0.9999 isn't 1

No, they are IDENTICAL. They are two different notations for the same number, EXACTLY the same, not "infinitesimally close".

not disagreeing about 1=0.999..., but the proof that goes like
x=0.999...
10x=9.999...
9x=10x-x=...
...
is wrong, right?

a =/= b exists c: a

use /math

thanks

people say it's hand-wavy
but on the other hand never give an example where that system fails, so fuck'em

To properly show that they're equal you need to construct the real numbers. And that's why nobody can do it, because there are no rigorous constructions of the """reals"""".

[math] \displaystyle
1 = \frac {3}{3} = 3 \cdot \frac {1}{3} = 3 \cdot 0. \bar{3} = 0. \bar{9}
[/math]

[math]1-\sum_{i=1}^{\infty}(9\times10^{-i}) = 0.1 - \sum_{i=2}^{\infty}(9\times10^{-i}) = \dots = \frac{1}{10^{n-1}}-\sum_{i=n}^{\infty}(9\times10^{-i}) [/math]
[math]\lim_{n\rightarrow\infty}\left[\frac{1}{10^{n-1}}-\sum_{i=n}^{\infty}(9\times10^{-i})\right] = 0-0 = 0[/math]
[math]\therefore 1-\sum_{i=1}^{\infty}(9\times10^{-i}) = 0[/math]

so 1 = 0.999...

this guy gets it

Specifically /calculus

>How can 0.999999...9999 be equal to one
Your "0.999999...9999" implies the nines end eventually. That's not the case with 0.99999...

>when they're separate numbers.
You might as well ask how "one" "1" and the Roman numeral "I" can be equal when they're all "separate numbers".

...

...

"Delta epsilon argument"

>3/3=0.9¯...

The 9's confirm

I think this is one of the strongest arguments against why we aren't in a simulation, as eventually a simulation would round that up to 1 at some point, yet from what observations can determine this is not the case in many irl instances

i told you not to post an answer you sons of bitches. now this schmuck's gonna post this thread again next week.

There are rigorous proofs delineating the real numbers, but it's about two months deep into a real analysis textbook's education, which requires lim sups and lim infs because you first need to prove rational numbers are dense, which requires a strong understanding of calculus and proof fundamentals to justify and force you to understand these things rigorously, which is a huge pain in the ass in a sub-30 minute video that would educate the plebs on the topic.

How do you know that limits like memory, which force us to do things like floating point rounding, exist for whoever would be simulating us, rather than only being part of the simulation?

Any evidence you gather from within the simulation that "this is how a simulator should work" seems pretty suspect then, no?

wat

If 1 and .999... are identical, and 1 and 1 are identical, why do you need to understand a lot of sophistry to prove the former but not the latter? Probably because the foundations of modern mathematics are as rigorous as the theological foundations assumed by your average Protestant preacher.

Also:
>"""""""""real numbers"""""""""

Formally, 0.999... != 1 and anyone who says that they are equal is being imprecise. Here is a rigorous proof:

Recall that the formal definition of 1 is [math]\{ \emptyset\}[/math] which has cardinality 1.
While 0.99... is a Cauchy sequence, i.e. a function [math]\mathbb{N} \to \mathbb{Q}[/math] mapping the natural number n to [math]\sum_{i=1}^n (9/10)^i [/math].
A function is a set of ordered pairs, with one ordered pair for each element of the domain. So the cardinality of the function is the same as the cardinality of the domain, which in this case is [math]\mathbb{N}[/math] which is countably infinite, so 0.999... is formally countably infinite.

This immediately proves that it is not equal to 1, because 1 is finite. But even if you choose to interpret 1 as the Cauchy sequence 1.00... they are still unequal, because two functions are equal if and only if they have the same domain and assignment rule.
In this case, 1.00... is the constant function that always maps to 1, while 0.99... is non-constant and never maps to 1, so they are still unequal.

You do understand that the pills won't help if you don't take them?

He took the !=pill and it's working

I reject the pills, they contain nothing but lies.

The difference is almost less than infinitesimal, therefore yielding a difference of zero and therefore yielding equality. That's a way to look at it.

If two things are different, then they are not identical.

You missed the part of the reals construction where you define the reals as an *equivalence class* on the set of rational Cauchy sequences.

And you've leapt to the erroneous conclusion that I was defining the """real numbers""", when in fact I had never used that word a single time in my post.

Equality is not equal to equivalence, and just because 0.999... and 1 fall in the same equivalence class (which is in any case an unsatisfying resolution as it amounts to saying "they're equivalent because I arbitrarily define them to be so"), it doesn't make them equal as members of said class.

*takes a dump in your thread*

clean it up, janny

are you also gonna argue that 1 != 2/2 ? because that's the same logic really

Indeed it is.
1 is a natural number while 2/2 is a rational number, so there's a type mismatch right from the beginning.

You can coerce 1 into the rational number 1/1 though (by first injecting [math]\mathbb{N}[/math] into [math]\mathbb{Z}[/math] and then into [math]\mathbb{Q}[/math] via division by 1), and then obtain that 1/1 is equal to 2/2 as rational numbers (a formal comparison would coerce both to sets and conclude that 1/1 is [math]equivalent[/math] to 2/2 as sets, but not strictly equal. Much like the other equality-abusing Veeky Forums meme 1+2+3...=-1/12.)

>1/1 is equivalent to 2/2
wrong, 1/1 equals 2/2. 1/1 is the equivalence class of the pair (1,1). so is 2/2. also stop being an autist.

[math] 0.\bar{9} [/math] isn't 0.999999...9999

How much of my post did you even read? Or are you being retarded on purpose?
I said it's a way of >>>>thinking

I did. I means they are not identical as some mathematicians suppose.

The thing about math is there is provably no rigorous base point, go google Godel's incompleteness theorem.

Math, to get around this, because it's still useful and stopping work at the ends of math because you're not sure about the base is lame, agreed when proving things you can pick axioms, which are "Things we're all going to say are true so we can have a conversation".

This is stuff we think ought to be true like "in 2D, if you pick two points that aren't the same, there's only one line that connects them."

Got proof of that? Nobody does, but seems pretty fucking true after 10 seconds if you draw two points on paper.

All this means is that axiom = """"""true"""""" thing we're going to treat as true.

If you accept the axiom choices someone uses as true, they have rigorously grounded their math within that set.

This relates to why we prove one but not the other.

The second one's a construction, which is basically a definition with rules stapled to it, one of which is reflexivity, meaning "it equals itself, if you care enough to ask". So, we don't prove 1=1 because, based on how we claim equality is supposed to work for reals, because that's like proving that the word "dog" means "dog": the proof is literally "Read the two words, bro", the longhand version being "words are reflexive and it's self evident that if you accept that, both sides of the equal side are the same thing"

None of the above applies to the other because 0.999... = 1 is a quirk of the construction which feels weird, so it's not sophistry to say "if we want to claim 0.999... = 1, we have to do some work to justify that".

Basically, you prove stuff when there's a good chance you're wrong, and you've successfully proved stuff when someone, following your argument, should. assuming they understood your axioms and agree you followed the rules of all other math you used in the proof, have no other possible fair conclusion besides that "your argument must be right".

In that sense, it is no different than making a game, 'Schmess' which is just like Chess, except the King can now move *two* spaces in any direction -- viola, there are new rules and moves in Schmess as compared to Chess. That being said, mathematics isn't supposed to be a mere game, where different axioms get you different results. It's supposed to tell us something absolutely true, not conditionally true -- namely, conditional on what axioms you choose.

If mathematicians really just think they are playing various games with various mathematical languages, then so be it, but I'd rather clue myself into what is fundamentally true about quantities, rather than play word and logic games, or just discover pragmatic applications.

this

Tbh I'd play schmess.

Just remember they are absolutely true things though - just absolutely true contingent on base assumptions.

I probably can't sell you on this if you mean it, and this is personal opinion, but axioms are as close to fundamental truth as things get.

Like, Planar geometry's axioms are, if you read Euclid, are almost literally the following:

>Lines go from one point to another point
>You can extend a line by drawing past the points on either side
>Circles exist and have a center and radius
>All right angles are the same
>If you kept extending two nonparallel lines, eventually they'll touch

So at least to me, on an intuitive level all of those are so in-your-face obvious that accepting these as being absolute truths about geometry feels fine.

Is it as good as fundamental truth? Nah. Is it, with some qualifications of it being very, very mildly limiting, just as good? Yeah.

It depends on which set, in hyperreal no in complex yes.

>all right angles are the same

Lrn2spherical or affine

>0.999999...9999
isn't standard notation,
nor is it what is shown in OPic,
Lrn2consistency

My main gripe with this is that it means an object with mass can reach the speed of light instead of arbitrarily close. Goddamn mathfags learn some physics.

>here are things that only work on a plane
>they don't work on a sphere

You can have right angles with different measures on a sphere you cuck. Perpendicular lines have different angles of intersection at different latitudes.

PROOF

journalofsci.weebly.com/home/simple-elegant-proof-of-0999-1

math is not defined by physics you tryhard

woah

and now

1 = 2

mind = blown

Christ Almighty

An infinite string is numbers is conceptually different than a finite one. It is mistake to equate the two.
0.9... approaches 1. It is definitively not 1, since 1 does not approach 1. It true that any number that isn't zero is too big given, but zero is also wrong, both given that it would no longer be approaching 1. This is like calc101.

We only sometimes equate the two in practice, but not in theory.

he was citing AXIOMS FOR ==> PLANAR

are you that kid who quit math because he couldn't digest the definition of a limit?

1x = 0.99999.....x
10x = 9.99999.....x
10x - 1x = 9.99999.....x - 0.99999.....x
9x = 9
x = 1

prove me wrong or scoot the fuck out of this thread.

you need to prove that the series 0.9999... converges beferoe you can do this

A limit is defined by the number it approaches, but it is not that number. By its very definition there is no point on the graph where the line reaches that value, so it cannot be conceptually interchangable will it.

Real witty retort btw, really got the noggin joggin.

By definition 0.9999.... is the limit of the sequence 0.9, 0,99, 0.999, ..., it's not the sequence it self. In analogy to graphs, 0.999... is the asymptote not the graph.

Proofs that 0.999... = 1 are always based on saying 1/3 = 0.999...

This is why it's easy to "prove" it, because you're using a flawed basis. 1/3 does NOT equal 0.999...

0.999 repeating is just a decimal representation of one third, but due to the fact that the decimal is infinitely repeating is CANNOT actually equal 1/3. It is just a representation.

>How can 0.999999...9999 be equal to one when they're separate numbers.
They're not. They're separate symbols, but they both represent the same number.

>Doesn't matter how many 9s you tack on to the end, you'll never actually reach 1.
That's irrelevant. Look up what the limit of a sequence is.

>1/3 = 0.999...
wew lad

>converges
- always growing, capped by 1
- done

This is a complete nonsense. Limit of a numerical sequence IS a number. It is the one and only NUMBER such that for every [math]\epsilon>0[/math] there exists a [math]\delta>0 [/math] and so on... but it's still a number. Period. And what do you think the symbol 0.9999.... stands for? It's a shorthand for [math]\sum_{n=1}^{\infty}\frac{1}{9^n}[/math], which is nothing else but the LIMIT of the sequence of partial sums, namely [math]\{ 0.9, 0.99, 0.999 \dots \}[/math]. And what does this limit equal to? Oh yes, 1.

[math]\sum_{n=1}^{\infty}\frac{9}{10^n}[/math], my mistake

0.9999 doesn't exist in the real world so it'll never make sense.

>since 1 does not approach 1
it does

What is usually meant by repeating decimals is 'that number which is the limit of adding on decimal places'. The limit of 0.3+0.03+0.003... is undeniably 1/3, so we can use 0.333... to denote a third.

With 0.999... it is the same story. The limit of 0.9+0.09+0.009+... is undeniably 1 and therefore the notation 0.999... denotes the number 1.

Now you may say things like "but that's not what I mean when I write 0.999..." but then you are just inventing another meaning for existing notation. You can define your own set of numbers in which there exists a number smaller than 1 but greater than all the reals that are smaller than 1, but it would certainly not have 0.999... as a real number by definition, and like I said, you'd just be redefining notation. What you mean when you write 0.999... may not equal one, but in the established mathematical language it does mean 1.

>>capped by 1
>you're saying by inspection you can tell that the number never reaches one
>but you want to prove the number DOES reach 1

Your hand waving arguement does not prove the series converges.

>prove me wrong
I'd be delighted to.

>9.99999.....x - 0.99999.....x =9

That would equal 9x, not 9.

Boom. Proven wrong.

You brainlets are always recycling the same proofs. Watch me drop a new proof I just discovered using my superior math brain.

In base 10, if you want a repeating digit (like a repeating x) you would do x/9

1/9 = 0.1111...
2/9 = 0.2222...
3/9 = 0.3333...
4/9 = 0.4444...
5/9 = 0.5555...
6/9 = 0.6666...
7/9 = 0.7777...
8/9 = 0.8888...

So what comes next?

9/9 = 0.9999...

We already know what 9/9 is. It's 1. Therefore:

1 = 0.9999...

>>but you want to prove the number DOES reach 1

nope, try again

Only wrong because he set up the equation wrong

Nobody understand how elegant and astonishing this is.

Your last line is flipped and it shouldn't be

That proof is shit. That implies there is a finite number of 9s. Because
>0.9 + 0.1 = 1
>0.99 + 0.11 = 1.1
>0.99999 + 0.11111 = 1.1111
And it repeats. You'd still be off by one decimal place no matter how small. That's why that proof doesn't work. Because to the nth decimal place is off 0.00...1 number.
>1 - 0.00...1(nth place minus one) = 0.99...9

>people show proof that .9999... = 1 using fractions, induction, etc.
>>not good enough, these explanations are too simple
>people prove .9999... = 1 using epsilon-delta and limits
>>No response
>>umm but they use different numbers so how can they be the same brainlet????

I want off this board.

0.0000...1 is equal to 1 - 0.9999...
0.0000...1 is equal to the chance of randomly picking 0.5 from a uniformly distributed random variable that can take values between 0 and 1 inclusive

if you roll this random variable, you will ALMOST SURELY not pick 0.5

however the chance that you will pick 0.5 is not 0.

therefore 1 - 0.99999... =/= 0
therefore 0.999999 =/= 1

only if you're expressing it as a limit of a series

0.33~ X 3 = 0.99~

0.33~ = 1/3

1/3 X 3 = 1

Therefore 0.99~ = 1

the chance that you will pick 0.5 IS zero.
Events with zero probability can happen.

>0.0000...1 is equal to the chance of randomly picking 0.5 from a uniformly distributed random variable that can take values between 0 and 1 inclusive
What the fuck is this bullshit claim

But look what happens when there's infinite digits.

Oh my god, it equals 1.11111....

It's almost as if infinity is different that finity.

The difference between 1 and 2 is 1
What's the difference between .9999999 repeating and 1?

Infinitesimal , or ?

equivalent vs equal...Heh

>a =/= b exists c: a

That's a qualitative argument for why you think it shouldn't be the same, not an actual mathematical argument. Infinitesimals don't exist in real numbers and probability theory is based on real analysis.

>mfw I type all that shit on a calculator and it's true
Mother of god...

1 isn't 1 either. It's just a visual representation of 1. And so is 0.999...
Both represent 1 equally well. And both act as a placeholder for 1 within mathematical axioms equally well. Therefor 0.999... = 1

Just like car = voiture

They suppose they need some bullshit to keep their job