/sci/ - Science & Math

>>6595834
If you have infinite decimal 3's, then no "1" is needed.

>>6595836

except for when you round all the 9s up to 10s and add the one at the front

>>6595840
The rounding happens at the "end" when the magic of infinity causes a 1 to be carried backwards to the front.

Are you denying that 1/3 + 1/3 + 1/3 = 1?

>>6595840
>>6595849
or you could even say that there's no need for a magical one.

1 can be represented both ways using our system.

>>6595849

>magic

good reason

i am saying that .3333..repeating is not exactly 1/3 and never can be

>>6595851
But it is. Do the long devision yourself, and you'll see you'll be trapped in an infinite circle of 3's.

Alternately, you can solve the summation of the ratio 3/(10^n) from n=1 to infinity. You'll find that it will come out exactly 1/3

>>6595854
>But it is. Do the long devision yourself, and you'll see you'll be trapped in an infinite circle of 3's.

agree, and if you do it 3 times and add up all your work you will get a 9 in every column forever

>>6595861
Therefore. .999 . . . and 1.0 represent the same number.

What's hard about this?

>still answering these threads

>>6595840
>you round
Rounding is an approximation. The answer is still .99999etc
/thread

>>6595861
>this post agrees that 1/3 is .33333 . . .

Then why is it hard to reach the conclusion that since 3/3 = 1, then 3 * .333. . . must also be the same number?

>>6595867
>Rounding is an approximation. The answer is still .99999etc

this

>>6595869
>>this post agrees that 1/3 is .33333 . . .

no it does not

>two different numbers are the same number

there are fractions that cannot be simply expressed in decimal form

22/7 for example

>>6595869

Because you can't convert between fractions and decimals in math proofs.

Its the first thing you learn in math grad school, or really any high quality math research. .333.... is not equivalent to 1/3 in pure math.

>>6595883

hmmmmmmmmm

>>6595834
And that's why DECIMALS DO NOT form the real number system. DECIMALS/~ form the real number system where the equivalence relation ~ is defined as (∀n,m∈Z) n*10^m ~ (.999..)*n*10^m.


>>6595869
>Then why is it hard to reach the conclusion that since 3/3 = 1, then 3 * .333. . . must also be the same number?

Because "1"≢".999...". If you show they "must" be the same number then you showed you system is inconsistent: Decimals ⊦ a≠b & a=b (Decimals/~ ⊦ a≠b & [a]=[b] which isn't a contradiction)

>>6595862
>What's hard about this?

It's illogical to derive an equivalence relation from a contradiction by assuming the contradiction doesn't exist. You must first define it. Anything else is mathematical nonsense.

If you can't understand such a simple concept then you're either too stupid to be on /sci/ or in high school and don't belong on /sci/.

>mfw /sci/ falls for such a shitty troll

How much longer do you think we will continue to have this debate?

>>6596039
Forever

Too many retards believe you can prove equality instead of define it.

>>6595834

Imagine I am able to get a gun with 30 decimals of accuracy. Suddenly I am able to shoot anything in the visible universe to the closest inch.

Add another 30 decimals of accuracy and I can shoot any electron in the visible universe.

So why does 0.99999.... == 1? It's because their is no difference in accuracy. They are the same.

All of you niggas should look up the definition of equality of two real numbers. Hint: the definition has nothing to do with digits and notation system.

>>6596100
The only concrete equality that the real numbers axioms define is that 0≠1, nothing more.

You need a model of the axioms to define other equalities and the decimal string model DEFINES 1=.999..., therefore you don't prove shit.

>>6595883
Allow me to fiercely disagree. n/7, where n is not a multiple of 7, is always expressable as a repeating decimal.

1/7 = 0.142857 142857 142857...
2/7 = 0.285714 285714 285714...
3/7 = 0.428571 428571 428571...
4/7 = 0.571428 571428 571428...
5/7 = 0.714285 714285 714285...
6/7 = 0.857142 857142 857142...

22/7 = 21/7 + 1/7 = 3 + 0.142857...

>>6596143

>implying that does not imply you are going to have to do some rounding and make the decimal and approximation

>>6596232
>rounding and approximations
go back to your rulers and stopwatches and voltmeters, kiddo. This is pure math we're talking about here, not middle-school physics.

>>6596238

i guess in pure math

3
3
+3
------------
10

>>6596249
you're not very good at trolling, are you.

>There are people on /sci/ RIGHT NOW who don't complete the rationals with Cauchy sequences to get the real numbers

>>6596255

oh sorry

.33333333
.33333333
+.33333333
-----------------------
1.0000000

>pure math

>>6596257
kek. do you think these middle-schoolers who can't understand basic math will be able to understand metric completions?

>>6596017
>Because "1"≢".999...". If you show they "must" be the same number
That's exactly what it is, they are different representations of the same number.
Do you also claim that fractions are inconsistent?

>>6596260
still wrong.

you have to put the "..." after the decimals. Otherwise they are not repeating.

0.3 + 0.3 + 0.3 = 0.9
0.33333 + 0.33333 + 0.33333 = 0.99999
0.33333333 + 0.33333333 + 0.33333333 = 0.99999999
0.333... + 0.333... + 0.333... = 1

.99999...=1

>>6596268

1.........=2

>>6596267

nigger for every 3 out to infinity... in the column you get a 9

stop rounding

>>6596273
wrong.

1.999... = 2

>>6596276
>stop rounding
It's not "rounding" if the difference between 1 and 0.999... is 0, fucktard.

<span class="math"> 0.99\ldots = 1[/spoiler]

Therefore <span class="math">1.99\ldots 8 = 2[/spoiler] and <span class="math">1.99\ldots = 2[/spoiler]

Therefore <span class="math"> 0.00\ldots 1 = 0[/spoiler]

But due to the Archimedean property of the reals, this means that <span class="math">x=0[/spoiler] for any <span class="math">x[/spoiler]. Contradiction.

QED

>>6596280
>1.999...8
That isn't even a number. You can't have infinitely many digits, followed by a different digit.

>>6596280
>0.99..=1
>Therefore 1.99...8=2
non sequitur

>>6596283
but other than this, you're correct.

>>6596284
Multiply both sides by <span class="math">2[/spoiler]. Are you saying that multiplication itself is flawed?

>>6596279

pls show me in which column out to infinity of

3
3
3

you get

10

at the bottom

kthxbye

>>6596279

in pure math the dots mean you go up a number or two if you feel like it

7.1111111111=7.2

for instance

>>6596287
0.999... multiplied by two is 1.999...

>>6596291
7.1111111111 ≠ 7.2
and 7.111... ≠ 7.2

but 7.1999... DOES equal 7.2

Are you noticing a pattern yet?

In a decimal system, there are two ways to represent a nonzero number: one with infinitely many trailing 0s, and one with infinitely many 9s.

Perhaps you should learn math before posting in this board.

>>6596294
This is false. What's more, I don't know a single person in math who doesn't even use LaTeX. Apply yourself.

>>6596301
>This is false.
Find me a math professor who says that 0.999... ≠ 1.

>protip: you can't, because math professors aren't fucking idiots, unlike you

>>6596303
Who are you quoting? Please keep this discussion to at least a reasonable level of decency, even if you can't do the same for intelligence.

>>6596300

can you show where in the OP the 1 comes from?

>>6596307
Can you find a professor who says that, or not? You're just avoiding the question at this point.

>>6596308
The 1 doesn't "come from" somewhere. It's already known that 0.999... = 1.

Just express 0.999... as an infinite series. Showing that the sum absolutely converges is almost trivial. After that, you can evaluate the sum using a simple formula for the geometric series.

let x = 0.999...
10x = 9.999...
10x - x = 9.000... = 9
9x = 9
x = 1

Q.E.D

>>6596312
>infinite
>converges

>>6596317
>I don't know what calculus is
back to preschool

>>6596317
....
I have no words for your retardation.

>>6596312
>After that, you can evaluate the sum using a simple formula for the geometric series.
This is what everybody says, but I have yet to see a single person actually provide a proof for said geometric series.

>>6596301
Nope, write 0.999... properly.
<div class="math">\sum_{i=1}^{\infty} \frac{9}{10^i}</div>

Now multiply that by two, then look at the decimal expansion.

>>6596322
>I've never taken 9th-grade math

here you go:
http://en.wikipedia.org/wiki/0.999...#Infinite_series_and_sequences
now fuck off until you get your high-school diploma.

>>6596322
>single person actually provide a proof for said geometric series.
Are you fucking kidding me? That' seem in HS.

Get out underage scum.

>>6596326
>>6596327
So you guys admit you can't prove it? Sure got you kikes with that one.

>>6596329

^ pls ignore the bait

>>6596332
And here we see how the Jews in academia stifle their opposition when it is pointed out that they actually have no idea what they are talking about.

>>6596329
For an infite series, the sum = a/1-r
where r is the ratio.
a = 0.9
r = 0.1
0.9/(1-0.1) = 1
happy now,retard?

>>6596335
I didn't ask you to apply it, I asked you to prove it. And please clean up your posts before pressing submit.

>>6596333
see
>>6596335

I hope you feel suitably idiotic

>>6596340
I can't imagine how you don't. How the hell is that a proof?

>>6596338
Just google it you fucking moron.
https://en.wikipedia.org/wiki/Geometric_series#Proof_of_convergence

>>6596343
So you admit that you cannot prove it, and must appeal to the higher powers to do it for you. This is how the decline of intelligence and good men happens.

>>6596312
>The 1 doesn't "come from" somewhere. It's already known that 0.999... = 1.

>it just is

wow is this really the "proof"?

3
3
3
----
10

BECAUSE I SAY SO PLEB

>>6596338
Okay, so for r<1
sum = a(1-r^n) / (1 - r)
as n tends to infinity
r^n tends to 0
so the sum to infinity = a/(1-r)
No latex because I'm on my phone. Fuck you.

>>6596344
I think you are misunderstanding something, and that is that you are not worth the latex to write said proof.

If you really want to know it, google it.

>>6596346
>Okay, so for r<1
>sum = a(1-r^n) / (1 - r)

This is why I hate people who say that <span class="math">0.99\ldots = 1[/spoiler]. Not a single one of them knows how to write an actually rigorous proof.

>>6595834
Just do it in base three.
0.1+0.1+0.1=0.2+0.1=1
Voila, solved.
(If you don't understand this you should be ignored.)

>>6596350
>he wants people to write a full rigorous proof on 4chan for his pompous ass
Seriously?

>>6596354

......your mom out to infinity on my based cock

>>6596356
Don't say something if you're not willing to prove it.

>>6596365
>prove 1 = 1
there's nothing to prove, it's just an identity.

>>6596370
No, <span class="math">0.99\ldots[/spoiler] is not defined to be equal to <span class="math"> 1 [/spoiler].

>>6596378
>identity
>definition
2 different things, faggot.

Are sin(x) and cos(x) defined by the equation <span class="math">\sin^2 (x) + \cos^2 (x) = 1[/spoiler] ?

>>6596379
If it's not a definition, then there is, in fact, something to prove. Are you sure you're reading what you write?

>>6596381
I meant that the proof is trivial.

Would you ask for "rigorous" proof of <span class="math">\sin^2 (x) + \cos^2 (x) = 1[/spoiler]? It's obvious, and if you're trying to demonstrate it, there's no reason to use "rigor" to prove it.

>>6596384
>I meant that the proof is trivial.
Then why can't you prove it?

>>6596389
It was already posted, and I'll post it again:

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

Now go back to highschool.

>>6596263
>they are different representations of the same number

"Different" means a≠b and "same" means a=b, blatantly this is a contradiction. The <span class="math">numbers ~ \bf{ARE}[/spoiler] the decimal strings/sequences/ω-tuples <span class="math">\underline{UNTIL}[/spoiler] you define an equivalence relation and put them into classes. Then they are different LABELS of the same equivalence CLASS that forms the (real) numbers, which is a perfectly logical and rigorous statement. You can't prove it exist because that's how you <span class="math">feel[/spoiler] what the numbers "mean".

>Do you also claim that fractions are inconsistent?

If you define fractions as a number divided by a nonzero, yes. They do NOT form a field.

If you define them as a number divided by a nonzero modulo the equivalence relation ~ where n/m~a/b iff a*m=n*b then no, they form a perfectly good field. But you must define this FIRST. You can't assume intuitive math is correct and then show that is the case.

All trolling aside, how do people not intuitively grasp this idea?

>>6596396
Obviously by "different" he meant a different string that corresponds to the same real number. Learn 2 reading comprehension.

>The numbers ARE the decimal strings
No.
The real numbers are equivalence classes of Cauchy sequences of the rational numbers.

>fractions do not form a field
Fractions are rational numbers, which absolutely form a field. Are you drunk or something?

>>6596392
So you can't prove it. You've shown again that you have to appeal to other people, who appeal to other people, and so on.

>>6596396
what's the difference between .9 repeating and 1?

>>6596303
No but EVERY mathematics professor that doesn't teach in high school will tell you you can not prove it which everyone here keeps trying to do.

>>6596404
What, you can't open the Wikipedia page and read it on your own? Is it too difficult for you or something?

>>6596408
http://en.wikipedia.org/wiki/0.999......

There's like 5 proofs listed on that page.

Doesn't anyone know how to use google anymore?

>>6596409
What, you can't prove it on your own? Is it too difficult for you or something?

I don't blame, you proving a falsehood is difficult for many people.

>>6596396
But the decimals, and the rationals, are indeed conventionally defined using equivalence classes, so not sure what your problem is.

But these classes aren't an arbitrary choice, they arise naturally (some might say intuitively).

>>6596410
>google
Google is a bad company. If you want to continue enslaving yourself to the Jews, then keep using it, but I have grown free of such needs.

>>6596406
a0 is 0 for 0.9999 and 1 for 1

clearly the sequences are different on the very first term.

>>6596410
None of them are proofs. I tried to change it but they refuse to believe their high school teachers lied to them. Read the section here
https://en.wikipedia.org/wiki/0.999#Infinite_decimal_representation

IT IS BY DEFINITION ONLY

>>6596411
Every proof varies depending on what may be assumed. Tell me what I can assume and I'll tell you whether a proof is feasible in a post.

>>6596419
>IT IS BY DEFINITION ONLY
every proof stems from definitions, we call them axioms

>>6596420
You don't even know how to write a rigorous proof yourself? Write it out as if it was undergraduate homework.

Where are you from that you add numbers left to right? I smell a terrorist

>>6596419
>a0 is 0 for 0.9999 and 1 for 1
what

No you silly man, how much less that 1 is .999...? Can you express this difference?

>>6596425
>You don't even know how to write a rigorous proof yourself?
I've a degree in maths
>Write it out as if it was undergraduate homework.
Which course? Real analysis would expect more than, say, maths methods. If real analysis, we'd have to be careful about what we could assume, which is what I'm asking.

>>6596434
>I've a degree in maths
This doesn't mean anything.

>If real analysis, we'd have to be careful about what we could assume, which is what I'm asking.
Were you one of those guys who constantly bothers the professor / TA about what you can assume? Disgusting. Think for yourself for once in your life.

https://www.youtube.com/watch?v=zSuUpdn4LnU
>mfw /sci/ can't even properly respond to the Truth

>>6596442
>This doesn't mean anything.
It means I have a degree in maths.

>Where you
No, you have that backwards. The professors bothered us about what we were assuming and what we shouldn't be assuming.

But tell me the framework of the proof you are asking for and I'll try my best if it's feasible.

>>6596312
>Just express 0.999... as an infinite series. Showing that the sum absolutely converges is almost trivial

No, it is not in fact trivial and assuming that is is exactly the point where you're making a mistake. To show it converges, you need to know all Cauchy sequences converge which is equivalent to showing the reals have the least upper bound property. To show the reals have lub property, you have to show that the decimals satisfy the real axioms. To show that, you need to DEFINE .999... <span class="math">=_{eq}[/spoiler] 1 as well as all multiples with an equivalence relation.

>>6596350
>Not a single one of them knows how to write an actually rigorous proof

Because you can't. Decimals are a <span class="math">model[/spoiler] of the real number axioms (i.e. strings/sequences modulo the above equivalence relation) and NOT something you can prove with arithmetic or series using axioms of the reals. You also CAN NOT prove it with Cauchy sequences or cuts either as you're assuming the decimal construction is correct to claim isomorphism to other constructions leading to a circular argument.

Too many poster on /sci/ are far too uneducated/underaged to realize that their high school math teachers lied to them with their fake "proofs" of this.

>>6596378
It is:
https://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Stevin.27s_construction
>It has been known since Simon Stevin[3] that real numbers can be represented by decimals. We can take the infinite decimal expansion to be the definition of a real number, defining expansions like 0.9999... and 1.0000... to be equivalent, and define the arithmetical operations formally. This is equivalent to the constructions by Cauchy sequences or Dedekind cuts and incorporates an explicit modulus of convergence. Similarly, another radix can be used

>>6596471
>To show that, you need to DEFINE
No, you just have to define what "..." means in sequences of rationals.

The equality to 1 follows from all the real number work you mentioned.

> A repeating or recurring decimal is a way of representing rational numbers in arithmetic. The decimal representation of a number is said to be repeating if it becomes periodic (repeating its values at regular intervals) and the infinitely-repeated digit is not zero.
sounds trivial to me

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

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

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

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

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

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

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

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

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

>Q.E.Fucking.D.

>>6596471
>This is equivalent to the constructions by Cauchy sequences or Dedekind cuts.
Exactly. We don't need to BOTH define using cauchy sequences AND ALSO define by defining the decimals. That is redundant.

>>6596432
No response? Yeah, because you can't obfuscate this question behind jargon.

>>6596403
>The real numbers are equivalence classes of Cauchy sequences of the rational numbers.

Just one construction of the reals and makes no sense to prove things about another construction until you've shown the other construction is correct (and proved that all constructions are all isomorphic).

>Fractions are rational numbers, which absolutely form a field. Are you drunk or something?

Nope, you don't have multiplicative inverses until you get a equivalence relation. Otherwise (1/2)=(1/2)*(3/4)*(3/4)^-1=(1/2)*(3/4)*(4/3)=(12/24) which is a contradiction because 1≠12 and 2≠24. You're <span class="math">assuming[/spoiler] without stating it which is incorrect.

>>6596414
>so not sure what your problem is

Because they are trying to prove it exist instead of defining it exist for decimals. That's absurd.

>But these classes aren't an arbitrary choice, they arise naturally (some might say intuitively).

But they should be called "justifications" of why they were defined and not "proofs" of the statement.

>>6596481
>assuming these high-school dropouts will understand your proof
i have bad news bro

>>6596491

It's algebra and ONE fucking limit. Are they really that far gone?

>>6596489
He's not saying they are unequal, he's saying they are defined to be equal. Everyone else is saying you can construct the real numbers in other ways, and the equality of 0.999... and 1 then follows from a much more limited definition of what infinite decimals are.

>>6596494

What do you expect? They're amerifats.

>>6596490
>But they should be called "justifications" of why they were defined and not "proofs" of the statement.
You could prove any other choice of equivalence class leads to inconsistencies.

>>6596499
>He's not saying they are unequal
Look at the OP's image.

He is saying just that.

>>6596525
You aren't talking to OP though.

mfw nobody can tell OP where the one comes from

>>6596481
>>The limit depends only on the r^n term. Restricting |r|<1, the limit resolves to
Do you have a single fact to back that up?

Also, your formatting is terrible.

>>6596481
>ar^3+....ar^{n-2}
Stopped reading right there.

>>6596559

You....you're asking me to prove that the limit as n approaches infinity of r^n=0 if the magnitude of r is less than 1? Or are you asking me to prove that for a continuous function (which S_n is in regions away from r=1) that the limit of the function is the function of the limit?

Because those are both so simple statements that if you need help with them, you really need to hit khan academy.

>>6595834
Yes, by summing 0.3333... three times you get 0.999...
Why is fine, because 0.999... is 1.

>>6596421
>every proof stems from definitions, we call them axioms

This stems directly from the definition as the definition is literally "all nonzero decimals that terminate are equivalent to the decimal with the last nonzero term decrement and terminated in '9's".

Also this doesn't stem from the axioms as the decimals/Cauchy sequences/Dedekind cuts are merely models of the real axioms

>https://en.wikipedia.org/wiki/Model_%28logic%29
>https://en.wikipedia.org/wiki/Interpretation_%28logic%29
>For a given theory in model theory, a structure is called a model, if it satisfies the defining axioms of that theory

1≟0.999... is a question of the model of the decimals and NOT the axioms of the reals. Therefore the proof can't be done based on axioms/convergence/series.

>>6596432
>No you silly man, how much less that 1 is .999...? Can you express this difference?

"how much less" is intuitive nonsense. You can't define them equal without a equivalence relation putting them into the same class.

>>6596481
>>6596491

This is literally the same false proof as multiplying by 10, subtract the original and divide by 9. It's not any more rigorous or impressive now that you're using "big" words like "geometric series" and "limit"

>>6596485
Decimals are a construction in and of themselves just like Cauchy sequences or Dedekind cuts. They ARE NOT short hand for another construction and you can't use other constructions to prove facts about them. It's not redundant, it's avoiding circular reasoning.

ITT people still don't get infinities and think that "after" an infinity of 9s there's "something" missing.
Monkey brains.

>>6596570
>khan academy.

>>6596578
>people still don't get infinities and thing that "after" an infinity of 1, 2, 3, ... there's a ω

>>6596577
>This stems directly from the definition as the definition is literally "all nonzero decimals that terminate are equivalent to the decimal with the last nonzero term decrement and terminated in '9's".
Unnecessary. You just define the infinite decimals (all of them, not just 999... ones) as limits of rational series.

For .999... there's no need for real numbers at all as it converges within Q

>>6596578
>ITT people still don't get infinities and think that "after" an infinity of 9s there's "something" missing.
>Monkey brains.

>Calling Newton, Leibniz, the Bernoulli's, and Euler "Monkey brains".

There's nothing wrong with having a nonzero number ε that is smaller than any 10^-n for all n. This is how calculus was originally derived and non-standard analysis is complete rigorous and logical. These are just not "real numbers" in the strict sense of the real axioms.

>>6596584
>limits of rational series

Not rigorous at all as you don't have limits in the rationals for all decimals. You need to construct the reals to define convergence in them.

>>6596582
you're likely being trolled by people who are better at you than math and who want to secure their future careers by deliberately trying to make others dumber.

>>6596595
>Not rigorous at all as you don't have limits in the rationals for all decimals
I don't think you know what rigor is, as I didn't claim they did have limits. That would be claiming Q is complete.
>You need to construct the reals to define convergence in them.
Convergence and non convergence exists in many metric spaces. For example 0.999... converges in Q. Reals are unnecessary.

>>6596471
>To show it converges, you need to know all Cauchy sequences converge which is equivalent to showing the reals have the least upper bound property.
Patently false. The Cauchy sequence in question converges in the rationals. It's just that the metric is not complete.

>To show the reals have lub property, you have to show that the decimals satisfy the real axioms. To show that, you need to DEFINE .999... =eq 1 as well as all multiples with an equivalence relation.
This isn't accurate. You could define the real numbers as the completion of finite decimals (since, i.e. dyadic rationals are dense in R). In the end, you need to define whatever set you're talking about. In the end, your argument boils down to "2+2=4 is only true for real numbers because we've defined the real numbers, 2, +, and 4 in this particular way." It doesn't tell you anything special about .9999... or 1.

>Decimals are a model of the real number axioms (i.e. strings/sequences modulo the above equivalence relation) and NOT something you can prove with arithmetic or series using axioms of the reals.
Maybe you mean to say that decimals aren't fundamental to the real numbers and have to be defined separately.

>https://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Stevin.27s_construction
You're going about this the wrong way. Fix a definition of real numbers first. Define what would satisfy you for something to be "true by definition" and see whether the theorem 0.999...=1.000... fits that criterion or not (hint: using Dedekind cuts or Cauchy sequences, this statement is nontrivial enough that we shouldn't call it true by definition).

>>6596577
>This is literally the same false proof as multiplying by 10, subtract the original and divide by 9. It's not any more rigorous or impressive now that you're using "big" words like "geometric series" and "limit"

Except I showed this without a single unjustified step, starting with a basic definition and using algebraic and calculus formalism. And if you're going to say my proof is false, then you need to point out which step of my proof was fallacious.

1 = 1 - 0.111... + 0.111... = 0.888... + 0.111... = 0.999
Easy.

>>6596750
Forgot the dots, damn

If you take two systems and compare every single aspect of their existence and each aspect comes up the same for both, then they are the same system -- even if they have two different names.

If you compare via subtraction you'll find:
1 - .999 . . . = 0.000 . . .
Which is the same thing as 0. That makes 1 and .999 . . . the same number.

>>6596567

>If I don't actually look at the proof, I can pretend it's not there!

>>6596855
>If you take two systems and compare every single aspect of their existence and each aspect comes up the same for both, then they are the same system
This is philosophy, not math.

>>6596956
>philosophy
>not math

Firstly, when algebraic/pure proofs don't work to convince someone, you need to start appealing to logic and emotion.

Second, while math isn't strictly a branch of philosophy, the two are very strongly linked. Deal with it.

>>6596956
The difference between 1 and 0.999... is infinitely small.

Things that are infinitely small can be ignored without affecting any system that interacts with them.

Therefore, treating 0.999... as 1 and vice versa does not affect any system that doing so interacts with.

>>6596567
>Proof coherently and correctly disproves his position
>doesn't bother reading it for no reason except that he either doesn't understand it, or can't accept that he's wrong, or he's afraid to be wrong.

>>6596559
Everything up to
>Noting that
is absolutely ancient mathematics. Infinite series were being worked with when Issac Newton was alive.

>>6596925
>>6596986
Are you guys just pretending to be retarded? I was insulting his piss-awful LaTeX, it's like he'd never touched it before.

>>6596481
This is fucking beautiful. Bravo, good sir!

>>6597005
Then you should have said that. The way you wrote that made it look like you were ignoring it because lul math iz hard.

>>6596959
Math is a subclass of logic is a subclass of philosophy, sure, but any non-autist (as in, not you) would be able to tell what I meant by philosophy.

>>6596977
>Things that are infinitely small can be ignored without affecting any system that interacts with them.
What are you, an engineer?

>>6597005

Unless the formatting actually presents a significant impediment to reading what I posted in that proof, then you're doing nothing but being a pedantic little bitch.

>>6597006
>Bravo, good sir!
Please keep yourself on reddit.

>>6597008
>>6597012
Just stop, you're making yourselves look worse.

>>6597013

You're the one on the science and math board complaining that you won't even consider a proof because the formatting isn't to your liking, like some sort of anointed god of the board.

Let's see your take on the proof then.

>>6597013
How would you have written the proof? Because that looks pretty fucking standard to me.

>>6597016
Are you just angry because I insulted your post? I never said a single fucking word about the proof.

>Because that looks pretty fucking standard to me.
I really hope you're just joking here.

>>6597023

More at being told I'm retarded.

Again, if the way I formatted it is so bad, let's see how it should look. You've got the floor.

>>6597028
>More at being told I'm retarded.
It's true. I doubt you've even looked at your post again, since it's pretty obvious you fucked up with it. I even pointed out the worst error,
>ar^3+....ar^{n-2}
First, using "." to make an ellipsis is retarded. Second, using four dots in an ellipsis is retarded. Third, trying to use an ellipsis there is retarded. A more professional of presenting it would be
>ar^{3} + \cdots + ar^{n-1}

>>6597028
>>6597019
>>6597016
>>6597012
>Falling for bait
Come on, /sci/, I expected better out of you.

>>6597037
That's a cool meme, but it's rude to interrupt a conversation when you don't have anything to add.

>>6597035

So, let me see here.... your complaint is the difference between <span class="math"> \cdots [/spoiler] and ... ? Well, I apologize, and will amend the code in the future. Since I was never taught LaTeX and have been picking it up as I go, I didn't know that command. I didn't imagine anyone would be so..."detail oriented" I guess I'll say, to notice something like that. But hey. Good on you!

>>6595834
>151 posts and 4 image replies omitted

Good job OP, 10/10.

>>6597040
>your complaint is the difference between ⋯ and ... ?
You didn't use "...", you used "....".

>Since I was never taught LaTeX and have been picking it up as I go, I didn't know that command.
Teaching yourself something isn't an excuse for being bad at it, it's a very basic command, and normal ellipses have never been proper to use in that sort of situation.

>I didn't imagine anyone would be so..."detail oriented" I guess I'll say, to notice something like that. But hey. Good on you!
This part of your post is hilarious, but retarded. What are you even trying to say? That you have no idea how to proofread your work for anything but "1=2" level errors?

>>6595890

Yes, you can. Both the fraction and radix representation of a real number are equally valid and equivalent representations are guaranteed to exist for any rational number. This follows from the fact that any real number may be represented as some infinite sequence of n-ary digits, and that the rational numbers are strictly a dense subset of the reals.

Since two such representations as 1/3 and .333... (in decimal) can be shown to be equivalent, it follows that algebraic manipulations of each much produce the same result. Hence, many of the proofs already posted here are correct, and .9999.... is in fact equal to the integer 1.

>>6595890

Yes, you can. Both the fraction and radix representation of a rational number are equally valid, and equivalent representations are guaranteed to exist for any rational number. This follows from the fact that any real number may be represented as some infinite sequence of n-ary digits, and that the rational numbers are strictly a dense subset of the reals.

Since two such representations as 1/3 and .333... (in decimal) can be shown to be equivalent, it follows that algebraic manipulations of each must produce the same result. Hence, many of the proofs already posted here are correct, and .9999.... is in fact equal to the integer 1.