/sci/ - Science & Math

>>11484068
.000...01 is the same as:
[math] \lim{n \to \infty} \frac{1}{10^n} [/math]
which clearly equals 0

>>11484243
I see. So the one just poof disappears like magic. All we have to do is make the denominator infinity. In fact, it follows from this that any number n over infinity equals 0. It then follows from this that any number n doesn't exist if there is infinity of real numbers. There is an infinity of real numbers. Modus ponens, any number n doesn't exist.
>clearly
meaning
>arguably
idiot

>>11484339
>I see.
Observe. All shitposters type like this.

>>11484351
That is actually my first time ever, and I've been shitposting a long time.

>>11484068
>Why isn't 0.00...001 possible
it is.
>what are the hyperreals

>>11484068
Because you would have to put infinite zeros before the one and so you would be unable to put the one. Also your notation is wrong since it implies that what you mean has an end. It doesn't have an end because there are infinite zeros.

>>11484411
It's like you didn't even read the post.
>It doesn't have an end because there are infinite zeros.
This thinking is so stupid. Can you seriously not conceive of there being infinite zeros before the 1? Why does your logic not apply to 0.999...=1?

>>11484420
I was pointing out that your notation is wrong because 0.0...01 implies there are finite zeros inbetween since there is an end. In 0.999... there is nothing implying an end.

>>11484420
because the 999... means there isn't an end it's just 9's forever and ever and ever and ever and ever...
the .000...001 makes no fucking sense and isn't comparable.

>>11484411
What makes you think there are infinite zeroes before the 1 and not just that the whole decimal expansion is infinite including the 1 at the end.

For the magnificent boundlessness that infinity is, you sure are selling it short with your weird arbitrary restriction of what it is or isn't allowed to contain.

>>11484471
It's that there is an end. When you write 1+2+...+100 you don't mean there are infinite numbers in the middle. It means that the pattern repeats until it reaches the end. On the other hand if you write 1+2+3+... you mean that the pattern repeats forever. If you want to talk about that which has infinitely many zeros before the one you have to find a different way to write it.

>>11484068
Because it isn’t. That’s literally the answer. It’s axiomatically true. You can’t have infinitesimals given the usual construction of the reals. All it takes is a slight definition change, though, and you get infinitesimals. You lose decimal representation for a lot of numbers though.

>>11484486
I'm just fuckin with ya you stupid nigger, infinity isn't even a valid math concept. Quit doing pretend-math in your head

>>11484498
>infinity isn't a valid math concept.
Are you still fucking with me? Also
>I was just trolling bro I'm not actually retarded.

>>11484357
imagine being a bad shitposter
how low is that

>>11484420
>an infinite, that stops
hurr durr

>>11484339
>So the one just poof disappears like magic.
Yes it does, you retard. That's how limits work. Have you taken Calc 1?

>>11484068
The numbers after the decimal point are indexed by natural numbers 1,2,3,... etc. For example for the number 0.5372 the list goes

1 5
2 3
3 7
4 2
5 0
6 0
...

with zero for every natural number larger than 4. So if you take 0.00..01 you get

1 0
2 0
3 0
..
? 1

What do you write instead of the "?"?

>>11485498
[math]\sum_{n=1}^{p} [\math]

1 0
2 0
3 0
..
[math]p [/math] 1

[math] p = \infty [\math]
what, you're allowed to use infinity in lieu of a real number for sums but you're not allowed to use infinity in lieu of a real number elsewhere?
higher math is such a disastrous inconsistent mistake.

>>11485498
[math] \sum_{n=1}^{p} [/math]
1 0
2 0
3 0
..
[math]p 1[/math]

[math]p = \infty [/math]

what, you're allowed to use infinity in lieu of a real number for sums but you're not allowed to use infinity in lieu of a real number elsewhere?
higher math is such a disastrous inconsistent mistake.

>>11484761
>Have you taken Calc 1?
Not yet fuck face. I'll have you know I'm taking Calc 0.999
Next semester I will. biatch.

>>11484461
I meant -0.999

>>11485605
>p=∞
not a natural number

>>11485635
then [math]\sum_{n=1}^{P} [/math] when p=∞ is an invalid concept?

>>11485683
even though the symbol "∞" is used, there's nothing actually "infinite" about the concept. it's just a limit of a certain sequence and the notion of limit doesn't use ""infinity"" at all. it's a common misconception that it does.

>>11485695
The only misconception is thinking ∞ works as a limit. Doesn't make much sense to have an unlimited limit.

>>11485620
So you're like 14? Why are you here? Do you have any interest in mathematics at all or do you just like pretending for some strange reason?

>>11485970
Yes, it actually does. It's you who doesn't understand the concept and you shouldn't speak about it as if you have some revolutionary objections to "higher math" when you haven't even taken the basics of what high school graduates should know.

When you have a limit that approaches infinity and your limit exists, that doesn't really mean that you need to compute your process an infinite number of times to reach your limit (which isn't physically possible). What is really occuring here is as your number of executions of whatever sum or limit calculation you are performing gets large, your window surrounding your limit or convergent sum becomes arbitrarily small in a way that is predictable and stable. For an infinite limit to exist, we actually need by definition the answer to be knowable in a finite amount of time/computations to some arbitrary degree of precision.

>>11485995
>says infinite
>really means finite
>continues to vehemently defend infinity as not being finite
>continues to use infinity explicitly finitely
bunch of underhanded cuntspeak if you ask me

>>11486001
>>continues to use infinity explicitly finitely
nope
lrn2read

>>11486001
No, you're just not understanding. There are plenty of cases where there is an unbounded truly "infinite" limit, but those cases are where the limit evaluation doesn't exist. You cannot for example say that the limit of x^2 as x approaches infinity exists because there will always be something larger and larger as you increase x one more above your previous value. Infinity doesn't "exist" in the way that conventional numbers do. There is no fixed point to which you can point to on a coordinate system and say that this point "is" infinity (except the Riemann Sphere but that's a totally different question). So for your limit evaluation to "equal" infinity as your variable approaches infinity just means that it's value gets arbitrarily large as you increase and will not ever settle on a fixed point.

You can however demonstrate that the limit 2^-x at infinity exists because as you increase x you get an arbitrarily small window surrounding zero.

Also, I'm a different person than the one you've been responding to.

>>11485605
18+ board. come back once you've taken your first calculus class and learned how limits work.

>>11486563
>we will set the limit to unlimited

gonna throw you a bone here. when you use infinity, you're just doing an x amount of finite steps, recognizing some kind of pattern (a function of consciousness, not mathematics), and then extrapolating an assumption of what continued work past x amount of finite steps might be """approaching""".

and in all that brain diarrhea of mind baloney are some hidden facets of unintelligible failed attempts at trickery, such as saying [math]\sum_{n=1}^{\infty} \frac{9}{10^n} = 1 [/math] which is a complete fucking waste of the whole point of the equation being the sole example of an actually algorithmic way of properly producing [math]0.\overline{999}[/math] as it's own, distinct identity.

people like you would argue there is no such [math]0.\overline{000}1 [/math] number, yet at the at same time you would rely on the existence of this number to add with [math]0.\overline{999}[/math] to claim [math]\sum_{n=1}^{\infty} \frac{9}{10^n} = 1 [/math] is a true expression.

>>11487679
>>11484680

>>11487724
>people like you would argue there is no such [math]0.\overline{000}1[/math number, yet at the at same time you would rely on the existence of this number to add with [math]0.\overline{999}[/math] to claim [math]\sum_{n=1}^{\infty} \frac{9}{10^n} = 1[/math] is a true expression.

>>11487724

>people like you would argue there is no such [math]0.\overline{000}1[/math] number, yet at the at same time you would rely on the existence of this number to add with [math]0.\overline{999}[/math] to claim [math]\sum_{n=1}^{\infty} \frac{9}{10^n} = 1[/math] is a true expression.

>>11487679
That's not how the argument for 0.999... = 1 works. There's no adding of 0.00....1 required. It's just a matter of there being no possible number between 0.999.... and 1 thus they must be equal. In fact the argument for 0.999... = 1 needs there to not be a 0.000...1 in existence for it to work. If there was such a number then 0.999... would not equal 1 because you could add some finitely small number to 0.999... before it would equal 1.

>>11484068
0.999 is considered 1 since its repeating and you cant know where the end is since as you stated on paper it has a infinite number of possible end points but with a repeating number its just the maximum since the math that produced it has a repeating remainder

so maybe its 1 to the power of 50 times 1000 decimal points later or maybe its 10 to the power of 95 times 4000 decimal points later . nobody knows where the proper end of a repeating remainder is. whats the last possible number? essentially the part that is missing to make it 1 is so low its considered effectively 1. a single trillionth of a part missing is inconsequential

>>11487791
[math]\sum_{n=1}^{\infty} \frac{9}{10^n} = 0.\overline{999}[/math] is the real truthful expression based simply on the math presented.

claiming it's equal to 1 rather than 0.999... necessarily requires the addition of an element equivalent to 0.000...1

>>11487761
it works precisely because inf never ends

>>11487791
>It's just a matter of there being no possible number between 0.999.... and 1 thus they must be equal.
But numbers are sequenced..??? What the fuck are you saying
If we have set {A,B,C} and we choose B and C. Are they equal because no item is between them?

>>11488191
The reals are uncountable and dense, they are not sequenced in the conventional sense. There are as many sequence entries between 0 and 1 as there are between 0 and infinity so sequencing the reals doesn't actually work. You can set a sampling period and sequence that way, but this will always be an approximation with an arbitrary and specified resolution.

>>11487813
Why do you suppose that 0.999... requires the addition of 0.000...1 and when would this addition occur? Where would this one even be? The whole point of the "..." is to denote that you can make it arbitrarily small and this will be true.

Think of it this way, if you want to get 1 and you're at 0.99 you add 0.01. The same can be extended for 0.999 and 0.001 and 0.9999 and 0.0001.

As you add an arbitrarily large string of 9's the absolute value of your added error between 0.999... and 1 approaches zero as the contribution of that 1 becomes smaller and smaller and smaller. If you were to endlessly add 9's to your string, your error would endlessly add 0's to the same degree of precision and never reach the point where a "...1" would be included to resolve to 1.000...

This is why all natural numbers (and any real number whose primary decimal expansion terminates with an infinite number of zeros) is said to also have a secondary decimal expansion with an infinite set of 9's as well.

A real trick number is 0.00....010....0
get your head around that poindexters

>>11488234
I don't know what you think you're trying to get at but you seem keen on trying to ignore the math equation [math]\sum_{n=1}^{\infty} \frac{9}{10^n} [/math] strictly equals 0.999... as a unique number identity which explicits exists as nothing smaller or greater or different than [math]0.\overline{999}[/math]

Address the example of this arithmatic or present a different math problem where you feel justified in saying 0.999... = 1, cause your logic certainly doesn't apply to what I'm talking about.

>>11484068
[math]0.000...1 = 0[/math]

>>11488249
0.00....010....0 = 0

>>11488234
>0.999... requires the addition of 0.000...1
1+0=1

>>11484068
Reminder the resolution of reality os the planck length
Fuck mathematics

>>11488234
>>11487813
what do you mean by the symbol [math]\sum_{n=1}^{\infty}\tfrac{9}{10^n}[/math] ? the limit of the partial sums ?

>>11488513
this wasn't for >>11488234, sorry

>>11488513
[math]\sum_{n=1}^{k} [/math]
k is the limit of the sum function. if it's set to infinity, then the only sensible extrapolation is that the partial sums tend towards infinite 9's in [math]\sum_{n=1}^{\infty} \frac{9}{10^n} = 0.\overline{999}[/math]

>n1 = 0.9
>n2 = 0.99
>n3 = 0.999
>n4 = 0.9999
>n5 = 0.99999
>...
>n∞ = 0.999...

Using the algorithm produces nothing but 9's, literally.

If the number were written in a book, the book would have infinite pages, and each page would be a string of 9's after the first page which starts as "0.9"

the very nature of the "infinite sum" exists uniquely and specifically to allow the number "0.999..." to exist as a unique and specific number that isn't 1, provided the assumption of using infinity as a limit is indeed true and correct.

It's a true and real value of "0.999..." that you can't get elsewhere, a way to create and craft that number specifically and intentionally when the existence of that number would only otherwise be in error of other simpler arithmetic.

as a very basic and uninformed notion, 1/3 = 0.333..., and if that decimal result is multiplied by 3 in the case of
>[math]3(\frac{1}{3}) = 3(0.\overline{333}) [/math]
it becomes a 'believable' result that
>[math] \frac{3}{3} = 0.\overline{999} [/math]
which in turn allows for a simple belief that 0.999... = 1

But this isn't the true math. This isn't what's really going on here, and it takes looking deeper into the actual division process to understand that [math]\frac{1}{3} = 0.\overline{3}[/math] is actually a [math]false[/math] statement, while the true statement is [math]\frac{1}{3} > 0.\overline{333}[/math], which itself can seem unintuitive at first without really taking the time to consider what is occurring in division and that the nature of a repeating decimal is merely an error in translation between fractional ([math]\frac{1}{3}[/math]) to decimal ([math]0.\overline{333}[/math]).

>>11488715
I can easily show you that 0.999.. = 1, but I need a confirmation.

Claim 1: Let [math]K \geq 0[/math] be a non-negative real number. Suppose that [math]K \leq x[/math] for any [math]x > 0[/math]. Then [math]K = 0[/math].
Proof: This is absolutely clear, but let's do it. If [math]K \neq 0[/math], then necessarily [math]K > 0[/math], which implies that also [math]\tfrac{K}{2} > 0[/math]. Choosing [math]x = \tfrac{K}{2}[/math] we get [math]K \leq \tfrac{K}{2}[/math], contradiction.

Claim 2: Let [math]x > 0[/math]. Then there exists some [math]n[/math] such that [math]\tfrac{1}{10^n} \leq x[/math].
Proof: Any [math]n \geq \log_{10}\tfrac{1}{x}[/math] will do the job.

Do you agree with this?

[math] \displaystyle
\boxed{0 < p < 1} \\
p^n-1 = (p-1)(p^{n-1}+p^{n-2}+ \dots +p+1) \\
\dfrac{p^n-1}{p-1} = \sum \limits_{j=0}^{n-1}p^j \\
\displaystyle
\lim_{n \to \infty} \dfrac{p^n-1}{p-1} = \lim_{n \to \infty} \sum \limits_{j=0}^{n-1}p^j \\
\displaystyle
\dfrac{0-1}{p-1} = \sum \limits_{j=0}^{\infty}p^j \implies \dfrac{1}{1-p} = \sum \limits_{j=0}^{\infty}p^j
[/math]

>>11488715
so because this error in translation between fractions to decimal exists, it accounts for the situation where [math]3(\frac{1}{3} = 3(0.\overline{333} == \frac{3}{3} = 0.\overline{999} [/math], making this specific "0.999..." value a false and incorrect value that exists merely because of understandable error in the difference between the fraction language and the decimal language. Much like how eskimos might have over 30 different words for "snow" but english doesn't have nearly as many, or how some culturally specific foreign words do not translate into another language that has a different culture lacking the reference element from the origin culture. There simply exists no direct translation between the fraction "1/3" to the decimal "0.333..." with current math, and since there is no direct translation, these two values are inherently not actually equal.

>>11488722
answer >>11488717

>>11488722
>eskimos
bi-polar ones?

>>11488723
schizo math.

>>11488727
Give counter examples then.

>>11488727
>i have no argument

>>11488729
nigga you directly responded to a post that clearly showed an allowance for 0.999... to exist as a seperate and unique identity from 1 using an infinite sum.

If you got a problem with that, then you got a problem with infinite sums (I do to, so lets talk about it)

>>11488735
nigga either agree that both claims are true or provide counter examples. everything else is irrelevant.

>>11488717
> rule: k must be less than or equal to x
> proof: [math] x = \frac{k}{2} [/math]
uhhhhm... bruh...

>>11488741
no you're just retarded. you don't actually know how to do math.
>>11488750

>>11488753
S E E T H I N G

>>11488750
https://www.tfd.com/contradiction

>>11488753
show me how it's done and give me a counter example then

>>11488764
>>11488766
yes you contradicted yourself. what did you prove?

the only thing you proved is that you contradict yourself.

amazing proof. truly astounding. thanks for the magnificent input to the thread.

>>11488770
>what did you prove?
"it's not true that Claim 1 is wrong"

>>11488776
okay, i'll bite. Why are you instantiating your "proof" of K < X supplemented by your desire to somehow proof that saying "K < X" isn't enough of a valid assumption statement.

where are you going with "K < X"
you didn't do anything with that statement. You just set up a scenario where K must be less than or equal to X, immediately followed by assigning X a value less than K, thereby creating a false statement.

why did you make a false statement?
what does your false statement have to do with this thread?

is there something else you wanted to do with K and X that related to the topic of the thread?

>>11488795
back up guys, it's a biting eskimo
lim-lim nanook at large

>>11488800
oh no i'm being accosted by an npc with AI-tier language comprehension. whatever shall i do.

>>11488795
obviously you disagree with the statement. support your opinion with a proof of the converse.
>is there something else you wanted to do with K and X that related to the topic of the thread?
both 1/3 = 0.33... and 1 = 0.99... follow from it.

>>11488805
>snowflake nanook sad

>>11488808
Maybe I can try to use your retard language to explain back to you the significance of the issue.

Let K = Fractional 1/3 ([math[\frac{1}{3}/math]). Let X = Decimal 1/3 ([math]0.\overline{333}[/math])
K < X.

But really i'm cheating you in the explanation cause one, i'm being a dick and using K and X to mean something relative to the problem you're having an issue comprehending, and two, arbitrarily assigning these values to variables has no inherent meaning behind the true motive of the issue.

An equality is only an equality if the steps stop or are deliberately accommodated to be infinite with pretext, as in an infinite sum.
1/2 = 0.5 is a direct equality because there is nothing lost in translating the fraction 1/2 to the decimal 0.5

1/3 is not a direct equality to 0.333--- because at every n'th step of trying to solve 1/3 via long division, 1/3 is still greater than the sum of partial sums.
>1/3 > 0.3 --->
>1/3 > 0.33 --->
>1/3 > 0.333 --->
>1/3 > 0.3333 --->
>etc.
>1/3 > 0.333... --->
because the steps never end, there is no true equality or translation between the sole fraction [math]\frac{1}{3}[/math] and the decimal attempt at rendering that fraction via division.

[math]\sum_{n=1}^{\infty} \frac{3}{10^n} = [/math] produces "0.333---" naturally with a strictly equal solution.

[math] \frac{1}{3} = [/math] produces "0.333---" as a strictly less-than decimal solution, or rather the solution to the infinite sum above is less than [math] \frac{1}{3}[/math] in it's state as a fraction.
This provides that "0.333..." is a unique number which can exist, and that 1/3's true value is greater than this specific unique number, in much a similar way as 1 is greater than 0.999--- when 0.999--- is from an infinite sum.
Questions of "greater by how much?" are irrelevant at face value without first deconstructing whether or not using infinity as a number to describe an amount of steps or an amount of decimal places is even intelligible to begin with.

>>11488879
Let [math]K \geq 0[/math] be a real number. Suppose that [math]K \leq x[/math] for any [math]x > 0[/math]. Then [math]K = 0[/math].

Do you agree or disagree with this statement ? It's a simple question which you have still not answered.

>>11484068
>if it has a limit, it can't be infinite
Its obviously wrong. Within every limit is a hidden infinite. Between 0 and 1, there are infinite amount of digits. 0.1, 0.01, 0.001, etc.

Its not wrong, its called infinite series

>>11488902
please use your statement to convey something meaningful rather than asking me whether or not you wrote a statement.
wheres the math.
get to the fucking point m8.

>>11488925
The statement implies 0.333... = 1/3. But there's no point going into it if you believe the statement is false (in which case I demand a counter example from you). I'll rephrase so you can understand it better:

Zero is the only non-negative number which is smaller than every positive number.

Do you agree or disagree with this statement? It's a simple question which you have still not answered.

>>11488935
zero doesn't exist, wildburger told me so

1 - 0.999... = 0.000...1

>>11488982
1-1=0

>mindblown.jpg

>>11488998
infinity - every number multiplied together = infinity

>>11489038
nah, inf-inf is undefined

>>11488278
That's not how decimal expansions work. There is no guarantee of uniqueness for decimal expansions and in fact all decimal expansions which terminate in either an infinite string of 9's or 0's are by definition non-unique.

Your series equals both 0.999... and 1 (because they are two representations of the same fucking number you retard).

>>11489093
no.

>>11489058
infinity - 1 = undefined
infinity - undefined = 1
1 + undefined = infinity

>>11489101
>>11489058
define "undefined"

>>11489107
either no answer
or several possible answers

>>11489101
inf-1=inf

>>11489107
An object of type undefined

>>11489113
Inf - all numbers multiplied together = inf

>>11489120
all numbers multiplied together = inf
inf-inf is undefined

>>11489125
>all numbers multiplied together = inf

>>11489125
>all numbers multiplied together
zero is there too, so it's 0

"There are infinite numbers" appears to be an affirmative statement.
We could say "the set of all numbers must have infinite size, infinite numbers in this set"

but what about "a set that has infinite 1's"? this is not an affirmative statement. it is an assumption that there are other applications of infinity than describing the amount of all numbers.
furthermore, if you add together the elements of an infinite set of ones, what is the sum?
is the sum infinitely large, but finite?
or is the sum infinity, following 10×1 =10, 1000×1 = 1000, infinity×1 = infinity?

Is the sum infinity? Or is the sum a number that has infinite digits?
Is infinity a number with infinite digits? If so, what does the number look like? 1093 has 4 digits. What are the first 4 digits of infinity?

infinity might not be real.

>>11489168
Infinity is a number with infinite digits and every digit is ∞, so the first four digits of infinity are ∞∞∞∞.

>>11489168
>what about "a set that has infinite 1's"
https://youtu.be/i7c2qz7sO0I?t=93

>>11489096
You can say no all you want but you don't have a compelling reason why you believe what you do.

>1 trillion has twelve 0's after it
>a number with 1 trillion 0's after it is still nowhere close to infinity

>>11489186
You can pretend you're right, but you aren't really right.

>>11489196
You don't even have to go very far, literally look at the Wikipedia for the term decimal representation.

https://en.m.wikipedia.org/wiki/Decimal_representation

There is literally a section about this. They have quite a few citations on the article 0.999... about this concept if you want something more rigorous.

>>11489204
Wikipedia is not a valid source of information.
1.000... = real
1.111... = real
1.222... = real
1.333... = real
1.444... = real
1.555... = real
1.666... = real
1.777... = real
1.888... = real
>>1.999 isn't real
2.000... = real

0.111... + 0.222... = 0.333...
0.888... - 0.777... = 0.111...
0.888... + 0.111... = 0.999...
0.888... + 0.110 = 0.99888...
0.888... + 0.11110 = 0.9999888...
0.888... + 0.111... = 0.999...

0.999... - 0.11 = 0.88999...
1.000... - 0.11 = 0.89000...
0.999... - 0.1111 = 0.8888999...
1.000... - 0.1111 = 0.888999...

there is a single significant digit difference between 0.999... and 1.000...

pushing it out of sight and out of mind does not prevent it from existing.

the mere assumption that infinite values must exist between 0.0 -> 1.0 necissitates that a smallest part S must exist such that S×∞ = 1

And so 1 - S = 0.999...

0 1 2 3 4 5 6 7 8 9

0.9 + 0.1 = 1.0
0.99 + 0.11 = 1.1
0.999 + 0.111 = 1.11
0.9999 + 0.1111 = 1.111
0.999... + 0.111... = 1.111...?
0.(n 9's) + 0.(n 1's) = 1.([n-1] 1's)
0.(∞ 9's) + 0.(∞ 1's) = 1.([∞-1] 1's)

subinfinite 1's

>>11489280
says a random idiot on 4chan

>>11489307
>0.999... + 0.111... = 1.111...?

1 + 1/9 = 10/9

>mindblown.jpg

>>11484339
Do you have schizophrenia? I couldn't follow this.

>>11489381
Wikipedia thinks 5 color images are examples of 4 color theorem. Wikipedia thinks NDgT isn't a television personality or comedian.
Wikipedia was never a viable platform because anyone could edit it, and it instantly became a corrupt platform the moment only select user and admins could.

you are an actually stupid nigger retard. Wikipedia is not a source of information for research, nor does it suffice as a collection of sourced researched because of the retardation of it's editors along with their biases.

put a bullet through your brain.

>>11484068
Pic is topkek

>>11484761
That’s not how limits work. By definition, as n approaches infinity, the function goes to 0, but never gets there.

if 0.999... = 1
and 0.000...1 = 0
then things cannot be infinitely large, or infinitely small.

>>11490792
if things cannot be infinitely large or infinitely small
then 0.999... does not have infinite 9's
and 0.000...1 does not have infinite 0's
meaning "..." merely defines an arbitrary but finite amount of repetition
and so 1 - 0.999... = 0.000...1 is a true statement
no more or less than 1 - 0.999 = 0.001

and the assumption that 0.999... = 1 is proven false.

>if 0.999... = 1
>and 0.000...1 = 0
>then things cannot be infinitely large, or infinitely small.
>if things cannot be infinitely large or infinitely small
>then 0.999... does not have infinite 9's
>and 0.000...1 does not have infinite 0's
>meaning "..." merely defines an arbitrary but finite amount of repetition
and so 1 - 0.999... = 0.000...1 is a true statement
>no more or less than 1 - 0.999 = 0.001

>and the assumption that 0.999... = 1 is proven false.

>if 0.999... = 1
>and 0.000...1 = 0
>then things cannot be infinitely large, or infinitely small.
>if things cannot be infinitely large or infinitely small
>then 0.999... does not have infinite 9's
>and 0.000...1 does not have infinite 0's
>meaning "..." merely defines an arbitrary but finite amount of repetition
>and so 1 - 0.999... = 0.000...1 is a true statement
>no more or less than 1 - 0.999 = 0.001
>and the assumption that 0.999... = 1 is proven false.

>>11490627
said the random 4chan idiot

>>11490792
>things
so like chairs? guess so

>>11490803
>meaning "..." merely defines an arbitrary but finite amount of repetition
wew lad

Honestly its time those chucklefucks pushing the idea of infinity were given a blind date with a brick wall. Its time to fight back and reclaim what our GUTS have told us all these years: Infinity does not exist. Its a mental proposition Just like the existence of flying pink elephants orbiting a star in the Andromeda galaxy is a mental proposition. Our GUTS tell us neither can actually exist. We must listen to our GUTS.

I have listened to my GUTS a lot. We often have little chats. My GUTS have told me there exists a finite number of things in existence. Therefore there exists a really BIG finite number which can not be counted beyond simply because there is nothing left to count. So nothing can be added to it. It comes right at the end of the number line after which there is just a black void. It sits there, being the very last number of them all. Humongously big, to be sure, but most definitely finite. I call this amazing big FINITE number the MOTHER FUCKER OF ALL NUMBERS ( MFOAN )

Trying to add anything to MFOAN gets you a very nasty electric shock applied through the testicles. We are not fucking around with this. We have a battery charger. Dont get us angry.

>>11490851
>chairs
anything you retard, including amounts.
there are no more an infinite amount of chairs as there an infinite amount of 9's in 0.999...

>>11490766
What function are you talking about? This thread is about numbers.

>>11490895
>there is a finite number of numbers

>>11491363
if you counted integers from the moment you were born to the day you died of old age, you would have only counted a finite number of numbers.

and that's more counting than you can really imagine, certainly within the realm of otherwise being "infinite". Cause that's just what infinity really means, innit?
"More than you can imagine"
it's still obviously just a really big but finite number, whatever you're imagining.

>>11490627
crank

>>11491410
don't reply to me you stupid cuck. I bet you surf wikipedia daily like a stupid bitch. God knows how many lies you believe in because you read it on wikipedia.

>>11491013
I'd agree with you, but then I would be wrong.

>>11491396
>retard, the post

>>11491438
You're the retard if you think you can grasp infinity.

>>11491448
>monkey can't reach banana
>monkey sadmadbad, scream nobanana is!
retard

>>11491418
schizo

>>11491448
you're the retard if you think modern math didn't already grasp infinity a century ago LMAO

>>11484068
stop using decimal notation and write down what you actually mean by .00...001

decimals are just representations of cauchy sequences for brainlets like you, they dont actually exist. So write what you mean as a cauchy sequence

>>11490792
>and 0.000...1
This DOES NOT EXIST you can not have an infinite amount of zeros and then a one somehow after it.

>>11490895
The universe is infinite, there are an infinite amount of particles in the universe and an infinite amount of space

>>11491759
>>11484680

>>11484068
x = 0.000...001

=>
10-x = 9.999...
(10-x)/10 = 0.999...

=>
1-x = 0.999...

=>
1-x = (10-x)/10
1-x = 10/10 - x/10
1-x = 1 - x/10
x = x/10
x * 9/10 = 0
x = 0

qed

>>11490895
Throw a ball at a wall. This ball represents your awareness of some number on the [x,y] plane where x is horizontal and y is vertical. Try as hard as you can to throw the ball from 1 meter away where it intercepts the wall at exactly [1m, 1m] in front of you. Realistically, not even the highest trained athlete, the most advanced robot, or anything could do this task exactly. Exact numbers are a scam and are unrealistic. More than likely, you could get close to the mark but you will never get there exactly. You might throw too high and to the left and you might get [0.732983457029384570293841908235701293857... m, 1.2820493857023945702934857029384502934857087 ... m]. Reality is infinitely exact as predetermined by Newton's Laws, unless there is some type of Quantum interactions/entanglement processes going on, where maths just dont exist anyways. Even so, reality does not bow down to our limited understanding of numbers as we see them.

>>11491457
>>11491589
face it, you're just a cuck with useless math knowledge that has no applications
>>11490813
>>11486001

>>11492240
Planck length, retard.

0 × infinity = 1

>>11492665
I guess if define infinity as 1/0, then the zeros weirdly cancel out? Doesn't make too much sense though because then 0/0 would also be 1.

>>11492627
>implying understanding 1st year calculus is useless math knowledge that has no application

>>11492679
Infinity is useless knowledge without an application. There is no implication.

>>11492686
>implying understanding 1st year calculus is useless math knowledge that has no application

>>11488982

0.999...
+ 0.000...1
= 0.999...1

>>11492715
Any realistic implement of calculus uses finite numbers.
You are just retarded and spent too mucb time suckling math professor penis for goodboy points that have no application other than certifying you to become a teacher to cheat other cucks

>>11492723
0.999...0 +
0.000...1 =
0.999...1

yes this is true and there is nothing wrong with this.

>>11492745
the definition of the derivative can't exist without limits involving infinity

>>11492760
>0=infinity

>>11492745
>Any realistic implement of calculus uses finite numbers.
yes, very good, that's kind of the whole point of calculus LMAO. good job on acknowledging its importance
>You are just retarded and spent too mucb time suckling math professor penis for goodboy points that have no application other than certifying you to become a teacher to cheat other cucks
S E E T H I N G

>>11492803
>claim to use infinity
>use it explicitly finitely

>>11492745
No. Implementations of calculus involving finite numbers are their own field entirely (Numerical analysis). Calculus doesn't work without having both infinitesimal and infinite values. Without infinitesimal values you cannot have an instantaneous derivative, only a sloped secant line/plane, and without an infinite number of divisions you cannot achieve an integral but rather a finite summation.

While numerical analysis is a very important part of real implementations of math, the way we assess how valid our numerical models are is comparison against idealized models done using real calculus. It doesn't do you any good to rule out infinites just because your autism can't work with them unless you believe them to be physically possible. They are a very functional tool within mathematics that has allowed for a metric fuck ton of relatively well put together scientific and technological development.

>>11492867
>calculus introduces infinity
>claims infinitesimal exists
>1 - 0.999... = 0.000...1
QED

>>11492867
that's not what I meant. the point of calculus is to solve problems which apparently require performing an infinite number of steps without actually having to perform an infinite number of steps is what I meant.

>>11492939
>infinite is finite

>>11492911
An infinitesimal doesn't prove that 0.000...1 exists, in fact it does the exact opposite. The "existence" of infinitesimals guarantee that you cannot have 0.000...1 because at any finite number of 0's there would always require one or more degrees of precision towards zero before reaching your infinitesimal.

You really need to stop thinking about infinity as a number because it's not and doesn't function like one. It's an analytical process. It is the process of "endless continuity" being applied to a numerical and algebraic system, not some point on a geometric surface you can point to. There will never be a point at which your infinite number of zeros has been exhausted such that 0.000...1 will be distinct from 0.000...0

>>11492939
Well yeah but they require a concept of infinite continuity to function and without this you're talking about discrete systems which require modifications in order to remain stable and consistent. I guess you could think about the purpose of calculus as being an analytical method for solving discrete algorithms which would otherwise take an infinite amount of time but that's sort of looking at it backwards. Calculus and the presupposition of a Cartesian plane with infinite divisibility/continuity predated just about every discrete mechanism for evaluation of those problems with the exception of Archimedes method of exhaustion. Newton wasn't expecting some sort of infinite number of discrete calculable steps in order to perform differentiation but to rather geometrically envision the average slope at a point as if the curve was a fluid body (hence his term for derivation being the "Method of Fluxion"). There was no required question of computability for this and in fact the idea of computability didn't really become a thing until the early 1900's.

>>11492630
planck length is not the "smallest unit" of spacetime and if you ever thought it was, you're a retard

>>11492797
>actually think h = 0
don't they teach you you can't divide by 0 in like first grade?

>>11492986
Thus there will never be a point at which 0.999... = 1.000...
since the 9's are continued endlessly.

>>11493196
Derivatives tend to zero, not infinity.
that was the whole point of
>0=infinity
cause the nigger was saying infinity was needed for derivatives, but derivatives deal with lim x -> 0, not lim x -> infinity.

>>11493046
Prove it.

>>11493231
>tend to
So you're saying they approach 0 and don't involve infinity so what's this finite value again?

>>11493234
spacetime is a continuous Minkowski space which is continuous (non-discrete)
discrete theories of the universe are midwit pseudo science.

>>11493242
lims tend to deal with integers and you're merely free to do arbitrary values between integers if you really want. If you're looking for a value at 0, you're probably going to deal with the math at 1 and -1, or if its not a parabola you're probably going to use a calculator and either have it solve it for you in obviously finite steps, or otherwise use whatever near-zero value your calculator will allow like "0.000000001"

>>11493266
Thats not a proof, that's a claim.
Prove it.

>>11493269
The hoops you're jumping through to avoid the concept of an infinitesimal which is part of the definition of the derivative are hilarious

>>11493316
The hoops you've been trained to jump through to defend your worthless math is the only funny thing here.

>>11493269
is this nigga for real ?

>>11493224
Are you literally retarded?

>>11493474
No, but you seem to be. You don't seem to understand what contuining endlessly means.
>0.9 < 1
>0.99 < 1
>0.999 < 1
>0.9999 < 1
>0.99999 < 1
>0.999999 < 1
>0.9999999 < 1
>this continues endlessly
>0.999... < 1

>as 0.999... approaches 1, the difference becomes smaller and smaller
not true. There are infinite digits, thus infinite degrees of accuracy.
Saying it approaches 1 is like saying 0.9 approaches 1.

If there were say 5 decimals of accuracy, basically finite, you could say its approaching the limit of accuracy.
>1 - 0.90000 = 0.10000
>1 - 0.99000 = 0.01000
>1 - 0.99900 = 0.00100
>1 - 0.99990 = 0.00010
>1 - 0.99999 = 0.00001
so if you took it at n=5, but then made the arbitrary restriction that you want n-1 decimals of accuracy (4), you'd have
>1 - 0.99999 = 0.0000
which mimics what is seen in
>1 - 0.999... = 0.000...
Basically, you need to shorten the degree of accuracy from the answer compared to the degree you used in the equation.

But if you have infinite decimal places, then there are just as likely to be infinite values between 0.999... -> 1 as there are 0 -> 1. That's just how infinity is defined to work. Theres no end to it and you can't really measure it.
It's even pretty easy to prove there are infinite values between 0.999... -> 1
>[math]\sum_{n=1}^{\infty} \frac{k}{(k+1)^n} [/math] provides any variety of "0.999..." values that grow endlessly at different rates; for any k > 0.
and if there are infinite numbers, then there are infinite k's to satisfy the equation, all of which provide answers that would classically equate to the same "0.999..." value, provided the understanding of the classical false assumption that this "0.999..." value "approaches" 1, which it really can't because there is no cardinality or compareability within infinity. There are no such thing as "infinitely large, infinitely small, infinitely far, or infinitely close". all those terms are completely meaningless. All numbers are equidistant from infinity if infinity is treated as a large value that is unimaginably far to the right on a number line. Neither incrementing or decreasing any number will get you closer to infinity than another number.

>>11484068
The first decimal number after 0 doesn't end with a 1

>>11494211
Lets call this concept "the big short", where we start with n decimals of accuracy but only use n-1 decimals of accuracy from the answer.

If n=infinity, the n-1 would inherently need to be sub-infinite, or rather, FINITE.
And we can easily tell this is historically being applied to [math]\sum_{n=1}^{\infty} \frac{k}{(k+1)^n}[/math] because not all k's produce clean strings of 9's. [math]\frac{9}{10^n}[/math] may produce nothing but 9's through partial sums as a clean example, but something more arbitrary like [math]\frac{837}{838^n}[/math] will produce a dirty result in it's partial sum analysis where its a string of 9's followed by a random string of garbage numbers. Thus, the big short is required to n-1 that ever-growing string of garbage numbers after the 9's such that "once infinite 9's are attained, shorten the remainder of the none-9 tailing string".
Without that short, without hiding the true digits behind "...", then there are easily an infinite amount of 0.999... numbers between 0.999... -> 1.

Higher math has a big problem with treating infinity as a finite object.

>>11494237
Whatever 0.999... value arises from [math]\sum_{n=1}^{\infty} \frac{837}{838^n}[/math] is inevitably going to be a number with exponentially more random decimal digits than 9's. It is already a complete mess merely by n=5

>>11484068
I mean isn't this the definition of limit. Like something infinitely getting closer to a value but not equal?
If 0.9999 = 1 then doesn't it break calculus?

>>11493984
>let's pretend finite is infinite

>>11494237
>Higher math has a big problem with treating infinity as a finite object.
projecting, the post

>>11494294
the opposite, calculus demands it

>>11494330
>>11494333
retard

>>11494345
>i have no argument

>>11494365
you have no argument

>>11494386
my argument: >>11494330
lrn2read

>>11484068
You clearly don't even know what the word "limit" means in a mathematical context so why the fuck are you making a thread
Google Dunning Kreuger effect, realize you aren't even qualified to ask questions, and then fuck off forever

>>11493984
Nice reasoning retard
>3 < π
>3.1 < π
>3.14 < π
>3.141 < π
>3.1415 < π
>3.14159 < π
>3.141592 < π
>this continues endlessly
>π < π

>>11494506
kek

>>11494294
only a function or a sequence can be "getting infinitely closer to a value but not equal". a number isn't getting closer to anything, it's not doing anything. it's an unchanging definite value. and 0.999... is not a function nor a sequence, it's a number. so how is this number defined ? it's the value which the sequence

0.9
0.99
0.999
0.9999
0.99999
...

is approaching. which is 1.
yes, it's true that the SEQUENCE 0.9, 0.99, 0.999 etc. is never equal to 1. but the symbol 0.999... is NOT THE SEQUENCE itself, it's the VALUE this sequence is getting infinitely closer to. clearly the sequence is getting closer and closer to 1, therefore 0.999... = 1.

>>11488234
people like this think 5 rounds up

>>11493279
By its own definition it cant be proven. Since any infinite Universe will be uncountable. On the other hand a finite, discrete, Universe can be felt in YOUR GUTS! Our GUTS have never been wrong.

Consider the following matrix. The Universe is either finite or infinite. Things are either discrete or non-discrete. A finite Universe can not be non-discrete. Yet an infinite Universe could be either discrete or non-discrete. As you can see finite beats infinite two to one. A clear win. Therefore finite it is. Just like our GUTS told us.

>>11494237
>Higher math has a big problem with treating infinity as a finite object.
name one instance where treating infinity as a finite object causes big problem

>>11494506
the endless digits of pi are still less than the whole true value of pi, you goddamned dumb nigger, else it wouldn't be irrational.

>>π < π
this is where you fucked up.

>>11494534
>closer
read
>>11494211
>>11494237
>>11494275

>>11494562
ironically it causes real rational problems because the techniques used to defend infinity are translated into other concepts involving number such as money.
it's the big short.
"say infinite", but "finite"
say "n", but mean "n-1"
if we can make up gay rules for infinity, we can convince people of gay rules elsewhere.
what's 1/10th of a cent? all we have is pennies!
>but some foreign money is worth less than ours, requiring more than their lowest denomination to equate our lowest denomination
make enough cracks for (n-1) to slip through, and nasty people get rich off accumulating the smallest bits.

like nasty fucking jews.

>>11494571
>irrational numbers don't have decimal expansion
ok

>>11494588
>>irrational numbers don't have decimal expansion
what the fuck are you even saying you weird nut.

>>11489168
digits are a social construct

[math]\frac{1}{3} > (0.\overline{333} = \sum_{n=1}^{\infty}\frac{3}{10^n}) [/math]
[math]3(\frac{1}{3}) > 3(0.\overline{333} = \sum_{n=1}^{\infty}\frac{3}{10^n}) [/math]
[math]\frac{3}{3} > (0.\overline{999} = \sum_{n=1}^{\infty}\frac{9}{10^n}) [/math]

>>11494588
just quoting >>11494571

>>11494572
Since [math]0.999\dots \geq \sum_{k=1}^N \tfrac{9}{10^k}[/math] and [math]\sum_{k=1}^N \tfrac{9}{10^k} = 1-\tfrac{1}{10^N}[/math], we have
[eqn]1 - 0.999\dots \leq \tfrac{1}{10^N}[/eqn]
and this holds for all [math]N[/math]. So unless you can give me a non-negative non-zero number which is smaller than [math]\tfrac{1}{10^N}[/math] for all [math]N[/math], the left hand side must be zero:
[eqn]1 - 0.999\dots = 0[/eqn]
mindblown.jpg

>>11494588
MATH_PI as a 32-bit float is only 7 decimals.
MATH_PI as a 64-bit float is only 15 decimals.
there aren't too many applications of PI that you'd need even 60 digits.

any practical application of PI will indeed use a minuscule finite amount of the trillions of digits discovered thus far.

>>11494615
>1 - 0.9 = 0 cause the left hand side is zero rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrfffffffffpblt
infinity is a not a number. infinity, such as in [math]\sum_{n=1}^{\infty}\frac{9}{10^n}) [/math], is a direction to continue endlessly.
if the starting point is n=1, a finite value, then continuing endlessly through finite values only ever renders more finite values. There is no "infinite'th" value. There is no leap from "finite" to "infinite" at some arbitrary point.

>>11494621
wait, you're saying that 0.999... is not a number ?

>"there is no largest number"
>YES THERE IS ITS CALLED INFINITY
>ITS DA BIGGEST NUMBER

>>11494628
I'm saying there is lost information cheated out of the comprehension of the number by trying to hide it behind "..."

On the contrary, there are as many individual unique "0.999..." numbers as there are integers, as detested in
>>11494211
>>11494237

>>11494629
what about infinity +1?

>>11494630
so you're saying that 0.999... is actually infinitely many numbers. yeah well sorry to break it to you, but they're all equal to the same number, namely 1. as detested in >>11494615

>>11494636
https://www.wolframalpha.com/input/?i=sum%5B9%2F10%5En%2C+%28n%2C1%2C57%29%5D
what the frickin HECK dudes why does [math]\sum_{n=1}^{57} \frac{9}{10^n} = 1 [/math]?????????????????

>>11494636
naw brah i explained to you why your post was wrong, didn't you read?
N has to be a finite number. at any test.

[math]\frac{1}{10^n}[/math] is a real number because N is a real number. There is no infinity'th N. All tests of N are endlessly finite. There is no point at which N increments from finite to infinite.

>>11485683
As P approaches infinity, friend. As P APPROACHES infinity.

>>11494640
I've never said N is infinity
1 - 0.999... <= 1/10^N for finite N, but for ALL of them
unless you can give me a non-negative non-zero number which is smaller than 1/10^N for all finite N, this implies 1 - 0.999... = 0.

>>11486001
You can't set variables equal to infinity. Your calculus and/or your professors are piss poor

>>11494643
nigga are you forreal.
>1 - 0.9 (n1) = 0.1
>1 - 0.99 (n2) = 0.01
>1 - 0.999 (n3) = 0.001
whethers its n=1, n=2, or n= any arbitrarily large finite number, that same n gets spit right back into the equation to provide for [math]\frac{1}{10^n}[/math]
since there is no n=infinity, there is no [math]\frac{1}{10^{\infty}}[/math]

>>11484339
>Modus ponens


hey friend :)) you should study analysis! or if that's too hard, calculus! they're really cool areas of mathematics that have the answers to your questions!

>>11494651
that's all true. and since

0.9 ⩽ 0.999...
0.99 ⩽ 0.999....
0.999 ⩽ 0.999...
etc.

we also have

1 - 0.999... ⩽ 1/10
1 - 0.999... ⩽ 1/10^2
1 - 0.999... ⩽ 1/10^3
etc.

in other words 1 - 0.999... ⩽ 1/10^N for all arbitrarily large, but finite values of N.

nowhere am I saying or implying that 1/10^inf is used.

so unless you can give me a non-negative non-zero number which would be ⩽1/10^N for all finite N, this implies 1 - 0.999... = 0.

>>11487679
That summation would never produce a .001. Every iteration simply adds another nine to the decimal. You may want to look at the difference between a summation and a limit. The entire concept of a limit is that it gets infinitesimally closer to a point but is never actually required to exist at that point. That summation you have there will always get closer to the number 1 every iteration. It won't eventually stop adding a nine to the end of the decimal. A limit simply describes the value a function approaches as the number of those iterations become arbitrarily large. You are describing two different things. One of them is based on the fundamental theorem of calculus. I'd suggest you Google that.

>>11487813
You clearly don't understand the concept of a "limit." I'm sure you'll do well in humanities, though, so that's good.

>>11489101
I kek'd one internet to you good sir.

>>11494663
>0.9 ⩽ 0.999...
it's fairly unambiguous my dude. there is no allowance for the statement "0.9 is less than or equal to 0.999..."

it simply isn't equal at all.

>he can't math without every single number being finite to rub his tummy to sleep
>Doesn't realize that every decimal is an approximation of a ratio
You certainly have implied a lot of 1/1 friend.

>>11494697
>0.9 > 0.999...
>0.9 = 0.999...
>0.9 < 0.999...
pick one

>>11494659
>>11494663
He should study psychology

>>11494651
1/inf=0

>>11494638
round-off error
because forcing WA to fiddle with real numbers was retarded from you

>>11494697
0.9999... = 1, no? so its bigger than >0.9

>>11494629
>BIGGEST NUMBER
no such thing
inf isn't a number

>>11493279
Relativity has been been proven millions of times

>>11493984
Can you actually prove that last statement? I get that you know how to assert things but can you actually demonstrate that 0.999... < 1 logically. It doesn't actually follow that your infinite sum will necessarily converge to being less than 1 and basic geometric series representations show that it does in fact converge to 1, so you're going to need to actually prove with some rigor that 0.999... < 1 for a non-finite number of 9's (hint: you won't be able to because you're wrong).

I'm with you, OP. Having two clearly different numbers be the same is obviously stupid and false. I see your proof >>11493984 and I agree with it. One caveat though. In that proof, you're using the obviously true theorem:

Let [math] x \in \mathbb{R} [/math] be fixed and [math] \{a_n\} [/math] be a convergent sequence such that [math] a_n \to a [/math] for some [math] a \in \mathbb{R} [/math]. Suppose [math] a_n < x [/math] for all [math] n \in \mathbb{N} [/math], prove that [math] a < x [/math].

I'm sure it's true, of course. Someone of your intelligence wouldn't use a theorem without a proof. I know a proof exists, I'm joy doubting you. However, for some reason I can't find any online so just to help me better understand your newfound truth, please help me out by providing one.

>>11494831
convergence is not equality. convergence is an assumption. sum k/(k+1)n "converges" on 1, but it equation itself does not equal 1.

convergence and divergence are used with implicit or explicit "infinity" involved. any finite limit would produce a real number that has an unequivocal real number value. divergence is when a value continues growing beyond an bound (tending towards infinity), and convergence is when a value continues growing (as with divergence) towards a value.

both are flawed with the assumption of "truly infinite" steps occurring however. divergence never actually reaches infinity and convergence never actually reaches it's limit

it's easy to point out the flaw in the logic right here
>0 -> infinity = divergence
>0 -> 1 = convergence
because there is the assumption that there are infinite numbers between 0 and 1, we come to the conclusion that
>"0 -> 1" and "0 -> infinity" contain the same amount of expansion
meaning
>divergence = convergence
>convergence = divergence
and then the value of either convergence or divergence needing to be terms that are used in any capacity disintegrates.

>>11495572
i'm not OP, there are many people who truly trust their rational thought and believe something's fucky with the way higher-mathfags use "infinity".

>>11484339
Do you literally just have a brain tumor?

0.999... != 1

unless
>0.999... has (n) decimal places
and
>1.000... has (n-1) decimal places, of which the elusive extra 9 being cut off from 0.999... turns into a rounding/carry value

if "..." means "infinite decimal places", then there are always more 9's.
if we take the above (n) and (n-1) assumption, and say n=3, it's easy to round and carry a whole value at any arbitrary n
>[ 0 . 9 9 9 ] = [ 0 . 9 9 | 9 ]
>[ 0 . 9 9 9 ] = [ 0 . 9 9 | (10) ]
>[ 0 . 9 9 9 ] = [ 0 . 9 (10) ]
>[ 0 . 9 9 9 ] = [ 0 .(10) 0 ]
>[ 0 . 9 9 9 ] = [ 1 . 0 0 | ]
this method doesn't make sense when using it with "infinite 9s" however. the action of rounding the last digit necessarily requires a last digit, which you wont be getting in an "infinite repetition".
There's also the issue presented in >>11494275 which allows a variety of "0.999..." values with trailing non-9 digits, where merely using (n-1) to cut the remainder will truly spit out some garbage number that may only be 5% 9s and 95% random digits. The (n-1) assumption here then becomes not necessarily "infinity-1", but rather "sub-infinite" or implicitly finite, with a further implied rule that this cut occurs between the 9's and random digits such that
>0.999...99999999999999999999999999999999975801975792912924484088945724023211623031968...
gets conveniently snipped to
>0.999...999999999999999999999999999999999...
and then
>0.999...

lots of unmentioned steps occur in the logical process of trying to justify the idea that [math]\sum_{n=1}^{\infty}\frac{k}{(k+1)^n)} = 1 [/math] for any k.

>>11496399
Suppose 0.999... != 1
Then, 0.333.... * 3 != 1
and 3/3 != 1
What is 3/3?

>>11496416
you're assuming 1/3 = 0.333...
see
>>11488879

>>11496430
the (n) and (n-1) assumption of >>11496399 can be further applied to most computational math. In the case of using a digital calculator, typing "1 / 3 = " will produce a result like "0.3333333" which leads one to believe that the three's continue endlessly and that this printed value is equal to "1 / 3" because you hit the " = " button.

Most calculators will reserve a final unprinted (sometimes it's printed) digit as a carry value, however.
"1 / 3 = " printing
"0.3333333"
will usually actually mean
"0.3333333]3" in the calculator's internal logic.
allowing you to take this result and "* 3" multiply it by 3 to compute
"0.9999999]9" and the final hidden digit is rounded up and turns into a carry to finally print
"1.0000000"

alternatively, if you intentionally type out
"0.3333333" and multiply it by 3, you'll get
"0.9999999" because you don't have direct access to insert a 3 into that final carry digit, meaning it computes on
"0.3333333]0"
"0.9999999]0" with no rounding.

Some calculators are obvious about this such as "2 / 3 = " printing
"0.6666667"
or maybe the last digit of pi it can print is rounded up from what a continued sequence would look like.

ultimately there is an invisible "number" between 1/3 as a fraction and "0.333..." as a decimal, preventing them from being absolutely equal. This invisible "number" simply doesn't exist in the decimal language, so it becomes a math error when translating from Fractional (1/3) to Decimal (0.333...)

>>11496430
1/3 = 3/10 + 1/30
= 0.3 + 1/30
= 0.33 + 1/300
= 0.333 + 1/3000
:
= 0.3... + 1/inf
= 0.3... + 0
= 0.3...

>>11496498
holy shit i've been avoiding this thread and I finally open it back up again and this is what I see

this is a lot of retardation and lack of knowledge in a single post

lmao

>>11496771
>i don't have an argument

>>11496521
>1/inf
Retard

>>11496521
infinity is not a number. There is no such thing as [math]\frac{1}{\infty}[/math].
If you continue on through your sequence, there is never a point where you increment from a real number to "infinity". There is no bridge between real numbers and infinity.

1/inf doesn't mean anything, it's complete nonsense.

>>11496811
>infinity is not a number.
it's use is less elegant than a number, but if all you need is it being big (protip: it is)
then there is no dispute of 1/inf=0

>>11496811
>There is no such thing as 1/∞.
sure bud
https://www.wolframalpha.com/input/?i=1%2Finf

>>11496811
>infinity is not a number.
its use is less elegant than a number, but if all you're using is it being big (protip: it is)
then there is no dispute of 1/inf=0

>>11496827
https://www.wolframalpha.com/input/?i=inf%2Finf
big think.
>>11496830
>if you increment a real number enough times, you will reach infinity, the biggest number of which no other numbers are allowed to be bigger
big think.

>>11496875
>i have no argument

>>11496875
>if you increment a real number enough times, you will reach infinity
[citation needed]

>>11496875
>inf/inf
so?

>>11496827
1/99 = n
99×n = 1

1/99 = n is asking "what number n, times 99, is equal to 1?"
solveable.

1/inf = n is asking "what number n, times infinity, is equal to 1?"
unsolveable.

IF the answer is 0, this needs to hold
In
1/inf = 0
0×inf = 1

However, 0×inf is NOT 1, so "1/inf = 0" is not a true statement, and thus must be a logically false statement.

>>11496891
1/inf=0
if it was >0, inf would be somehow bounded

>>11496886
Its what >>11496521 said
Their formula was [math]\sum_{n=1}^{\infty} \frac{3}{10^n} = \frac{3}{10} + \frac{3}{100} + \frac{3}{1000} + ... [/math]
Which they erroneously extended to a make-believe scenario where n=infinity.

>>11496891
>this needs to hold
inf isn't a number

>>11496897
Inf is already bounded if you assume there are infinite numbers between 0 -> 1.
In order to reach or surpass 1, would require your assumption be false.

>>11496900
"reaching infinity" means the size of the number, not the amount of digits, retard

reaching 1/3 is not "reaching infinity"
neither is 0+0+0+... "reaching infinity"

read a book

>>11496907
retard, infinity is not between 0 and 1
don't confuse cardinality with value

>>11496914
Then are not infinite values between 0 -> 1, which must mean only finite values exist in that space (you retard)

>>11496909
stfu retard schizo no one can understand your chimp language.

>>11496917
>only finite values exist in that space
yes
there are an infinite amount (aleph-1) of those finite values in that space

>>11496954
So you're flip flopping and now agreeing there must be infinite values between 0 -> 1?
that means you cannot reach 1, since you've placed 1 /after/ an infinite amount of real values starting from 0.

>>11496917
>no one can understand your chimp language
it's pretty basic stuff
https://www.youtube.com/watch?v=i7c2qz7sO0I

>>11496956
>infinite values
infinite amount of finite values

>>11496956
>you cannot reach 1
not if you try to list every finite one
but only a retard like you would try doing it like that

>>11496961
You're the retard trying to treat infinity like a number, not me. I'm just using your own retarded poorly-thought out logic to prove counter-points.

Every justification for infinity can ironically be used to show infinity does not actually exist, almost as if it's a physical constant of the universe that infinity doesn't exist.

>>11496973
>treat infinity like a number
[citation needed]

>>11496973
>does not actually exist
math isn't physics
zero doesn't actually exist, pi doesn't actually exist

Daily reminder that
[eqn]1 - 0.999\dots < \tfrac{1}{10^n}[/eqn]
holds for all finite values [math]n[/math]. So unless one of you schizos can give me a non-negative non-zero number which is smaller than [math]\tfrac{1}{10^n}[/math] for all finite [math]n[/math], this implies
[eqn]1 - 0.999\dots = 0.[/eqn]

>>11496988
you are doing bozo math and making assumptions.
How many 9's are in your "0.999..." expansion?
If you answer infinite, you're retarded.
If you answer it is endless, then WHAT is it endlessy of?
Finite values.

do not merely instantiate a random value of "0.999..." and pretend anyone knows what you're talking about. Where did you get that 0.999... value?
What math did you do to produce the 0.999... value you're referencing?

>>11497096
>How many 9's are in your "0.999..." expansion?
I don't know exactly, but more than any number you can come up with

>>11497096
>Where did you get that 0.999... value?
>What math did you do to produce the 0.999... value you're referencing?
[math]0.999\dots[/math] is a number which is larger than [math]0.\underbrace{99\dots9}_n[/math] for any finite [math]n[/math] and it is the smallest number with this property

>>11497120
get your shit together, latex
[math]0.999\dots[/math] is a number which is larger than [math]0.\underbrace{99\dots 9}_{n} [/math] for any finite n and it is the smallest number with this property

>>11485498
>What do you write instead of the "?"?

Let me turn that example back at you. The other infamous "proof" that this is paired with makes the opposite assumption:

1 .9
2 .99
3 .999
4 .9999
5 .99999
6 .999999
...
? .999...

What do you write instead of the "?"?

In the .000...1 example, you say that expression is meaningless because you can't assign it a natural number. But .999... can't be assigned a natural number in this format either. I see this alot in these discussions. "Infinity only works when we want it to." There is just no rhyme or reason to how infinity is applied.

>>11484497
>You can’t have infinitesimals given the usual construction of the reals.

And you can't have fractions given the usual construction of the naturals. That doesn't prove that 1/2 doesn't exist. It is obnoxious to insist that the discussion take place in a number set that is DESIGNED to omit the correct answer.

>>11497125
Any larger finite n also satisfies the rule of being larger than any finite n.

>>11497168
>1 .9
>2 .99
>3 .999
>4 .9999
>5 .99999
>6 .999999
>...
this is not what the post was saying AT ALL. read it once again carefully.

>>11497177
x is larger than any number
x is a number
x is larger than itself
hurr durr

>>11484339
>I see
no you don't

>>11497179
>this is not what the post was saying AT ALL. read it once again carefully.

It's saying that the "last digit" can't exist. But the same logic PREVENTS .999... from equaling one.

>>11497359
oh sweet summer child,
0.9... reaching 1 is BASED on it never ending, which in turn means a last digit doesn't exist.

>>11497359
>But the same logic PREVENTS .999... from equaling one.
no, it doesn't at all

> Why isn't 0.00...001 possible?
How did you manage to type that string of 10 characters if it isn't possible? Or are you saying that it is the name of something, and if so then which thing?

>>11497394
>no, it doesn't at all

Prove it.

>>11497454
digits after a decimal point in a number are indexed by natural numbers. there's no ∞-th position at which you can place a last digit.

this is the whole content of the post. so answer me this question. does the existence of 0.999... require placing a digit at the ∞-th position of a decimal expansion of some number ? it doesn't. therefore there's no contradiction.

>>11497488
>does the existence of 0.999... require placing a digit at the ∞-th position of a decimal expansion of some number ?

If you want it to equal one it does. Otherwise there are a finite number of 9s and it is unambiguously less then one.

>>11497594
0.999... doesn't have 9 at the ∞-th position.

there's a digit for each natural number, and this digit is always 9. and yet this doesn't mean there's a digit at the ∞-th position.

>>11497237
>larger than any number
doesn't exist.

>>11484339
schizo take meds

>>11497845
it's in the definition of infinity

>>11497636
Any test of n->inf will always be a real number, so there will also always be a real number amount of 9's at any "observation" of it.

The issue can be similarly attributed to quantum mechanics and trying to measure where an electron is. It can be anywhere in the atom, and as soon as a measurement is made, then it suddenly appears in only the single place it was measured.
If the field of where an electron could be is considered an infinite decimal expansion of random probability, then measuring where the electron is at a given moment is equivalent to checking any n'th decimal place of 0.999...

But again, this... isn't really quite proper language because the notation of the ellipses involved in "0.999..." isn't honestly defined.
[math]\sum_{n=1}^{\infty} \frac{k}{(k+1)^n}[/math] produces a single coherent repeating string of decimal 9's only when k=9 or 99 or 999, etc.; otherwise the result is a string of 9's followed by any variable length of constantly changing digits.
I dunno about you but it seems disingenuous to me to consider a number like "0.99999999...9997482974462737462628374646383722262538222639995958272917262543019374937364..." as "0.999..."
and it requires some disingenuous assumption that n->inf /will/ indeed produce n=inf, aiming to provide the idea that "whatever comes after the 9's doesn't matter because there are infinite 9's".

The issue is, there is no reason at all to ever take "infinity" at face value and actually have "infinite" accuracy. The degree of it all is just too incomprehensible to ever utilize. The only thing that rationalizing n=inf and the ignoring of trailing digits encourages is the concept that "it's okay to ignore small decimals," which can be turned into a money theft tactic and was even the plot of Office Space.
USD value may only end at two decimals of accuracy in the common market, but may easily have 16 decimals in relation to trading equivalent dollar values from foreign exchange.
Fractions of a cent add up.

>>11484068
>chinese rogan and jones
kek'd
is that an app that can do this?

>>11498200
Infinity doesn't exist then.
You're telling me that you don't find it ironic that infinity is used to define the largest value of which none can be greater?
The whole point of it all is that there isn't supposed to be a largest number, not that there is one. "Infinity" is ironically a finite labelled end to the expansion of numbers when you treat it as a number.
It's disingenuous.

>>11498261
the symbol ∞ is literally defined to be an object such that x < ∞ where x is any real number. the question "does ∞ exist?" is meaningless. does it exist in the real world? it doesn't, but neither do numbers, perfect lines etc. does it exist in mathematics ? of course it does, I have just defined it. the question "is ∞ a number?" is unsubstantial, because it's literally just semantics.

why do we use "∞" ? because it's convenient. for example when some quantity Q grows beyond any bound, such as 1/x^2 around zero, we will write Q -> ∞. but it always means what I've just said - that it grows larger than any arbitrarily large, but ALWAYS FINITE value.

So, suppose you do want a number system that includes decimal expansions indexed by transfinite ordinals. Can you define addition and multiplication? Do they have identity elements? Which elements have additive or multiplicative inverses? Are addition and multiplication associative? Commutative? Do they distribute over each other? Is there an order relation defined on them? Do bounded sets have least upper bounds?
These are the sorts of questions you should be asking.

For example, what is 10 * 0.00.....1 ?

>>11484339
Based

>>11498254
got b& for being too funny
and "racist" I guess

>>11498279
Infinity is meaningless because any justification for it's presumed existence can be turned right back around to show that it can't exist.

The best possible definition is when trying to decribe how many numbers there are, to which the answer is "infinite".
Infinite, as an adjective.
Not INFINITY, a noun.
"Infinite" in this context is analagous with "unlimited", and these words are used to describe "how many numbers there are".
Nothing about this imposes or assumes "infinity" is a number that must be larger than all other numbers.
And since we know any and all numbers are "real", then any indication of something "contuining infinitely" must come with the restriction that any test or observation of the result must be at a finite point, which eludes the notation of "..." for 'repeating decimals'.

>>11498426
have you actually read the post you're replying to ?

>>11484068
Infinite means without end, the 1 in question is at the END of the string of zeroes, thus it can't exist QED. This is basic upper level math stuff. You not understanding how the math works is not an argument against it.

>>11498491
Of course he can't read, he believes .999...=/=1

>>11498537
>n1: 1.0 - 0.9 = 0.1
>n2: 1.0 - 0.99 = 0.01
>n3: 1.0 - 0.999 = 0.001
>n4: 1.0 - 0.9999 = 0.0001
>n5: 1.0 - 0.99999 = 0.00001
>n6: 1.0 - 0.999999 = 0.000001
>...
do you agree that this pattern continues, and that for any n, the decimal expansion will also have n amount of digits?

is it possible to increment n to infinity?

ATTENTION BRAINLETS

STOP TRYING TO WRAP YOUR HEAD AROUND .9999... OR SHOW IT DOESNT MAKE SENSE OR DOESNT EQUAL 1 WHEN YOU

1) DONT KNOW WHAT THE REAL NUMBERS ACTUALLY ARE
2)DON'T KNOW WHAT A CAUCHY SEQUENCE IS

The real numbers ARE BY DEFINITION the collection of all CAUCHY SEQUENCES of fractions. The problem arrises because you can have 2 different Cauchy sequences that are equal and everybody knows you cant have numbers make sense if two distinct numbers ARE THE SAME. So for real numbers to even begin to make sense we need to make a whole bunch of cauchy sequences EQUAL.

THIS IS WHY THE TWO CAUCHY SEQUENCES 1 AND .999... ARE THE SAME IN THE REAL NUMBERS. IT IS LITERALLY A RULE OF THE REAL NUMBERS THAT THEY MUST BE. A RULE AS BASIC AND FUNDAMENTAL AS 1+1=2.

>>11498624
[math]\frac{999999999999999999999999999999999}
{1000000000000000000000000000000000}[/math]
shut up retard.

>>11498574
>do you agree that this pattern continues, and that for any n, the decimal expansion will also have n amount of digits?
yes
>is it possible to increment n to infinity?
yes, but not in the way that you're thinking

>>11498792
>yes
n + ??? = ∞

>Between every two real numbers exists an infinite amount of real numbers.
>By definition, not one number exists between .999... and 1. The same goes for .000...1 and 0.
>Therefore they are the same numbers.

>>11498983
this implies 0.999... is not a real number.

yet [math]\sum_{n=1}^{\infty} \frac{9}{10^n} [/math] is an equation to cleanly reproduce 0.999...

so if 0.999... can be intentionally constructed as a real number, that also supplies there are infinite real numbers between 0.999... and 1, which can be proven via >>11494211

>>11484339
>All we have to do is make the denominator infinity
you dont make the denominator infinity moron. "Infinity" isnt a number, it's a concept. And the value gathered from the shown limit is of infinitesimal magnitude. that means for ANY application the value is zero because if you terminate and use the value at any 0.000...01 you are magnifying its value by an incalculable order of magnitude.

>>11499025
>its too small so it doesn't matter
At what point is the decimal too small?
60 decimals?

>>11499011
that summation equals 1, retard. you have had 300 replies to figure this out. that summation = 0.999... = 1

>>11499040
7
I'm a carpenter and have lost some fingers along the way

>>11499441
Equality is not convergence.
Convergence is barely a valid concept itself cause it's treated implicitly as the reverse to divergence even though both tend towards a number they can't actually reach.

>>11499441
Its actually amazing you've been here for 300 replies and a variety of well rounded grounded posts that logically demolish the concept of infinity yet you're still holding tightly onto it as if it weren't mumbo jumbo.

>>11484068
[math]\forall \epsilon > 0 \, \exists N \, \forall n > N \, \left|\frac{1}{10^n} - 0\right| < \varepsilon[/math]. Ergo the limit is 0.

>>11499675
>rounded grounded posts
kek no

>>11484339
>I see. So the one just poof disappears like magic
YES

YOU DUMB FUCK

>>11499667
>m-muh feelings

>>11499751
Yes, your feelings seem awfully tender about this subject considering it got nothing to do with feelings.

>>11490813
>if 0.999... = 1
>and 0.000...1 = 0
The first is true. The second is true in a conceptual sense, but technically 0.0...1 isn't defined -- our definition of a "repeating decimal" representation of a real number doesn't include an ending number, because an endless ordered list of decimals, well, has no end. That's kind of the whole point we're making here.
>>then things cannot be infinitely large, or infinitely small.
Unclear what you mean by this, much less how it follows from the first statement. I'd love to hear more clarification on these points. My best guess about what you're saying is that, since 0.0...1 = 0, there's no such thing as an "infinitely small" number. I'd argue, however, that there is a definition of an infinitely small number: it's uniquely zero. The definition I think of when I hear "infinitely small" is "the limit approaches zero", since if it approached any other number I could pick another number that's smaller than it (by, for instance, halvingit) -- not exactly infinitely small then.

>then 0.999... does not have infinite 9's
>and 0.000...1 does not have infinite 0's
Again, "infinity" isn't a number, and the definition of 0.999... isn't actually "infinite 9's after the decimal place". It's "the real number that the sequence 0.9, 0.99, 0.999, ... converges to", which conceptually is the same thing but is more precise to do analysis with. It also makes clearer the issue here: we're not actually interested in some "infinite" number, we're interested in the limit of a sequence...

>>11490813
>>11499827
>meaning "..." merely defines an arbitrary but finite amount of repetition
...which is, exactly as you say, *defined* as (formally) whether for all ε > 0, we can choose an integer N such that for any integer n >= N, the nth term in the sequence (here, 0.999...9 with n '9's) differs by less than ε from the value we're assigning it (that is, 1). This is dealing only with finite numbers, because (at least in the system of real numbers we're working with here) those are the only kind that exist.
>and so 1 - 0.999... = 0.000...1 is a true statement
>no more or less than 1 - 0.999 = 0.001
And so we come to our conclusion. Again, you're close. 0.000...1 exists for a finite number of zeros. But I think you'd agree it's pretty trivial to find, for some ε > 0, some n such that 10^-n < ε; that is, for any ε, eventually the sequence 0.1, 0.01, 0.001, 0.0001... will have a smaller number, and all subsequent numbers in the sequence can be trivially shown to be smaller than this number. And from this, the definition of a limit puts the sequence 0.1, 0.01, 0.001, ... as converging to 0, and 0.9, 0.99, 0.999... as converging to 1.

>and the assumption that 0.999... = 1 is proven false.

A quick note: if 1 - 0.999... = 0.000...1, and if our hypotheses are correct, then it's equivalent to 1 - 1 = 0. Your point here is, if anything, a point of guidance in our favor.

-----------

The above's an analytical, by-the-book answer: this is the definition of a limit, and infinite decimal representations are defined in terms of limits, so that's that.

I would mostly like to ask you: if you disagree with this analysis, then not only where, but also why? What does the concept of 0.000...1 mean, if not zero? If it's a specific number that's not zero, then you could find an even smaller number by halving it; that's not "infinitely" anything to me. Just because the sequence 0.1,0.01,0.001,... is all positive doesn't mean that the value it converges to is positive.

>>11499827
Lot of text but i'll undo it with this:

Whats the real number that 1/3 converges to in decimal?

>>11499809
your opinion has no support from mathematicians, you have no references.
it's all just m-muh feelings, tinfoil hat autistic screeching

>>11499835
0.3...
in base 3 it is 0.1
in base 12 it is 0.4

>>11496310
it's easy to point out the flaw in the logic right here
>0 -> infinity = divergence
>0 -> 1 = convergence
because there is the assumption that there are infinite numbers between 0 and 1, we come to the conclusion that
>"0 -> 1" and "0 -> infinity" contain the same amount of expansion
meaning
>divergence = convergence
>convergence = divergence
and then the value of either convergence or divergence needing to be terms that are used in any capacity disintegrates.

What exactly do you mean by "expansion"? If you mean the cardinality of the set of real numbers between 0 and 1 equals that of the set of all positive real numbers, then yes, that's one of the weird, fucky results you get by playing with the concept of infinity. It's harder to show with the real numbers, but pic related is a neat way of thinking about the countability of rational numbers, aka there are "as many" rational numbers as there are integers.

In any case, though, I think you're misapplying the definition of a limit. 0 -> infinity means that, for a given positive epsilon, we can choose integer N such that n > N implies the nth term in the sequence is greater than epsilon; similarly, 0 -> 1 means that the normed difference between the nth term and 1 is less than epsilon. Unbounded growth and bounded approach are very different, even if we use the same notation.

In any case, your concept of "expansion" is far too nebulous to do any real math with. One might even say its "value" as a term that is "used in any capacity disintegrates".

The thing is, the limit definitions above might be conceptually fucky, but they're *useful*. They're how we prove and how, over the past hundreds of years, we've steadily built upon the concepts of calculus. I don't care how weird they might seem (which is really more a question of familiarity, anyways); utility and elegance are paramount, and in actual practice our current definitions have both, which is exactly why we use them.

>>11499835
It can be proven that the real number produced by the division 1/3 does not have a finite decimal representation. Luckily, that's what we have real numbers *for*: they can be defined as the set of all convergent sequences of rationals, "filling the holes" of mere ratios of integers. So 0.333... is a perfectly reasonable representation of 1/3, if we keep in mind its actual definition.

As another person notes, 1/3 could be defined with a finite sequence in another base, like 0.1 in base 3. Similarly, 1/2 in base 10 is 0.5, but 0.1111... in base 3.

>>11499837
>well trained retards do not agree with me therefore its not true
coomthink

>>11499840
Yea but in base 10, 1/3 converted from fractional to decimal is a repeating number. Or rather, [math]\frac{1}{3} > 0.\overline{333}[/math] because there is no direct equality, in decimal.

I think ya'll need to take a step back and realize that decimal and fractional are two different languages and not merely equations vs answers. They might be two very similar languages, but they're not the same, and fractional has easy ways to express a value ([math]\frac{1}{3}[/math]), while decimal has no way to express that same value and must settle for less on [math]0.\overline{333}[/math].

I'm bringing this up because the identity of that value in decimal leads thlo the common misconception/autocorrection of
>[math]\frac{1}{3} = 0.\overline{333} [/math]
>[math]3(\frac{1}{3}) = 3(0.\overline{333}) [/math]
>[math]\frac{3}{3} = 0.\overline{999} [/math]
>and thus 0.999... = 1
But again, 1/3 isn't the only identity in "0.333..." or, multiplied, "0.999..."

to covey this easier, imagine there was a value we could add onto the decimal 1/3 so that it cleanly and completely equals the fractional with nothing lost in translation, let it be
>[math]\frac{1}{3} = 0.\overline{333}_4[/math]
and assume the arithmatic is understood such that
>[math]\frac{3}{3} = 0.\overline{999}_{4×3>10} =
1.0 [/math]

But then take [math]\sum_{n=1}^{\infty} \frac{3}{10^n}[/math], and see it print out a different identity in solely just [math]0.\overline{333}[/math] with no extra bit required to "equal" like 1/3 needed.
The direction of the sum to reach equality is to solely print 3's, while the direction of the conversion of 1/3 to decimal equality requires an extra bit, comparing
>[math](\frac{1}{3} = 0.\overline{333}_4) > (\sum_{n=1}^{\infty} \frac{3}{10^n} = 0.\overline{333})[/math]

and this refocuses the issue from the actual math involved to the reasoning behind whether or not "infinity" is valid.

>>11499956
>coomthink
nice reference, tinboi

>>11499956
>in base 10
which is the Chosen one, because...?

>>11499956
wall-of-text schizo

>>11499956
An easier way to boil it down is saying, what if relative to decimal expansion, "10" decimals was sub infinite and the 11th was infinite.
Such that
>[math]\frac{1}{3}> "0.3333333333_3"[/math]
but using the same above addition of a magical bug to equate translation
>[math]\frac{1}{3}= "0.3333333333_4"[/math]

this is the proper way to think about the way infinity is actually being implied. It makes an arbitrary assumption that after some finite n, anything after may as well be infinite. Whether that's (arbitrarily large ) -> (infinite) or (10 decimals) -> (11th infinite) doesn't really matter cause they function identically. Both operate under n->(n+1), yet the former makes the insane assumption that there exists an n such that n+1 = infinity, and when you ask what that n is, well, you don't get an answer. You get a middlefinger and a "trust me dude".

And this is further proven the case because [math]\sum_{n=1}^{\infty} \frac{k}{(k+1)^n}[/math] is equal to any variety of differing values than a simple string of logical 9's for [math]k \rightarrow \infty[/math] when k isn't 9, or 99 or 999 or other 9's. Functionally something k=10,502 would produce partial sums with 1000 digits of random numbers for every four 9's, so a classical identity of this expanion from k=10,502 would necessarily need orders of infinity greater "length" to accomodate it's 9's and random digits than k=9.

>>11500008
and now for a sip of tank cleaner

>>11500014
You're the one who already drank too much chlorine if you believe in infinity.

a lot of confusion comes from the fact that people are confusing SEQUENCES with NUMBERS

it's absolutely meaningless to say ""0.999... approaches/converges/gets closer to something"". why is that? because 0.999.... is a definite NUMBER, a single point on the number line. it doesn't approach or do anything, it's sitting still. only a SEQUENCE of numbers, indexed by naturals, can approach something. for example

>n=1: 0.9
>n=2: 0.99
>n=3: 0.999
>n=4: 0.9999
>n=5: 0.99999
...

this is a SEQUENCE of numbers. one number for each natural number n. and 0.999... is defined to be unique NUMBER that this SEQUENCE is approaching. but 0.999... itself doesn't approach anything, because it's a number, not a sequence.

I'll say it once again. 0.999... DOESN'T APPROACH anything because it's a fucking NUMBER. the aforementioned SEQUENCE approaches this NUMBER, that's literally the number's definition.

similarly 0.333... doesn't approach anything because it's a fucking number sitting still on the fucking number line. it IS approached by the sequence

>n=1: 0.3
>n=2: 0.33
>n=3: 0.333
>n=4: 0.3333
...

but 0.333... is NOT THE SEQUENCE ITSELF. it's the NUMBER that this sequence is getting close to.

it can be easily shown that the first sequence approaches the number 1 and the second sequence approaches the number 1/3. therefore 0.999... = 1 and 0.333... = 1/3.

>>11500057
...my stomach hurts, means it's working !

>>11500057
>spends a lot of time explaining 0.999... is a number, not a sequence

>ends it with 0.999... = 1

>>11500155
number = number

try again, retard

>>11500017
your tummy ache is making u cranky

>>11500165
>spends a lot of time explaining b is a constant
>b=a
thyme again, retabr.

>>11500057
So you just make the false assumption that 0.999... instantly has infinite 9's.

k

>>11500170
>So you just make the false assumption that 0.999... instantly has infinite 9's.
quote where did I make that assumption

>>11500168
>implying two constants cannot be equal

>>11500170
LOL yes
that's exactly what "..." means,
infinite from the get go
it's not a cute little tug boat chugging along

>>11500252
>i press da = button on calculator and it instantly say 0.333...!!!

>>11500280
>my ruler proves earth is flat
found the retard engineer