sum_{k=1}^{n}{9\cdot 10^{-k}} - 1| < \varepsilon
[/math]

proof:

Let [math] \varepsilon > 0[/math]. Choose [math] N \in \mathbb{N} [/math] s.t. [math] 10^{-N} < \varepsilon [/math]. Then, for all [math] n > N [/math]:
[math]
|\sum_{k=1}^{n}{9\cdot 10^{-k}} - 1| = 10^{-n} < 10^{-N} < \varepsilon
[/math]
Q.E.D.

>> No.8779419

>>8779184
>really makes you think

Or maybe not. Go back and read the context of what I was replying to.

>> No.8779428
File: 20 KB, 240x210, 0c05974f4c2ef0d8f816ee0069c769a55cc392846a1858c4adec4b22faa93420.jpg [View same] [iqdb] [saucenao] [google]
8779428

>this bait thread is still up
Really makes you think

>> No.8779462

>>8778222
infinitesimals aren't real numbers

>> No.8779475

>>8778155
Wow you're an idiot

There are no such thing as infinitely small numbers. Infinitely small numbers are equal to zero, learn basic limits

>> No.8779490

>>8779419
>>8779475
almonds are being activated

>> No.8779492

Omg your level in math is so low, people... With all your non-rigourous proofs, people who say that 0.999...≠1, etc. You clearly don't apprehend math the right way.
I knew american students weren't good at math but I didn't expect them to be that bad.

>> No.8779531

>>8779492
Agree in general on the low, disappointing level, thought this is a MATH&science- board. There are probably PhD students and 8th graders on the same thread discussing with each other.
But be aware, there is indeed non-standard analysis where you have indeed "infinitely small (and large) numbers".

> ...non-rigourous proofs
>>8779355
...here you go.

>> No.8779578

>>8773372
fpbp

>> No.8779579
File: 2.71 MB, 341x305, 1481733267859.gif [View same] [iqdb] [saucenao] [google]
8779579

>>8774651
How about this one to fil this shitty thread :

1/9 = 0,111111...
2/9 = 0,222222...
3/9 = 0,333333...
...
9/9 = 0,999999...

>> No.8779583

0.888...=1

>> No.8779595
File: 99 KB, 1059x747, time to skip this thread.jpg [View same] [iqdb] [saucenao] [google]
8779595

>>8773852
Now "imagine" that :
[math]-1 = i2[/math]

>> No.8779770

>>8778605
>It's like saying "1/2 = 0" without bothering to mention that you are talking in the set of whole numbers.
Not really, no, since there's no 1/2 in the set of whole numbers.

>> No.8780107

>>8774529
This is wrong. The sqrt is a function. There is only one sqrt for every number.

>> No.8780247

>>8779355

I guess this is the damage that those garbage proofs do. I can look at a computer or an airplane and see that they work without understanding how. So I can trust that the people who made them know what they are doing.

But when so many people are unable to question the assumptions behind the "non-rigourous proofs", how am I to know that the same flaws aren't hidden in the symbols you use?

Look at the poor answers to >>8778000 and >>8778258
They sound more like politics then math. And then...

>>8779579
See? The problem with this should be obvious from my previous posts, but it keeps coming back. They ask me to "think" but won't do it themselves. (I'm questioning all those equalities.)

>>8779770
Even when people are proving my point, they think they are opposing it. There are no infinitesimals in the Real set for the same reason there is no 1/2 in the Whole set....

>>8779462
The Real set isn't any more real than the whole set. People can't tell the difference between the two meanings of the world "real". I wish the "Real set" had a different name.

>> No.8780265

Here is another example of the limits of "Real" numbers:

Imagine you have a perfect random number generator. It will generate a real number between zero and one.

What are the chances of this machine generating a 2? Obviously zero, because that is outside it's range.

But what is the chance of it generating .5?

That works out to 1/infinity, or zero. But it is a DIFFERENT ZERO, because .5 is possible, but 2 is not!

The two probabilities are clearly different, but the Real Set lacks any ability to describe how.

>> No.8780301

>>8780265

Oh BTW, that random number generator can never exist because if it did, it would invariably generate an infinite amount of information and thus destroy the universe. That should tell you how "Real" real numbers are.

>> No.8780475

>tripfag spergs over nomenclature

lel

>> No.8780616

>>8780247
>>8780265
I get your point and you are right. I understand the problem.

Some people here seem to think (after they meditated a few minutes about it, if at all) that mathematicians came along with some theories out of nowhere. But this is not true, sometimes it was a decade-long (or even longer) struggle (fought by brilliant minds) to formulate sound concepts and definitions. Take your probability example, probability theory went to such a struggle. And measure theory, as part of it, wasn't easy to formulate as well.

According your p=0 problem (short, not entirely formal correct answer):
You have to define a sample space [math] \Omega [/math], in your example, [math]2 \notin \Omega [/math], so it makes no sense to ask about the respective probability at all). Also P(0.5) = 1/infinity is not proper formulated. You would instead argue that {0.5} is a borel null set. My point is (but it is yours too, if I understood you correctly): Mathematics has to be done carefully and well defined.

Yes it could not exist in the real world (as far as we know, church-turing isn't proven and can never be), but that is "normal" in mathematics. A "point", "line", "circle" etc., according to their math-definition, also don't exist in the real world.

>> No.8780923

>>8780616
>borel null set

The problem with a borel null set is that the person you are talking to will think you are calling them names.

>> No.8780960

>>8780616

And in any case here we have Real numbers being the opposite of reality. And as I predicted, you DIDN'T answer the question using real numbers.

I started with an infinite set, (The Reals between 0 and 1) and described a finite set within it. When the infinite set is described in a finite way, (range of 1) the finite set becomes an infinitesimal, but retains properties distinct from zero.

> 2∉Ω, so it makes no sense to ask about the respective probability at al

This doesn't make sense to me. How can a simple "what is the probability" question not have an answer?

>> No.8781004

0.999...= 1 - 0.00...001
= 1-10^Inf
=1-1/(10^inf)
=1-0
=1

QED

>> No.8781179

>>8780960

>And in any case here we have Real numbers being the opposite of reality. And as I predicted, you DIDN'T answer the question using real numbers.

I'm not sure if I get your point: There are no "real" numbers. All math concepts are abstract like the examples "point", "line", "circle" etc. But this is philosophy. We observe the world --> we translate them and make abstractions that are in a way "ideal" --> we are playing around in these acstract worlds and eventually get answers to questions --> we translate the answers back to the real world.

>This doesn't make sense to me. How can a simple "what is the probability" question not have an answer?

(I'll try to explain it, still in a pop-sci way:)
You have to define a "model" in which you do probability calculation. First, you need a sample space [math]/Omega[/math], which events are possible? In your example with the RNG, would it make sense to ask for the probability, that it will rain tomorrow? Or course not, it is the same with 2, it is not an event that exists in your model. So how can you ask for its probability? Then you need a set [math]\mathcal{F}[/math], build out of [math] \Omega [/math], that satisfies some properties (for example: if "A" (e.g."Tomorrow rains") is an event and "B" (e.g. "Tomorrow I will play WoW") is an event, then "A and B" ("Tomorrow rains and I will play WoW") must also be an event).
Third, on this [math] \mathcal{F} ][/math] you define a measure [math]P[/math] which assigns each element in [math] \mathcal{F} ][/math] a value in [0,1], the "probability". For example you could define P(x)=1 if x=0.5 and P(x)=0 otherwise. Now the probability of your 0.5 is 1. Normally with Reals, you take a measure on borel sets, another math concept.

>> No.8781587

>>8781004
that's a real quick way to get a backhand raising to the power of infinity

>> No.8781633

>>8773303
1 - 0.9 = 0.1
1 - 0.99 = 0.01
1 - 0.999 = 0.001
1 - 0.9999 = 0.0001
1 - 0.99999 = 0.00001

See a pattern yet?

1 - 0.9999999999999.... (repeating infinitely) = 0.00000000000000...(infinite number of zeroes)...1

>> No.8781657

>>8781633
>0.00000000000000...(infinite number of zeroes)...1
What do you think this notation means?

Protip: it doesn't mean anything.

>> No.8781985

>>8781179
>I'm not sure if I get your point: There are no "real" numbers.

The point was that other posters were saying "infinitesimals aren't real." and acting as if that were the end of the discussion, as if all the other objects you mention had a property that infinitesimals don't.

>(I'll try to explain it, still in a pop-sci way:)

I got all that from the other post. But I still don't see what is wrong with saying "The probability of a "2" from a device that states the chance of rain is zero because that answer doesn't make sense in context.".

>>8781657
>Protip: it doesn't mean anything.

Shut up you Borel Null Set.

>> No.8781986

>>8781985
>infinitesimals aren't real

its pretty obvious they arent

otherwise they would be in R

>> No.8782019

>>8781986

But why is that even relevant? Infinity and the imaginary unit aren't real. This constant confusion between "real" and "The Real set" is about 2/3s of the problem.

>> No.8782023

>>8782019
no its not its this >>8780475

>> No.8782037 [DELETED] 

[math]9*\sum_{0}^{inf}10^{-n}[/math]

>> No.8782042

>>8780247
You can't do real number operations with infinitesimals

>> No.8782054

>>8781657

witchcraft version of mathematics ;).

>>8781985
> But I still don't see what is wrong with saying "The probability of a "2"...

Maybe I should have taken a more absurd example then the one with rain tomorrow. "What is the probability that the second moon orbiting earth will explode?", nothing, because earth doesn't even have a second moon. The event is not just not possible, it doesn't even exist.

>>8782019

Yes that, and
2. that some are argue with non-standard analysis. But there are more then one non-standard model and it is not a priori obvious what things like "converges" or "continuous" mean in such a model.
3. and that most seem to be unaware that [math] \mathbb{R}[/math] is defined (in the view people have in this thread) as equivalence classes of limits of series in [math]\mathbb{Q}[/math]. So it doesn't matter if 0.9999... and 1 are _exactly_ the same, they lie in the same equivalence class* and we use "=" to denote this. Therefor 0.9999... = 1 (meaning "rational series which converge to 0.9999... also converge to 1").

* which depends on the metric. There are also other complete spaces build on [math]\mathbb{Q}[/math], for example p-adic space [math]\mathbb{Q}_p[/math] where for example 1 + 1/2 + 1/4 + 1/8 +... =/= 2.

>> No.8782067

>>8781587
And it's equally as valid as all of calculus

>> No.8782075
File: 298 KB, 600x878, Pope Real set.jpg [View same] [iqdb] [saucenao] [google]
8782075

>>8782054
>"What is the probability that the second moon orbiting earth will explode?"

That one is actually LESS absurd. It's probability is not even zero because something definable as a second moon might exist someday and then explode. (I don't think "moon" is all that strictly defined, so a satellite might count - and someone could have named one "second moon".

But getting back to the original point, you have these two different situations. "Can't happen" and "can happen with zero probability" that can't be explained with real numbers. To me, it is the second, not the first, that is absurd. But it becomes simple if you say that the chance of the RNG producing .5 is an undefined infinitesimal.

> So it doesn't matter if 0.9999... and 1 are _exactly_ the same, they lie in the same equivalence class* and we use "=" to denote this. Therefor 0.9999... = 1

I expect this is probably true, but it makes the Real set a totally inappropriate tool to discuss the equivalence of 0.9999... = 1. None of these "proofs" prove anything because the Real set is defined to exclude the very subject that is being discussed. It's like proving that all cars have the same top speed by using a racetrack that enforces a speed limit.

>> No.8782077

>>8779428

B≈∞

B = Baited

>> No.8782081

>>8774651
the thing is 1/3 =/= 0.3333...
We only use the decimal representation as a guide but we intuitively know that there has to be a number greater than 3 somewhere in that number x to make 3x = 1. Therefore let's just call 1/3 = p. p exists on the number line but is not equal to 0.333....
Real close though
Or just leave it as 1/3

>> No.8782097 [DELETED] 

[eqn].999... = .9 * \sum_{0}^{\infty}[/eqn]

>> No.8782098

>>8782081
>1/3 =/= 0.3333...
prove it

>> No.8782105

>>8782098
You can complete come infinite number of tasks in convergent series for example
However no matter how many times you multiply 3 by 3, you will never get anything but a 9. The operation is divergent and you cannot complete the infinite number of steps
I believe this notion will eventually crumble in math but not much will change, it is still useful as a convention

>> No.8782106

>>8778641
>just as .999... can only approach nine
ooh watch out, science is in town

>> No.8782110

>>8782105
*some

>> No.8782111

>>8782105
ty, your nobel prize will come in the mail

>> No.8782112
File: 78 KB, 725x353, Screen Shot 2017-03-27 at 3.38.07 AM.png [View same] [iqdb] [saucenao] [google]
8782112

>>8782098

>> No.8782124

>>8782111
It is useless to try and converse with you like adults

>> No.8782131

>>8782075
>But it becomes simple if you say that the chance of the RNG producing .5 is an undefined infinitesimal.

I understand your problem. Mathematicians ran into the same. That's way they came up with measure theory. And as said before, this was not done "easy peasy lets do it this way". It was a delicate process to define all this things and concept in a sound and consistent way.

>>8782075
>None of these "proofs" prove anything because the Real set is defined to exclude the very subject that is being discussed.

No, they ("infinitesimals") are just handled in a well defined and consistent way (although not the only possible well defined and consistent way). And although it seems confusing, this way is the "intuitive" way if you think about it carefully. Imagine you are god and you actually can draw a "Point", a "Line" etc. in the mathematical sense. Draw a Line with length 1cm and divide it into 3 parts of equal length. Now measure the length of such a part. You will get 0.33 cm and something. Then as a god, you zoom in and see, it is 0.333 cm and something and you repeat this process and (because you are a god you can "count to infinity") you get 0.33333....cm. So divide 1 by 3 (i.e. 1/3) is equal to 0.333... .There can no (in this daily-life-math-model) thing be in between 1/3 and 0.333..., because the third part of you 1cm line has a beginning point and an ending point, hence it's length can be described with one single value.

>> No.8782160

>>8782131
>Imagine you are god and you actually can draw a "Point", a "Line" etc. in the mathematical sense. Draw a Line with length 1cm and divide it into 3 parts of equal length.

Should I draw the line [0,1] or [0,1)? I think we are going around in circles now.

>> No.8782174

>>8781633
>0.00000000000000...(infinite number of zeroes)...1

So what about
0.00000000000000...(infinite number of zeroes)...01
That's less right? But you could just call that
0.00000000000000...(infinite number of zeroes plus another zero)...1
But infinity plus one is still infinity, so really nothing has changed, so is it equal or lesser?

>> No.8782193

>>8782160
Good answer :) . But for our purpose, it doesn't matter. For now and in this example, don't think in intervals, just think of a 1 cm long "Line".

Lets try it another way: You insist that 0.9999.... + "infinitely small number" = 1. Lets call this "infinitely small number" [math] \mu [/math]. You can do this and you end up with non-standard analysis and a set which is not [math] \mathbb{R} [/math], but something like the Hyperreals or the Surreals (to be honsest, I'm not an expert in non-standard analysis). But now, you run in a bunch of other problems, for example:
Is [math] \mu = -\mu [/math]?
Is [math] f(x) = x+\mu [/math] continuous at 0?
What is [math] \sum_{i=0}^{\infty}{\mu}[/math]?
etc.
As said, it can be done, but it is delicate and non-trivial and much more non-intuitive as it seems on first sight and definitely less intuitive then the "equivalence class based on euclidean metric" - concept.

>> No.8782358

>>8774587
>9x=9
wrong right there
9x = 8.999...

>> No.8782392

This is grade 9 bullshit

>> No.8783156

>>8782174
>But infinity plus one is still infinity
[math]/omega + 1 \neq \omega[/math], you simpleton. Obviously 0.0...1 with [math]\omega[/math] zeroes before the 1 is ten times as large as 0.0...1 with [math]\omega+1[/math] zeroes before the 1.

>> No.8783171
File: 32 KB, 498x341, bioelectricity.jpg [View same] [iqdb] [saucenao] [google]
8783171

>>8773303
>0.999...=1
0.33333... = 1/3
0.3333... * 3 = 0.99999...
1/3 * 3 = 3/3
3/3 = 1
Therefore 0.99999.... = 1

Your story checks out

>> No.8783187

how about this?
infinity = 1

>> No.8783188

>>8783156
>Obviously 0.0...1 with ω zeroes before the 1 is ten times as large as 0.0...1 with ω+1 zeroes before the 1.
Both numbers are infinitely small. There's no such thing as an infinite number that is "ten times bigger" than another

>> No.8783197

>>8774587
>10x=9+x10x=9+x
>9x=9
the fuck are u doing?

>> No.8783203

>>8783171
>0.3333... * 3 = 0.99999...
nice try shekelstein

>> No.8783207

>>8783188
But 0.0...1 is just 0.0...01 shifted to the right by one decimal point. Shifting to the right is exactly the same as division by 10. You can also check this by adding 10 copies of the latter together and seeing that you get the former.

>> No.8783213

>>8783207
0.0...01 is just 0.0...1 shifted to the right, I mean. Whoops.

>> No.8783216

>>8783207
>0.0...1 is just 0.0...01
0.0...1 is = 0.0...01

You can't add a 0 because, by definition, infinity is a number without limit. Adding an extra 0 implies that the infinity reached its limit, which makes it not infinity, but some arbitrarily large rational number.

>> No.8783219

>>8783203
It's true, unless you're going to argue that 3*3 isn't 9

>> No.8783236

>>8783216
>You can't add a 0 because, by definition, infinity is a number without limit. Adding an extra 0 implies that the infinity reached its limit, which makes it not infinity, but some arbitrarily large rational number.
Very true.

For exactly the same reason, you can't add a 1 either, and 0.0...1 is not actually a thing.

>> No.8783240

>>8783236
>For exactly the same reason, you can't add a 1 either, and 0.0...1 is not actually a thing.
Yes, 0.000....1 is = 0

>> No.8783253

>>8783216
>>8783236
>order types beyond [math]\omega[/math] don't exist
Seeing how you are clearly still in high school, I will explain things more simply.
Consider the set [math]S = \{1-2^{-n} : n \in \mathbb{N}\} \cup \{1\}[/math]. This set has an infinite increasing subset consisting of [math]\{0, \frac{1}{2}, \frac{3}{4}, \ldots\}[/math], yet 1 is greater (farther to the right) than each element in this infinite sequence. Does that mean that [math]S[/math] is not actually infinite, but rather some "arbitrarily large" set with a 1 on the end? No, it just means that infinite does not imply unbounded. Similarly, there are infinitely many 0s in 0.0...1, and they are all followed by a 1. (In the same way that [math]\omega[/math] follows all the finite ordinals.)

>> No.8783278

>>8783253
>order types beyond [math]\omega[/math] don't exist
They exist, but they don't play a role in representing the real numbers.

You can define an "extended digit sequence" that maps each element of [math]\omega + 1[/math] to a digit, and define 0.0...1 as [math]s(n) = 0, s(\omega) = 1[/math]. Great. Now what real number does that digit sequence represent? If it involves a term like [math]10^{-\omega}[/math], then you might find that your silly tricks are all for naught.

>> No.8783299

>>8783278
>Great. Now what real number does that digit sequence represent?
1-0.999...

>> No.8783335

>>8783299
Cool! Now can you render that as a Cauchy sequence, or a Dedekind cut?

>> No.8783493

>>8778424
>You have simply declared that a string of numbers can be infinite, and then declared that nothing can come after that. Why? Why not?
Look up the definitions of "sequence" and "limit," wiseass

>> No.8783538

[math]
\displaystyle\sum_{i=1}^{\infty}\frac{1}{2^i} = 1
[/math]

Prove me wrong faggots

>> No.8783723

>>8773303
equals means they are the same
not 99.999...% the same

>> No.8783728

>>8783723
That's ok, because 100% the same and 99.999...% the same are the same amount of sameness.

>> No.8783853

>>8783723
"Equals" means belonging to the same equivalence class w.r.t. some equivalence relation.

>> No.8783884

>>8773303
If there's an infinity small gap between two things, is there even a gap?

>> No.8783901

>>8773552
>>8773617
>>8773619


It is literally equal to one. No mathematical distinction, just two ways of representing the same value. The point you aren't understanding is the "..." it represents and INFINITE convergence, i.e. in the time-independent mathematical space you are working in, that convergence elapses instantaneously, equaling the value to 1.

>> No.8783938

>>8783723
This, seriously

>> No.8784178

>>8783723
Good thing that they're 100% the same, then

>> No.8784461

>>8783171
>>8783171
>>8783171
Jesus, I just came back to this thread and nobody even noticed my solution.

I already proved this shit here.

>> No.8784516

[math]1-.999... = 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000... 1[/math]

>> No.8784521

>>8773303
haha u fkn normies
there's only "no mathematical distinction")
(>>8783901) after a shitton of nontrivial formalization which defines the meaning of infinite decimals desu

>> No.8784551

I argued this with my 7th grade algebra teacher. She said I was wrong but I was all like, "Nuh uh, my Dad told me so." Then when I saw him I was like, "Dad my teacher said you are wrong." Then he wrote out the proof (which is irrefutable in the framework of real analysis) and he wrote QED at the end. Then next week I brought it into class and I was like, "Check my dubs." I really liked her handwriting. I never used to cross sevens before her class but I liked those crossed sevens that she would write on the board I started crossing 7 and Z after that.

>> No.8784658

>>8783335
[math] 0.999... = \sum_{n=1}^\infty \frac{9}{10^n} [/math]

>> No.8784663

>>8783335
>>8784658
This series obviously gives you a Cauchy sequence converging to 0.999...

>> No.8784700

>>8784658
[math]
\displaystyle
0.\bar{9} = \sum_{n=1}^\infty \frac{9}{10^n}
[/math]

>> No.8784710
File: 106 KB, 953x613, 0.999 = 1.jpg [View same] [iqdb] [saucenao] [google]
8784710

>> No.8784716

>>8784710
>you might think infinitely small numbers (0.000...01) exist. They don't

but they do

>> No.8784726

>>8784716

They do not.
Just think about it - if the 0s go on forever, there's no room for the trailing 1. It cannot exist, there is no room left.

>> No.8784730

>>8784726
https://en.wikipedia.org/wiki/Infinitesimal#Number_systems_that_include_infinitesimals
yeah okay buddy educate yourself before you make statements like that

>> No.8784740

>>8784730

You can cite wikipedia all you want, but infitesimals, like infinity, are not actual numbers. They are a concepts. That is why we use limits to describe them.

Also, I happen to be a graduate in math major, so kindly fuck off with your rhetoric.

>> No.8784748

>>8784740
>You can cite wikipedia all you want, but infitesimals, like infinity, are not actual numbers.

That depends on which field you are working in. Since you neglected to specify one, any and all are valid.

>> No.8784757

>>8784748

The crux of this thread argues that [math]1 \ne 0.99\bar{9}[/math]

Which can be debunked with a simple proof:
[math]x=0.99\bar{9}[/math]
[math]10x=9.99\bar{9}[/math]
[math]10x-x=9.99\bar{9}-0.99\bar{9}[/math]
[math]9x=9[/math]
[math]x=1[/math]
[math]\therefore 0.99\bar{9}=1[/math]

>> No.8784763

>>8784757
Incorrect. Your statements have no value. Refer to the post that you quoted.
>That depends on which field you are working in.

>> No.8784767

>>8784757
Or, even more simply:

[math]\frac{1}{3}=0.33\bar{3}[/math]
[math]1=\frac{1}{3} \times 3 = 0.33\bar{3} \times 3 = 0.99\bar{9}[/math]

>> No.8784781

>>8784763
>incorrect

The burden of proof is now on you to show, precisely, which step is wrong. Go on, try. Make sure to apply for a noble prize for your work. I'm sure the committee would have a grand time giving you the laughs and cheers you deserve for your insightful and revolutionary findings.

>> No.8784787

>>8784781
Like I said, you never defined the fucking field you're working in. If you consider the surreal numbers, 1=/.999.. is a PERFECTLY VALID statement.

>> No.8784802

>>8784787
Why does it matter to you what field of math I am working on? Mathematics can be branched out into a large and diverse set of schools. In any case, that statement still holds true, even in the field of surreal numbers. If not, prove which of the steps is faulty. If it's wrong you should easily point out which one and precisely cite in which context from a credible source.

>> No.8784826

>>8784767
[math] \displaystyle
1 = \frac {3}{3} = 3 \cdot \frac {1}{3} = 3 \cdot 0.333... = 0.999...
[/math]

>> No.8784829

>>8784826
prove 1/3 = 0.3333...

>> No.8784832

>>8784829
https://www.mathsisfun.com/long_division.html

>> No.8784840

>>8784832

It's funny. When you do the long division, no matter how many 3s you write down, there'll always be that remaining 1 you have to carry over. You'll never be able to write in proper, decimal form the 1/3.
But when you multiply it out and get that 0.999..., that 'infinitely small 0.000...1' 'magically' fills in that gap.
The 0.00...1 is just an artifact of the decimal system. It doesn't have any real value

>> No.8784979

Jesus, unbelievable. Use a fucking formalism to discuss math.
Something like that:
>>8784826
>>8784757
etc. (99% of this thread)
is not a formal proof. It is like "0.9...=1 because a cat". This is:
>>8779355
The only not shit-posters since then are these two
>>8783253
>>8783278
Again: [math]R[/math] is usually build on [math]Q[/math] with results of dedekind cuts or equivalence classes of limits of cauchy sequences. Argue in that formalism.

>> No.8785398

>>8784726
"There's no room left to write it" is not a valid argument. These are abstract constructions, like all other numbers. And believe it or not, order types greater than [math]\omega[/math] still exist, as mentioned many other times in this thread.

>>8784740
>infitesimals, like infinity, are not actual numbers. They are a concepts.
What the fuck do you think numbers are if not "concepts"?

>> No.8785589
File: 112 KB, 395x578, 1475883703383.jpg [View same] [iqdb] [saucenao] [google]
8785589

Math is fucking retarded.

>> No.8785613

>>8784829
prove it isn't

>> No.8785630

>>8784829
https://www.wolframalpha.com/input/?i=.333...&lk=3

>> No.8785657

You know the "proof" with
x = 0.99...
10x = 9.99...
10x - x = 9
x =1
doesn't hold. Because 9x isn't equal to 9, if you multiply x directly instead of (10-1).

Here is one way how:
9 x 0.99...
9 ( 9/10 + 9/100 + 9/1000 + ...)
81/10 + 81/100 + 81/1000 + ...
Rearranging this sum as a decimal addition:
8.1
0.81
0.081
0.0081
...
Even an infinity of steps later "the last" decimal of this sum can be nothing but 1. So the sum is equal to 8.99...991

We can prove this another way:
n being the number of 9s after the decimal
n=1 9 * 0.9 = 8.1
n=2 9 * 0.99 = 8.91
n=3 9 * 0.999 = 8.991
n=4 9 * 0.9999 = 8.9991
...

The result of the multiplication is always 8.(n 9s)1


Anyone can tell me what am I doing wrong?

>> No.8785663

>>8785657
oops
>8.(n-1 9s)1*

>> No.8785857

>>8785657
>what am I doing
rubbing 1 out?

>> No.8785868

>>8785657
>Because 9x isn't equal to 9, if you multiply x directly instead of (10-1).
This proves 9x = 9 because 10x-x = 9, you brainlet.

>> No.8785872

>>8785657
>Anyone can tell me what am I doing wrong?
Yeah, there is no "last digit" of an infinite sequence.

>> No.8786162

>>8785872
I'm not saying there is, see the infinite numebr fo 9s in between the 8 and 1. I just know the identity of it
9 times 0.999... will be in the form 8.(9)1

>> No.8786301

>>8785872
What is the last element of [math]\{1-2^{-n} : n \in \mahtbb{N}\} \cup \{1\}[/math]? Your answer: "there is none". Actual answer: 1.

>> No.8787079

>>8785398

If a series of never-ending 0s terminate with a 1, then the 0s aren't actually never-ending. Because it terminates. With a 1.

>What the fuck do you think numbers are if not "concepts"?

Numbers are mathematical objects used to count, measure and label. Infinity is an abstract concept to describe something without a bound.

>> No.8787237

>>8787079
>If a series of never-ending 0s terminate with a 1, then the 0s aren't actually never-ending. Because it terminates. With a 1.
Once again, please learn what ordinals are before commenting on things you don't understand. Just because [math]\omega+1[/math] is a successor ordinal does not mean it is finite.

> Infinity is an abstract concept to describe something without a bound.
The countable ordinals are infinite but bounded above by [math]\omega[/math]. The extended real number line is infinite, but bounded on both ends. Infinite does not imply unbounded.

>> No.8787247

>>8775311
embarrassing bait
but nice dubs

>> No.8787250

>>8778029
you mean base 10
what the fucqk is a "2"

>> No.8787255

>>8783197
he subtracted the [math]\mathcal{x}[/math] from both sides

>> No.8787259

>>8780247
The set of Real Numbers, denoted [math]\mathbb{R}[/math], is unambiguously defined you fucking pseudointellectual faggot

>> No.8787265

>>8787259
Sure.... Don't look at the job listing on indeed... no positions are being filled.... nobody is hiring... yes..... perfect excuse to live off your parents trust fund forever like a literal baby, a trust fund that probably doesn't exist

>> No.8787270

>>8778210
Go ahead and introduce your axiom. Once you realize that your axiom is false (as in its negation can be proven) in the theory you added your axiom to, you'll realize (or you would if you had more than two braincells) that you can prove any sentence expressible in your theory within your theory, and it is therefore absolutely fucking worthless and no one is going to give a modicum of a fuck about it.

>> No.8787278
File: 91 KB, 550x385, 1490300965703.png [View same] [iqdb] [saucenao] [google]
8787278

>>8778735
I wish pseudointellectuals like you that comment on math as if they had a fucking clue when their understanding is that of a glorified middle schooler would be round up and publicly executed.

If there are any genies out there pls read this post thx

>> No.8787312

>>8787250
>what is binary
??????

>> No.8787327

>>8784710
The last part of the image is the most useful, I always found that displaying it as an infinite series was the best way to convince people that it is exactly equal to one.

>> No.8787720

>>8774587
Lol lets just make 10x equal to 9.999... for no reason lmao XDd

>> No.8787775

I'm going to type out as many digits of that number as fit into a 4chan post, and you will tell me whether you actually believe that this is equal to 1. Because it's not, it's obviously never ending and therefore NEVER REACHES 1.

0.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999