/sci/ - Science & Math

>infinitesimally small

so, 0.

>>10437705
t. Retarded fucking idiot

>>10437694
My thoughts regarding the picture:
x=0,99999....9999
10x=0,9999....9990
10-x=8,9999....9981
Therefore 9x=/=9

>>10437694
What do you mean by “infinitesimal”?
What do you mean by “distance”?
Why do you say “always” when numbers are eternal and not time-dependent?

You should try working on your formal jargon because I have no idea what you are trying to convey

>>10437729
What about the right side?
9 x 1/9

>>10437705
Wow it's almost like there's a distinction you're purposefully choosing to ignore, a number greater than zero cannot also be zero

>>10437755
infinite isn't a number
infinitely small also isn't a number

>>10437708
>>10437729
So what number is in between .999... and 1?

>>10437761
Again, a value greater than zero cannot equal zero

>>10437767
Again, infinitesimal is not a quantity

>>10437777
I never said it was. That was OP, I'm telling you 0 =/= >0

>>10437764
(0.999.... + 1) / 2

>>10437785
That’s 1, anon

>>10437785
What does the decimal expansion look like?
>>10437784
What does the decimal expansion of 1-0.999... look like?

>>10437797
It's
1 - ((1 - 0.999...) / 2)

>>10437802
>What does the decimal expansion look like?
I typed it out but too long to post here

>>10437807
What does 0 have to do with this now?

>>10437808
At least your response was remotely clever

1=3/3=3*(1/3)=3*0.333...=0.999...

>>10437802
A number greater than zero

>>10437817
Proof?

>>10437818
0.999...<1

>>10437817
What number?

>>10437823
But that's a false statement anon

>>10437823
that is only true if you can show that there is a number between 0.999... and 1.

>>10437815
>1/3 = 0.3333...
>being this stupid

>>10437835
prove it wrong
even better, give the "correct" answer

>>10437842
the correct answer is: 1=1

>>10437694
Hey OP, can you help me solve this?
[math]\sum_{i=1}^{\infty}9(10)^{-i}=?[/math]

Infinitesimals don't exist in the field of real numbers since the real numbers are defined by the completeness axiom. Everything bounded above has a least upper bound. What could the least upper bound be other than 1. Every real number is defined in terms of cauchy sequences, where for all epsilon we can find an element of the sequence such that all successive elements will be within epsilon of its limit. If you were correct the irrationals would also never equal what they are. Stop posting this garbage.

>>10437708
how is that /pol/?

>>10437708
anti-/pol/tards are in a constant state of seething about /pol/

>>10437694
and your penis is infinitesimally small. why is this still up for debate?!?

>>10437694
A distance is something finite and measurable so if you say that there is a distance between 0.9... and 1 you are saying that there is a numer x that satisfies 0.9...+x=1
So please find x :)
And don't tell me x=0.0...01 because as soon as you put a digit 1 after zeroes your are telling that x has finite digits, but x must have infinite digits since 0.9... has infinite digits, and so the only possible answer is x=0.0...=0.
And pls don't even tell me that x=1-0.9... come on don't be a brainlet ;)

>>10437983
yeah, 0.000... is patrician-tier

>>10437877
its funny because literally no one likes /pol/, i guess we're all seething together, its not so bad desu

>>10437785
fucking this

>>10437764
1/10^∞

[math]\frac{1}{\aleph_0}[/math]

>>10437694
>another 1=0.999 thread

>>10437983
You seem smart, can you help me with a really simple problem? Current through a 300mH coil increases at a rate of 50uA per 10us, how much is the induced voltage?

It should just be V=L (di/dt), which I computed as 1.5V but the correct answer is 1.5kV

>>10438048
post in >>10412555 or >>>/wsr/

>>10437764
0.000......01

>>10437729
yeah but its 0.999.....
not 0.999...999
Then nines do not end.

>>10438057
That implies that .999... terminates

>>10438194
No it doesn't. Go back to >>>/x/ fucking retard

>>10438199
Yes it does. Go back to >>>/lgbt/ fucking retard

>>10438044
Based tired old frog poster

>>10437694
9/9 equals 1
9/9 does not equal a "0.999..." that is different to 1
Q.E.D

>>10437694
0.999... x 10
= 9.999...
9.999... - 0.999... = 9
Oopsy

>>10437694
0.999... is and infinity sum that converges toward 1. how stupid can you be

>>10437694
>>10438350
[math]0.\bar9=0.9+0.09+0.009+\cdots=\dfrac{9(0.1)}{1-0.1}=\dfrac{0.9}{0.9}=1[/math]
(Using a geometric sum)

Never post here again

That .1 you think exists is actually accounted for. There is no end to the number but if you pretended there were and you wanted to add that ...1 and equal 1 you wouldn't be able to without going over one because in reality there is no goddamn ...1 because again there is no end to the number.

The number .999 aint shit it's just a way of saying 3/3 which equals one

>>10437877
/pol/ is an example of an idiot

>>10438448
cry some more

>>10438199
Yes it does you fucking brainlet

>>10438780
Why are you even talking still when you already lost? Rope yourself you fucking baffoon.

>>10438804
This is my first reply in this thread you braindead subhuman
.999... /= .000001
.999... Doesn't end

>>10438350
1/9 = 0.111...
+
8/9 = 0.888...
=
9/9 = 0.999...

>>10438861
Thank you for confirming what I just said
Learn to read next time

If 0.999...=1
then 0.999...8=0.999...
and 0.999....7=0.999...8
and so on
therefore 0.000...=1

>>10438882
>...8
stop

>>10438893
Who cares about that ...8? its never coming anyways. Might aswell throw a ....73 in there

>>10438882
>and so on
flesh this out a bit
https://youtu.be/BROS4TUg-WU

>>10438882
>then 0.999...8=0.999...
Isn't correct

>>10438900
You can't terminate a recurring decimal at an arbitrary point and say it's equal to another recurring decimal.

>>10438916
is too

>>10438910
I didnt make it through the whole video, got stuck on 9:59:59...

>>10438927
How about 0.999...=1.000...?

>>10438940
As long as you don't terminate either decimal that equation is correct

>>10438947
and 0.999...888...=1.000...?

>>10438947
oh bullshit, terminate on the right just as you please

>>10437694
Life gets so much easier when you figure out how to differentiate semantic arguments from ones of substance. "numbers" are symbols for abstract concepts that can describe lengths and amounts

Does the symbol 0.999... represent the infinite series 1/(10^k), or does it represent it's sum? If it's the sum, it's 1, but if you wanted to say 1 why not just 1? If it's not the series, then it can't be compared to any real number, but then why is it written sort of like a number? Either way, who cares? Not me.

Most philosophy is rubbish, but study pragmatism.

>>10438952
yes
0.999...888... = 0.9... = 1

>>10438960
>who cares? Not me
>writes a wall of text
kek

>>10438962
0.9...8...7...32...1...76352...42...=1?

>>10437694
Both 'proofs' use infinite series in their premises. That's literally the problem.

>>10438810
0.999.... == 0. ... 01. Nice try you worthless sack of shit.

>>10438962
You're wrong anon
>0.999...888...
Implies the number is decreasing, which doesn't make it .999...

As mathematics student i can tell you 0.9999is a defined number and there is a space between 0.9999 and 1 for example 0.99995 but if there is no space between 2 numbers p.e. 0.9'(' stands for ongoing ) there is no countable number between 0.9' and 1....further the left side of the picture is the right method to get the value for x ...the ' is resembled by the .... (Sry for bad english im german)

>>10438967
yes
0.9...8...7...32...1...76352...42... = 0.9... = 1

>>10438971
anything after '...' is zero

>>10438970
>0.999.... == 0. ... 01
No it doesn't
.999... Goes on for eternity, if you know the end of the sequence then it is just an infinitesimal number, therefore it isn't .999...
You are either baiting or genuinely retarded

>>10438964
It's boutta get even more ironic

>>10438960
I'm gonna go ahead and prove the second option I said was true and 0.999... is not a number to save time because that's where this debate always leads eventually

I hope we can agree that
if 0.999... is a number, 0.999... is irrational

From the definition of equality of real numbers (no number in between)
if 0.999... is a number, 1=0.999...

So if 0.999... is a number, 1 is irrational
1 is not irrational

Therefore 0.999... is not a number

>>10438952
You're looking at it the wrong way
Imagine you've got 2 cars driving next to eachother on a long road, printing 9's onto the path behind them
>0.99999...
>0.99999...
These cars are printing the same number
At some point, the 2nd car decides to stop printing 9's and start printing 8's
>...999999...
>...998888...
The two cars are now printing different numbers, and the gap between those numbers will continue to grow forever.
>>10438955
0.999... = 1 = 1.000... (closest number to the right of one)
Though that might be going a bit to far for the people in this thread

>>10438978
0.9...a cat...mitch hedberg...3...(you)...=1?

>>10438981
No, it represents the number repeating itself

>>10438988
>0.999... is irrational
retards say the darndest things

>>10438988
>.999... is irrational
But can't you represent it as a fraction, 3/3?

>>10438994
0.1 = 10^-1
0.01 = 10^-2
0.001 = 10^-3
:
0.000...1 = 10^-inf = 0

https://de.m.wikipedia.org/wiki/0,999… answer of your question I guess

>>10439001
>0.000...1 = 10^-inf = 0

>>10438997
>>10438999
Ok that's wrong but the original point still stands.

Decimals are an engineering format, fractions are the true representations of numbers. There's no such issue in fraction form and no need for this ... bullshit.

>>10439009

>>10439023

[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]

>>10439046
the fuck is this supposed to prove??

>>10439104
https://en.wikipedia.org/wiki/Geometric_series

>>10439111
I know what a geometric series tell me what it is you're trying to prove buddy.

>>10439122
0.9...=1

>>10439128
How so? Your final equation holds

>>10437694
1 - 0.999 = 0.001
The one who agree with this, please suck my cock

>>10439186
>0.001
>my cock
sounds about right

>>10437767
How much greater than 0 is it?

>>10439164
[math]
p=0.1 \\
\dfrac{1}{1-0.1}=\frac{10}{9} = 1 + \frac{1}{9} \\
\sum_{j=0}^\infty 0.1^j= 1 + \sum_{j=1}^\infty 0.1^j \\
9+1=9+9\sum_{j=1}^\infty 0.1^j \\
1=9\sum_{j=1}^\infty 0.1^j \\
[/math]

>>10439265
>>10439046
Oh so it's literally just the 0.000...1=0 circular reasoning argument but with Latex?

>>10439291
point at the failing line
instead of making hand wavy overemotional gasps

>>10439296
line 4 of your first post.
p^n goes to 0 with infinity.

I know what you're going to say, and the problem is actually on the right hand side of the equation. You can't keep it in series form once you take the limit. You have to take its sum using the identity you used in the line above.

>>10439314
>line 4 of your first post.
>p^n goes to 0 with n->infinity.

0<p<1
so I'm good
if that's your objection, I simply don't care
bye

>>10439326
>I don't care that I'm wrong
good for you
bye bye

>>10439314
that's just a variation of 1/inf
https://www.wolframalpha.com/input/?i=1%2Finf

>>10439353
Other guy is a retard who didn't read the second part of my post. Don't be like other guy. Read the second part of my post.

>>10439364
>can't keep it in series form once you take the limit.
[citation needed]

>>10437694
you cant compare rational and irrational numbers

>>10439388
well fuck, there goes my theory that 3 < pi

>>10437694
OP is stooped.
0.9> = 1.0>
Because the point is infinitly small the point doesnt exist.
like relativity no one fucking notices that small of a disturbance.
BTFO half cup fool nigger.

>>10439404
the pills won't help if you don't take them

>>10439370
Ok forget the second part of what I said actually. it is p^n that's wrong.

0<p<1
c=p^n as n->inf
b=p^-n as n->inf
cb=(p^n)(p^-n) as n-> inf
cb=p^0 as n-> inf
cb=1
if c=0, cb=/=1
therefore c=/=0

inb4 you just post wolfram instead of actually using reason like a human being

>>10439407
tell me how large the distance between 0.9> to 1.0>
write it.
now.
write it.....
OP is STOOPED!

>>10439408
1/inf
>just post wolfram
it's done already
>reason like a human being
I'm sure WA doesn't calculate it, it reads it from a database that a human mathematician created.

>>10439408
that scribble, my eyes
use latex you beast
or at least write it on paper using normal math notation and send the picture

>>10439419
appealing to authority in an argument of logic is nothing short of conceding defeat

>>10439428
no

>>10439430
>>10439431
>I know more math than WA but can't figure out how latex works
k

>>10439435
I've written countless homeworks and lab reports in latex
But why the fuck would I take the time just so a couple NEETs on 4chan can read my post in 5 less seconds?

>>10439439
>why write math when talking math
k

>>10439443
who gives a fuck as long as the information is there

are you gonna make a reply or not?

>>10439444
>as long as the information is there
it isn't, it's just madman scribble

>>10437694
what's an infinitesimally small distance

>>10437729
so if you cancel the nines, x=
nothing? is x an empty number?

this just means that infinitesimals don't exist because space is continuous, it's the same thing as with Zeno's paradoxes

>>10439445
are you retarded? What don't you understand, besides the fact that you're wrong?

>>10439452
>>10439435

>>10439453
Just admit you're wrong, freak.

>>10437694
there are no infinitesimals in R
in the filed of the reals the convergent series 0.99999... is equal to 1.
your argument would be valid in the hyperreals (*R)

>>10439458
sure, me, WA, wikipedia, every math course in every university everywhere

We're all wrong, thank god we are now saved by the 4chan mathjesus, who writes to us in zodiac killer scribble

>>10439479
I'm glad you've finally admitted defeat, oh good champion of math

>>10439482
I see your deduction capabilities are intact.

>>10439222
Infinitesimally.

>>10437831
0.00..1

>>10437834
No, because 0.00..1 is the next number after 0. By your logic 1 = 2 because there is no number between them (in the domain of natural numbers).

>>10437864
>every real number is DEFINED
It's really fucking dumb when people use model processes to try and explain reality.

We didn't DISCOVER the Bayer Process. We INVENTED it. We MADE A PROCESS. It produces a result. This result is a direct consequence of a process, not a fact of the nature of aluminum. For fuck's sake. We didn't "bring out the natural aluminum." We fundamentally modified a reality.

>>10439580
By your logic Z is R

>>10439587
>philosophy

>>10439592
How so?

>>10437708
>living rent free in your head

Will people ever stop falling for this bait?

>>10439622
Not him but you're assuming that
[math]|(0,1)|=\aleph_0[/math] (this is wrong btw)
Which is probably what he's getting at

>>10439909
>let's pretend finite is infinite

>>10438984
Stop trying, just admit you've lost.

>>10439909
>that autistic looking handwriting
You should have been drowned at birth

>>10439944
where did i pretend that finite was infinite?

>>10440063
autistic looking handdrawing for an autist (you)

>>10440025
prove him wrong instead of spouting ad hominems

>>10439909
8.99... = 9 though
Because
[math]8.\bar9=\\8+0.9+0.09+\cdots=\\8+\sum_{i=1}^{\infty}9\cdot0.1^i=\\8+\dfrac{9\cdot0.1}{1-0.1}=\\8+\dfrac{0.9}{0.9}=\\8+1=\\9[/math]

>>10440083
Nayeth! Thyne logic hath no bearing on the purity of my argumentation!
0.000... - 0 > 0 since therewith following mine ellipsis there beeth an 0.000...1!

>>10439206
Then lick my boiclit

>>10440091
0.00...1 isn't a number though
If it were then >>10439632 would be true because you could create a bijection
[math]f:\mathbb{N}\rightarrow[0,1][/math]
By having f(1)=0.00...1, f(2)=0.00...2, etc.
This is wrong though, see https://en.wikipedia.org/wiki/Cardinality_of_the_continuum

>>10440067
by showing finite examples
and alluding that the infinite case is similar

>>10440110
not alluding but reasoning that the infinite case is similar...

or do you see any fallacies?

>>10440083
how does the 3rd row equal the 4th row?

Yes

for All natural numbers n serie [math]\sigma_{i=1}^n 9/10^n [/math]
is less than one. But the limit is one.

>>10440134
It's a geometric sum

>>10440130
infinity changes everything
https://youtu.be/i7c2qz7sO0I?t=40

>>10439458
Congratulations on proving every mathematician ever wrong internet warrior.

>>10437694
>.33333... = 1/3
>3×(1/3) = 1
>3×(.33333...) = .99999... = 3×(1/3) = 1

>>10440179
This is what I was thinking, is the whole .999... = 1 issue just a result of using a base 10 system? I've never thought of how the choice of base affects the "decimal" representation of fractions.

>>10440249
>just a result of using a base 10 system
nah, in base 2
0.111...=1

in base 16
0.fff...=1

>>10437729
so what part about infinity didn't you understand?

>>10438040
this

>>10439580
>next number after 0
wanna know i know you never studied math seriously?

>>10440365
What would be a good explanation of why this is false? Would it be sound to say it is because there exist a number between any two other real numbers?

>>10440080
I already did. Why are redditors always this incompetent?

>>10440409
basically yes. also that decimal expansion doesn't work in a way that you can write 0.000...1, there is no such thing as putting a 1 after "infinite" zeroes.

>>10440435
why the quotes?

>>10440444
because infinity is not a number or a quantity per se

>>10440454
not a number, but the definition does use the word quantity

https://www.wolframalpha.com/input/?i=infinity
An unbounded quantity that is greater than every real number.

>>10440489
https://en.wikipedia.org/wiki/Infinity
the wikipedia definition uses neither.

>>10440422
No you didn't
refer to >>10439445

>>10437694
>Why is this even still up for debate?
It's not. 0.999...=1


>>10437877
What horrible thing happened in your childhood that you think everyone who doesn't like you is just offended? You're not even offensive, just annoying. That's it. There's no deeper layer to it.

>>10440520
Nigger, you're retarded. I'm telling you, 0.999 isn't equal to 1

>>10440655
You didn't say .999 in any of your posts

>>10437694
So are you pointing out that arithmetics are basically flawed?

Yes we can agree with this.

So this agreement would end these stupid threads forever?

>>10437983
>please don't give me correct answers
Ok. (the leading zeros are for formatting)
x=0.999
01x = 0.999
02x = 1.998
03x = 2.997
...
08x = 7.992
09x = 8.991
10x = 9.99
There's one less trailing decimal so if x=0.999... then 10x has one less after infinity then 9x does. While related this isn't a answer to the question you asked so you can still pretend you're correct without issue.

this is why i hate decimals

>>10437694
Oh

>>10441103
0.11111111... * 9

https://www.wolframalpha.com/input/?i=0.999......

>>10437694
I fucking love the demonstration that 0.999 is equal to 1 without buying into it. It's like ye old troll physics comics but if someone shitposted in academy. And people are SPASTIC about it.

0.999... is not a number. It is a nonsense phrase created with symbols.
If it is a number, then show it to me. Show me an infinite string of 9s after a decimal point. "..." is not an infinite string of 9s and it is not a sufficient substitute.

>>10439587
Based retard. The reals have certain mathematical properties. It is a complete metric space which means it's defined in terms of the completeness axiom, which means that every bounded sequence has a least upper bound. The hyper reals (the field extension where infinitesimals are defined). This means every cauchy sequence is convergent which means that the argument is true.

Any 2 different reals have an uncountably infinite number of numbers between them (nonzero measure). Are there an infinite amount of numbers between the limit of the sequence and 1? No, there are zero

>>10441380
Pi is not a number. A whole lot of numbers after a decimal and then a "..." because of weak excuses like "we can't calculate that" is not a number. I like my numbers with a beginning and an end.

>>10441380
>If it is a number, then show it to me
1

[eqn] 0.9999.... = 0.9 + 0.09 + 0.009 + ... [/eqn]
[eqn] = 9(0.1 + 0.01 + 0.001 + ...) [/eqn]
[eqn] = 9 \sum_{n = 1}^\infty (\frac{1}{10})^n = 9( \sum_{n = 0}^\infty (\frac{1}{10})^n - 1) [/eqn]
[eqn] = 9(\frac{1}{1 - \frac{1}{10}} - 1) [/eqn]
[eqn] = 9(\frac{10}{10 - 1} - 1) = 9(\frac{10}{9} - 1) [/eqn]
[eqn] = 9(\frac{1}{9}) = 1 \qquad \square [/eqn]

There are no assumptions in this proof other than that you need to accept that 0.999... can be expressed as a series which turns out to be a geometric series whose limit is easy to calculate.
Anyone who thinks they can dispute this has no hope achieving anything with mathematics.

>>10439909
nigger why do you write your 9 like a g

>Arguing based on a fundamental misunderstanding of definitions
LMAO at you brainlets

>>10441527
That's literally all this board is along with a handful of self-hating actually knowledgable people

>>10441406
correct

>>10441416
1 is a number, 0.999... is not

>>10441548
>1 is a number, 0.999... is not
Impossible, they're the same thing. They're either both a number or neither of them are. So you're saying 1 is not a number?

>>10441548
it's not a number? well 1/9 is not out of |N nor Z but it is in Q and |R

>>10441650
Man, I know you're right but fuck you. Your way of writing [math] \mathbb{R} [/math] and [math] \mathbb{N} [/math] triggers me more than faggots saying 1≠0.99...

>>10439632
>>10440106
>>10440106
>>10440365
What if we constructed a countable set of numbers that includes all reals?

>>10440505
And we all know Wikipedia is correct about everything.

>>10437694
https://www.math.brown.edu/~res/INF/handout3.pdf

How about you spergs learn what a real number is?

>>10442070
>And we all know Wikipedia is correct about everything.
about math it typically is, yes
wikipedia math articles are often edited by career mathematicians and professors

>>10442081
hello, undergrad
did you feel smart last week when the prof introduced dedekind cuts?

real men use cauchy sequences

>>10442070
>What if we constructed a countable set of numbers that includes all reals?
why don't you ask Cantor and see what he says

>>10441236
You probably added lots of 1 to represent 0.111... but you have to divide 1/9 to get infinite ones
I as well get 0.999... if I just write myself lots of ones

>>10439265
I would have just said that the constant coefficient is [math]0.9[/math], not [math]9[/math], but overall well done. I love seeing the truth [math]\LaTeX[/math]ed out.

>>10439408
>he thinks the multiplication rule of limits holds for infinite limits
Based retard

>>10437694
ok so what's 1-0.999...?
1-0.9
=0.1
0.1-0.09
=0.001
...
you keep doing this but you never stop, it's 0.0000... by induction all the next ones are 0 too so it's just 0

>>10439408
c=p^m as m->inf
b=p^-n as n->inf
cb=(p^m)(p^-n) as m,n-> inf
cb=p^(m-n) as m,n-> inf

>>10439408
>[math]0\times\infty[/math] cannot equal [math]1[/math]
[math]0\times\infty = \lim_{x\to 0} x\times\frac{1}{x} = 1 \\
0\times\infty = \lim_{x\to 0} x^2 \times\frac{1}{x} = 0 \\
0\times\infty = \lim_{x\to 0^+} x\times\frac{1}{x^2} = \infty \\
0\times\infty = \lim_{x\to 0^-} x\times\frac{1}{x^2} = -\infty
[/math]
They're called indeterminate forms, retard. I bet you think [math]1^\infty = 1[/math] as well.
Leave immediately and do not come back until you have taken a high school calculus course.

>>10439408
>[math]0\times\infty[/math] cannot be [math]1[/math]
[math]
\displaystyle
\begin{alignat*}{2}
0\times\infty&=\lim_{x\to0}x\times\frac{1}{x}&&=1\\
0\times\infty&=\lim_{x\to0}x^2\times\frac{1}{x}&&=0\\
0\times\infty&=\lim_{x\to0^+}x\times\frac{1}{x^2}&&=\infty\\
0\times\infty&=\lim_{x\to0^-}x\times\frac{1}{x^2}&&=-\infty
\end{alignat*}
[/math]
They're called indeterminate forms, retard. I bet you also think [math]1^\infty[/math] cannot be anything but [math]1[/math].
Leave immediately and do not come back until you have taken a high school calculus course.

>>10437705
dy/dx is undefined, dx is 0

10s = 9.999....
s = 0.999......

10s = 9.999....
-s = 0.999...
9s = 9.000...
s = 1.000....

This is how numberphile explained it, but a better way to explain is

10s = 9.999....
s = 0.999.....

10s = 9.999...
- 0.999...s = 0.999...
9.00000000000000000......1s = 9
(9.00000000000......1s)/(9.000000......1s) this is sort of like the same question as ∞/∞ but we could argue 9/9.0000000000....1 = 0.999999..... but 0.000000000....1 = 0 because 0.99999 = 1, an infinitesimal gives you the illusion that 9 = 9.0000001 but really 0.999999 strings on for infinity so this one googleplexth is undefined.

>>10437705
fpbp
>"To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be." -Leonhard Euler

I will say this one last time, you FUCKING, fucking retards.

0.333333..... is an incomplete expression. It represents 1 as being constantly divided by 3, therefore 1/3.

Multiply 0.33333... and you get 0.9999....
They are both incomplete expressions. It's just a inherent problem of the numbering system humans invented.

>>10444219
Worst post ITT

>>10444219
>It represents 1 as being constantly divided by 3
No, [math]\cfrac{1}{3\cdot\cfrac{1}{3\cdot\cfrac{1}{\cdots}}}\ne0.\bar9[/math]
If you take the partial fractions you'd see it doesn't converge

>>10444256
sorry, meant that it approaches zero

>>10444256
>>10444262
also meant [math]0.\bar3[/math] lol

>>10443493
Just use convergence test and geometric series formula

>>10443493
Missing a for all n bigger than N there buddy.

>>10442604
Do you think it's necessary to learn aboit dedekind cuts because some nice idea is used in the proof or something like that? I was only taught the method with the equivalence classes of Cauchy series recently.

>>10437694
>1/3 = 0.333333...
>2/3 = 0.666666...
>3/3 = 0.999999... = 1

>>10445991
>1/3 = 0.333333...

>>10437994
/pol/ lives rent-free in everyone's minds. they don't even think about you

Get on the next level.

...9999=x
...99990=10*x
...99999=10*x+9
x=10*x+9
0=9*x+9
x=-1
so
...999=-1

You don't trust me? Do ...9999*....999 and admire the result.
Or just do 0-1 in a naive way.

>>10446361
wew you're retarded

>>10444265
Neat

>>10437694
that statement isn't even true now fuck off
9(.999)=8.991
the x in 10x=9.999 is different than x in 9x=9 why the fuck you tryin to say 1=.999

>>10447305
What are you talking about?
I think you've misunderstood
x = .999
10x = 9.999
10x - x = 9
9x = 9
/9 /9
x = 1
Therefore .999 = 1

>>10446728
thats literally valid in the p adics you fucking undergrad

>>10447373
>valid in the p adics
So it's nothing

>>10447374
its not even prime you moron
nicely done showing off that you dont know what youre fucking talking about though

>>10447387
You are the biggest brainlet ITT
Not even worth my time to explain how wrong you are
Never post on /sci/ again for the sake of the rest of us

>>10447371
because you are trying to say x=.999=1 by using two different equations, the fuck you smoking bro?

>>10446728
And you failed.

>>10447462
anon...
Please explain your reasoning to me

>>10437705
>>10444206
Although, zero is an absence of quantity

>>10447462
It's the same equation
You're allowed to add, subtract, divide, multiply, and take the root of one side assuming you do the same to the other side and maintain the equality
You should know this from elementary school

>>10448888
Quads of truth and this anon is still wrong hahahaha

>>10446224
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...

>>10444206
Based Euler

>>10437694
exactly 1=/=.999 you just fucked up OP

>>10444206
>Filename
KEK

>>10437694
There's a lemma stating that two real numbers are unequal to each other if you can find at least one number between both of them.

You wont find one between 0,9999... and 1

>Why is this even still up for debate?
Because /sci/ still takes the bait after 9 years of existence.

>>10439446
how is that induction?

>>10451937
>debate
>de bait
>bait
hehe

>>10451904
it's shit

anon's lemma: two numbers are equal if they're equal

top b8

>>10439046
>>10439408
>>10443493
>>10443809
>>10443628
>Unironically using limits which are just as unconstructive as infinitesimals, R and all the other 0.999... garbage

>>10454272
fuck off back to >>>/x/

>>10454288
Not so fast... This guy want to have a talk with you.

>>10437694
Kek

>>10454293
pedos don't count

>>10454296
A: "your proposed mathematical therorem doesn't sufficently hold"

B: "Wow you're a pedophile"

>>10454293
>[math]\pi[/math] doesn't real!!! REEEEE

>>10454304
>bus stop masturbator screams at people walking by

sorry, no Fields medal for you

>>10438396
>9(0.1)
please, for holy fuck's sake, don't do this. reads like the function 9 applied to the parameter 0.1. what does it cost to add the multiplication operator?

>>10437877
Every response before this verified your claim.

>>10437694
[eqn]0.999999...=0.9+0.09+0.009...=\sum_{n=1}^{\infty}9\cdot10^{-n}[/eqn]
I'm sure you agree so far. However, here's the thing about infinite sums, they
are — by definition — equal the limit of their partial sums.
That means:
[eqn]\sum_{n=b}^{\infty}f(x)=\lim_{i\to\infty}\sum_{n=b}^{i}f(x)[/eqn]
So that [math]0.999999...[/math] is in fact equal to the limit of the following sum:
[math]0.9[/math]
[math]0.99[/math]
[math]0.999[/math]
...

Which is obviously equal to 1. So begone, brainlet.

>>10437854
-1/12

>>10437694
>>10454405
WTF my latex got messed up but it worked in the box where you try it. I hate this.

>>10454405
>equalthelimitoftheirpartialsums
thats a weird name for a variable

If I had a cake and a paper thin slice of cake was taken from me, I might as well have a full cake.

>>10454319
Is this a hot new meme or are you really that autistic

>>10454415
Use \text{*put text here*} if you want to write in math mode

>>10437877
Anti-polsmokers are just autists that fear political discussion. They use pol an excuse to invalidate whatever that person just said if it isnt not political. Ironically anti-polsmokers make things political.

>>10454272
Literally all the things you listed are rigorously defined you retard lmao

>>10439587

>we invent, not discover, emergent processes in already existing systems

People invent the signifiers to describe the process. The process itself already exists in potential.

>>10455910
>already existing systems
That we invented.

>>10455887
>Literally all the things you listed are rigorously defined

>>10455887
>all the things you listed are rigorously defined

>>10455887
>rigorously defined

Just go full Pythagoras, it solves the problem

>>10437694
Is there number between it and 1? No? It's one. Welcome to real numbers.

>>10456627
Always post a jpg image of a charlatan and crook if you want to get believed on /sci..

>>10456621
>>10456625
>>10456627

>>10454315

>>10456712
Fuck you and your bullshit opinion

>>10456731
>i have no argument

>>10437751
By infinitesimal they mean infinitely small.
By distance they mean difference (think [math]\|x-y\|[/math])
By always they mean that no matter how many more digits you consider, you always get some difference; that is, you never get to exactly 1.

Where they are wrong, however, is in how infinite sums work. You can get arbitrarily close to 1 by factoring a sufficient amount of digits, so it equals 1.

>>10441017
>after infinity
F

>>10437694
Neither of those two statements are contradictory.

x = 0.999... = 1

Why does this literal zero effort shitpost continually get responses and why am I responding to it too?

>>10443493
>He has to use first principles

>>10444206
wow Euler was based

>>10453044
it's pronounced inductriontation

>>10437850
then 1 = 3/3 = 3/3
1/3 = 0.333...
1/3 * 3 = 0.999...
0.999... = 3/3 = 1

>>10456627
>even this schizo retard can't deny the undeniable prowess and rigor of based Rudin

>>10457610
>>10456733
ok let's say we have this:
[math]\lim_{x\rightarrow\infty}-\lim_{y\rightarrow\infty} , (x,y) \in \mathbb{N}[/math]
The problem with this is that it doesn't has any meaning but it would if we wouldn't write it up like an actual retard.

This functions can be executed by something in reality, f.e. you could try to count from x to inf without ever raising y or you could go like x+1 and y+1 or the other way -> y+1 and x+1. The way how the function is executed is hidden because inf says just "heh I'm totally random until I'm too big for you little boy".

And you actually didn't mean inf when writing this, you're meaning "arbitrary close to inf", not inf itself. If so then just write it down:

[math]\lim_{x\rightarrow k}-\lim_{y\rightarrow a} , (x,y) \in \mathbb{N} , (k,a) \in \textup{arbitrary high}\in
\mathbb{N} \textup{ simultaneously counted per cycle}[/math]

>>10437694
The decimal system was a mistake

>>10438078
Funniest thing I've ever seen on this board

>>10457832
Literally illegible. Take your pills, Wildasberger.

>>10437694
it's not 1, but for all intensive purposes its 1.

>>10458161
For all intents and purposes*, it is 1, because it is 1.
Also, if you study it intensively, you will see that it is 1.

>>10457832
>you actually didn't mean inf
wrong

>>10458161
>it's not 1,
it's exactly 1

>>10454272
Constructivists literally don't believe in excluded middle lmao gtfo with your nonsense axioms