/sci/ - Science & Math

>>12053975
>comic sans
Immediately disproved.

a delicious thread to hide in my facemouth, yum yum yum

This argument comes down to whether you want to use the word "equals" when doing limits.

Oh man you're SO close to getting it with argument 2, but you still miss it

>>12053976
>>12053977
>>12053998
>>12054020
>no argument
onetards btfo

>no number between 0.999 and 1
There’s also no number between 1 and 2.

>>12053975
>Argument 2: 'onetards' agree with reality (physicists, engineers) not theoreticians, therefore they are wrong.
...

For argument 1, the result is there is a RATIONAL number between any 2 distinct reals. So which two integers is 0.999... a ratio of?

>>12054195
It's a windup, mate. Read proof#3.

>>12053975
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...

>>12053975
What is 1/3 in decimal if it’s not .333333...

What is the philosophical motivation for this?

>>12054412
too btfo .9999... retards with one easy trick
.3333.... = 1/3
.6666.... = 2/3
1/3 + 2/3 = 1
.3333... + .6666.. = 1
>noooo 1/3 does not equal .3333...
what does it equal then in decimal?

>>12054401
>>12054424
What is pi in decimal? Don't give me a shortened version I want the exact value in decimal

>>12054432
pi is not repeating dumb fuck. So it can be shorten to .3333... like 1/3

>>12054446
Can’t I mean

>>12054446
Wtf are you even saying. You admit that not all numbers have decimal representations then? What is your argument even?

>>12054469
No not all numbers do like pi can’t really expressed in decimal only rounded. 1/3 can be expressed like .333333....

>>12054473
>1/3 can be expressed like .333333..
Except that's wrong and those are not equal. Circular reasoning is so ingrained in the mentality of onetards that you can't even imagine making a real argument

>>12053975
>Here the mistake is that 1/3 does not equal 0.333...
So what exactly is 0.333... if not the result of dividing 1 by 3?

>>12054424
You will also appreciate the two-lines Fermat's theorem proof.

>>12054432
you see, it's the "..."
that makes it non-shortened
>calling infinite "shortened", what an idiot

>>12054668
The result of dividing 1 by 3 is 1/3. Do you onetards try to be this retarded or what?

>>12054668
0.333...+0.000...1/3=1/3

>>12053975
Today I will remind them

0.9999=1 for most practical purposes. If you need to go further, you are a nerd.

>>12054724
0.000...1/3
express this as a decimal please
>>12054722
well obviously I meant the result expressed as a decimal

>>12054765
Irrelevant. Once again onetards have nothing to say

>>12054811
Express the square root of 2 as a decimal please

>>12054923
Literally all your arguments are stupid if they can be applied to something which is obviously false.

>>12054926
It's impossible to represent an irrational number in a system which is only capable of representing rational numbers.

>>12054926
1.414... where the nth term in the decimal expansion are given by calculating a finite number of terms in a Cauchy sequence that converges to [math]\sqrt{2}[/math]

>>12054929
That also works for 0.(9)=1.(0) crowd.

>>12054938
"1.414... where the nth term in the decimal expansion are given by calculating a finite number of terms in a Cauchy sequence that converges to √2" is not decimal notation

>>12054955
what?

>>12054956
yes it is, I told you how to calculate the nth term in the decimal expansion, you can do this to generate arbitrarily many terms in the decimal expansion

A reminder that you can represent decimals as infinite list without maximal index and infinite list with maximal index. In the second case it is possible to have number standing just before 1.

>>12054395
PWNed!

>>12054957
>Literally all your arguments are stupid if they can be applied to something which is obviously false.

>>12054964
>1/inf
Wow, such rigor, much self-praise.

>>12054961
So? Decimal notation consists of a series of numbers, a decimal point and another series of numbers plus an optional line indicating a repeating series of numbers. A formula isn't decimal notation.

>>12054962
Yes, but the second case isn't the real numbers.

>>12053975
So close. Algebraic manipulation does not work for non-convergent sequences/series. All internally consistent expressions of 0.999... converge, and the number they converge to is 1.

>>12054968
That's indeed what I wrote, but it doesn't help me understand what you mean by "That also works for 0.(9)=1.(0) crowd.".

I don't really get this. 1/3 would only work in a number system based around 3's. In a 10 digit system, the best you can do is an approximation, which is .333 to infinite. It should be understood that .333 is a representation of 1/3, but if you're interested in being exact, 1/3 should be used. There's a reason we use different systems, as they allow specificity and flexibility depending on the task.

>>12055002
The "Literally all your arguments are stupid if they can be applied to something which is obviously false." also works for 0.(9)=1.(0) crowd.

>>12054985
>All internally consistent expressions of 0.999... converge
All of them? That's a very bold claim.

>>12055012
I would say this is incorrect. The decimal notation we were taught in school allowed for putting lines above digits to indicate them repeating indefinitely, and this allows you to exactly represent fractions like 1/3 (as 1.3 (repeating the 3 indefinitely)) and 1/7 (as 0.142857 (repeating the 142857 indefinitely)). However it should be noted that in this system, representations aren't unique, so the real number 1 can be represented both by 1 and by 0.9 (repeating the 9 indefinitely)

>>12055020
Are you actually interested in having a discussion with me or do you just want to throw poop around?

>>12055038
You can also proclaim that 0.999999 is equal to 1, and that will be true when no precision is needed, but not true metaphysically.

>>12055042
Precision? We're talking about exact representations. And 0.9 (with the 9 repeating indefinitely) is mathematically exactly equal to 1.

Please, I challenge anyone thinking this is false to give a precise definition of the number "0.999..." which I can't prove to be equal to 1.

>>12055041
Your argument is "I don't like some arguments. If I'll apply arguments which vaguely look like them to the different task, they supposedly will be wrong. Therefore initial arguments are wrong". With such a method you can disprove absolutely everything.

>>12054978
A told you how to calculate the sequence of numbers you're referring to

Since there are an infinite number of digits in the sequence following the decimal place, I can't include the entire sequence, but I can tell you how to calculate the nth term, which is a way of specifying every term in that sequence.

That is to say,
[eqn]
\sqrt{2} = 1.d_0d_1d_2...
[/eqn]
where [eqn]d_n = \lfloor 10^n \times x_{2^n} \mod 10 \rfloor, \\
x_0 = 1, \\
x_{n+1} = \frac{x_n}{2} + \frac{1}{x_n}
[/eqn]

Here, I'll do it for the first few terms in that sequence
1.414213562373095...

I've written done this in haskell so you can calculate the full expression since haskell supports infinite lists which you can evaluate here, but note it will take an infinite amount of time and memory to run
https://repl.it/repls/SuperficialSadQueryoptimizer

>>12055047
It's not a proof, more of showing you how stupid you sound. Also, please tell me which of the arguments isn't a literal exact translation of your original arguments.

>>12055046
Here: a number is an infinite sequence of integers. If two integers in this sequence differ, then two numbers are not equal.

>>12055053
This is very great, but it is not decimal notation. Based Monad Wizard though.

>>12055055
None of them because they talk about different things.

>>12053975
>he doesnt accept the proof of [math] 1 + 2 + 3 + ... = -\frac{1}{12} [/math]
fucking retard
>hurr moving parentheses make bad
https://en.wikipedia.org/wiki/Riemann_series_theorem
you cant move parentheses moron

>>12055038
>The decimal notation we were taught in school allowed for putting lines above digits to indicate them repeating indefinitely, and this allows you to exactly represent fractions like 1/3
To me, that's more of a compromise in a system that can't really represent that particular number, so a vinculum is used to make it work so we're not constantly jumping between decimals and fractions. A base 3 system could represent 1/3 accurately, but then a vinculum would be require for other fractions that don't work in base 3. What really matters is that people agree about the representation, which they generally do unless they feel like arguing on /sci/.

>>12055056
I assume this is supposed to define the real numbers 0 <= x < 1 ?

>>12055064
Let's define "=x" as "equals x or equals x+1". Then 2+2=5.

>>12055063
??? How is saying "0.999... is real because (0.999... + 1)/2 exists" different from saying "1.000... is real because (1.000... + 1)/2 exists" ???

>>12055056
youre still wrong, a number can be represented as an infinite sequence of numbers 0 to 9
two numbers are equal if their difference is 0

>>12055068
lets define (You) as "a retard", then youre a retard.

>>12055066
You can also add the index of the decimal point. Or take another sequence for the integer part.

>>12055068
This Anon is actually right. Ramanjuan Summation isn't the same thing as normal Summation.

>>12055071
They both can be real?

>>12055073
That would allow for infinite numbers like ...111. Certainly a cool system, but not the same as the real numbers.

>>12055078
Real numbers will be their subset.

>>12055076
Yes, and they're both equal to 1.

>>12055083
Is 0.9 equal to 1? Is 1.0 equal to 1?

>>12055081
No. A subset would imply them being defined the same way. You could define a mapping though and depending on how that mapping worked, either 0.999... wouldn't map to a real number just like ...111 or it would map to 1.

>>12055084
no, yes

>>12055085
...111 would be larger than every finite number, unlike 0.999...

>>12055084
>Is 0.9 equal to 1?
Obviously, no
>Is 1.0 equal to 1?
Obviously, yes
You know the answers to these questions. Are you trying to tell me because these hold for all finite lengths, they hold for infinite lengths too? That argument doesn't work. All the approximations of pi are rational and yet pi itself is irrational.

>>12055089
No, whatever ...111 mapped to wouldn't be larger since it wouldn't even be part of the real numbers. You can't compare real numbers with things that aren't real numbers.

>>12055086
Ok, let's go a bit further:
Is 0.99 equal to 1? Is 1.00 equal to 1?

>>12055097
Ah, so we can't say that 2>1.5?

>>12055090
Now you can see why applying argument fit for a number starting with 0. to the number starting with 1. is being faulty.

>>12055098
Read >>12055090, faggot

>>12055109
Read >>12055108, faggot

>>12055102
No, I'm not saying that. 2 can be defined by a Dedekind Cut and is thus a real number. It can also be defined as a natural numbers, but comparing the natural number 2 with the real number 2 makes no sense.

>>12055108
Do you have some sort of brain damage? Whether a number has 0 or 1 as integer part is literally completely irrelevant. I could make the argument that 0.000... is not equal to 0 too.

>>12055113
>It can also be defined as a natural numbers, but comparing the natural number 2 with the real number 2 makes no sense.
But it's pretty obvious that two apples are more than one and half.

>>12055119
Since 0 is equal to 0.0 or 0.00, that will not be very convincing.

>>12055128
So? Maybe it's not equal in the limit. As I said, properties can change in the limit.

is this the best /sci/ meme?

>>12055136
Yes, you can also define functions which will have vastly different value at the point than their limits at that point. That still will be pretty arbitrary and not some metaphysical truth.

>>12055108
if the distance between .999... and 1 is 0, then theyre the same
regardless of what they start with

>>12055123
What even is your point here now?
"I can easily compare two apples to one and a half apple
thus
it makes sense to compare natural and real numbers
thus
it makes sense to compare any two numbers from any two arbitrary number systems
thus
0.999... defined as a sequence can be compared to the real number 1 and is not equal to it"
?

>>12055144
The distance is infinitesimal though.

>>12055141
The irrationality of pi is not a metaphysical truth?

>>12055147
no its not
the distance between 1 - 1/w and 1 is infinitesimal
.000...1 is not an infinitesimal

>>12055145
>it makes sense to compare any two numbers from any two arbitrary number systems
Then what was the demand to bring the number equal to 1/3 about? Fractions are fractions, decimals are decimals.

>>12055147
There are no infinitesimal real numbers.

>>12055152
>.000...1 is not an infinitesimal
Then what is it? Finite number? Infinite number?

>>12055158
Great, you can also have a system where numbers less than 1/1000000 don't exist. It will even be practically useful.

>>12055152
>.000...1

>>12055138
Yes, you can discuss it forever.

>>12055153
>Then what was the demand to bring the number equal to 1/3 about? Fractions are fractions, decimals are decimals.
That wasn't me. >>12054765 was my first post in this thread.

>>12055171
Ok, so if you think that there is no need to map everything on everything, then the problem does not even exist.

>>12055161
Please, define that number. It's not valid decimal notation. It's not a real number. It's not even validly definable by an infinite sequence of integers.
Retard.

>>12055175
It's an infinite sequence of zeros followed by 1. If you don't like infinite sequences, then stop discussing them, be they 1.(0), 0.(9) or 0.(0)1.
>It's not valid decimal notation. It's not a real number. It's not even validly definable by an infinite sequence of integers.
How do you declare infinitesimals with these things?

>>12055165
>equating some arbitrary system with the real numbers
Holy fuck, you're actually retarded. Centuries of mathematical work has been based on the real numbers. It is the standard system. Nowhere in this thread have you mentioned or defined a system which would allow for infinitesimal numbers.

>>12055174
Actually, I'm interested in you number system now if we can at least agree for the reals. What would the difference of 0.999... and 1 be in your number system? 0? So, two numbers that aren't real could have difference 0. Doesn't seem very useful or nice at all.

Can we stop having these discussions as it just makes these retards orgasm more? We needn't stoop down to their level.

>>12055184
>It's an infinite sequence of zeros followed by 1.
>infinite
>followed by
Please google what the word "infinite" means. (Tip: It comes from the prefix "in-" and the word "finite", meaning "not finite". Without end. You can't follow something up if there's no end to it)

>>12055187
In practice you constantly use systems with limited precision. That's actually pretty obvious unless you can count to infinity and remember infinite digits of pi (in which case you must be a God).
>It is the standard system.
Standard system may be simply convenient, not metaphysically true. Just like you can ignore relativity effects in most cases, but that does not disprove the relativity theory.

>>12055200
There is an infinity of numbers in the interval [0,1], yet it still has an end.

>>12055201
>In practice
>Standard system may be simply convenient, not metaphysically true. Just like you can ignore relativity effects in most cases, but that does not disprove the relativity theory.
What? Are you basing your arguments on engineering practices or """metaphysical truth""" now? These are completely different things.

>>12055194
>What would the difference of 0.999... and 1 be in your number system?
0.000...1 of course.

>>12055205
Yes, we call the interval [0,1] infinite because if we were to put the numbers in there into a sequence (which isn't possible anyways) it would have no end.

>>12055207
If you argue then it's just convenient to equate 0.(9) to 1.(0), then no problems, continue to do that if you wish. If you argue that it is an Ultimate Metaphysical Platonic Truth, then we have some questions.

>>12055211
THATS NOT A SEQUENCE YOU FUCKING IDIOT REEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE 0.000...1 IS NOT A THING. NOT A THING. NOT A THING.

GOD I HATE YOU FAGGOTS IM LEAVING THE THREAD NOW

>>12055214
Anon, stop arguing with him. They try to understand it. You're honestly just wasting your time on this thread.

>>12055214
You can also replace some zeros in 0.000...1 with infinite sequence of digits you like more.

>>12055224
>try
*never try

>>12055221
It's a number, which is also infinitely small. But bye-bye!

>>12055214
>>12055224
Is the set of negative integers infinite?

>>12055061
how is it not decimal notation? it's a sequence of integers of numbers, a decimal point followed by another sequence of integers
just because one has to be creative in order to specify that sequence of integers doesn't mean it isn't a sequence of integers.

Anons, embrace the ultrafinitism.

0.(9) < 0.(9)9 < 0.(9)(9) < 0.((9))

>>12055281
>0.(9)9 < 0.(9)(9) < 0.((9))
not real numbers

>>12055310
What about 0.(...(9)...)?

>>12055310

>>12055335
>omegas
not real numbers

>>12055342
They are more real than reals.

>>12054187
Heh. I like this criticism of naively using the word number without defining it.

>>12055349
so what?

>>12053975
Sigh, why do I even...
Counterargument 1: So this number is bigger than 0.999... but smaller than 1? How would you write this? 0.999...+?
What about the next number in the sequence?
Counterargument 2: Being an engineer is not even an argument.
Counterargument 3: That example doesn't converge. 0.999... does. Pay attention next time in Analysis 101.

>>12055433
Stop falling for bait and let the thread die.

>>12055433
1: 0.(9)5
2: If you are en engineer, then it's OK to ignore precision questions.
3: What does your converging definition say about infinitesimals?

>>12055422
Ask that about >>12055342

Open Python and enter
>print(0.9999999999999999999==1)
Get
>True
Q.E.D.

>>12055551
stfu and stop bumping

It's impossible to calculate a difference between the 1 and whatever the object described by 0.999... can possibly represent.

1 - 0.999... = 0.000...
No difference = same thing.

>>12055566
nineist btfo

>>12055568
0.000...1

>>12055568
>t. retard falling for bait

>>12055575
kill yourself you retarded janny wannabe

>>12055574
no, that would be the answer to 1 - 0.999...9 not 1 - 0.999...

1 - 0.999...9 = 0.000...1
1 - 0.999... = 0.000...

>>12055599
The more you talk about how 0.99=1, the more they will post, the more they love the attention.

>>12055604
stfu

>>12055600
Well, 0 is 0, 0.0 is 0, so is 0.000, but is 0.000... 0?

>>12055604
>0.99=1
Truth.
I mean, knowing the difference between 0.9 and 0.95 can be useful in some rare cases, but 0.99 is 1 by all means.

>>12055627
There's never any difference between 0.000... and 0.

>>12055645
It's essentially either trolls or NEETs living in their mom's basement, wearing a t-shirt covered with cheeto dust. The amount of effort they put in to 'not' understand math, they could've put all of it into actually understanding it.

>>12055661
Conclusions looking for corroboration is the post-modern scientific method.

>>12055661
It's fun to troll dorks who finished half of analysis 101 and now think that they are the source of divine truth.

>>12055677
It is. But these threads bait newfags who then spend their effort convincing these monkeys rather than doing actual /sci/.

>>12055335
holy shit that is obsessed with 3s

>>12053975
how many people with any math background beyond calculus believe this shit

>>12055686
They actually create threads like this, so they want that.

>>12055711
All of them.

>>12055686
Spending a bit of effort is justified when you get fun stuff like >>12055221 in return.

what number is before 0.999...?

you cant put anything there

the truth is that infinity is a meme

>>12055748
Can't put any number right before 1 either. Same thing.

>>12053975
0.333=1/3 is trivially easy to prove
Literally just do the long division lmao

>>12055762

0

this post was made by integer gang

>>12055792
Sure, if you have infinity of time to spare.

bros what if 1/3 is not infinite but we just havent discovered it?

>>12055826
1/3 is surely finite, just eat 1/3 apple.

>>12054504
>can be expressed as
>BUUUTTT IT'SH NAWT EEKWAL
and?

>>12056026
Ok, then 1 can be expressed as 0.999.... (or 0.999 if needed), but isn't equal to that.

>>12056037
But it is.

>>12056066
>BUUUTTT IT'SH EEKWAL

infinitesimals are purely algebraic objects, and as such, never turn up in limits
limits are geometric, infinitesimals are not
the limit of [math]1/2^n[/math] is literally just 0, by how limits are defined

an infinitesimal is a nonzero element which squares to 0
which means it satisfies [math]a^2 = 0[/math]

[math].000...1[/math] cannot even be an infinitesimal, since it does not square to 0
[math]a = .000...1[/math] means that [math]a^2 = .000...1[/math]
so [math]a^2 = a \implies a^2 - a = 0 \implies a(a-1) = 0[/math]
so [math]a[/math] is either equal to 0 or to 1

and it cant be 1, since [math]a[/math] has a 0 in the ones place
problem?

>>12056081
.999... is not 1 minus an infinitesimal
[math]1 - 1/\omega \neq .999... = 1[/math]

>>12056105
>an infinitesimal is a nonzero element which squares to 0
>.999... is not 1 minus an infinitesimal
>1/ω
So what is it? 1/ω or thing which squares to 0?

>>12056105
>limits are geometric
How are they not analytical objects? Genuinely curious, I don't have a stake in this contest of yours.

>>12056127
geometry and analysis are the same in my mind
>>12056119
if you want to write it as epsilon then be my guest
one over omega or epsilon are both not .999... anyway

>>12056145
[math](1/ω)^2=1/ω^2[/math], not 0.

>>12056165
yeah, its just a habit of mine to write 1/w
but its still fine, since 1 - 1/w is not .999... anyway

>>12056178
You say that 1-0.999...is not infinitesimal because it is not 1/ω. But you also say that infinitesimal must always square to 0. So what is it?

The answer can literally be either or depending on how you want to define the conditions of something "equaling" another thing.
I don't even know math but I'm apparently better at it than all of you retards.

/thread

>>12056194
the retards in this thread cant fucking handle a definition of infinitesimal besides "squares to 0"
im not going to get into dealing with it more properly
>>12056215
>The answer can literally be either or depending on how you want to define the conditions of something "equaling" another thing.
no it cant
we already have a definition of that, you dont get to change it willy nilly you dumb pseud

>>12056235
>cant
>we already have a definition of that, you dont get to change it willy nilly you dumb pseud
Sorry kid, but just did.
/thread

Reminder that 0.999... is still not equal to 1, no matter how much you want to believe otherwise.

>>12053976
Wasn't the Higgs Boson announcement also in Comic Sans?

>>12055738
I feel bad for him lol.

I have speaking to God recently. God tells me that infinity does not exist ( except for Him, obviously ).

>>12053975
0.99999999... is the cauchy sequence that converges to 1, however this is not equivalent to 1, rather 1 is the least upper bound of the former sequence, though on its own makes an equivalence class that could be represented as [1]

>>12056635
God doesn't exist. I'm sorry anon, you might be schizophrenic.

>>12056635
>I have speaking to God recently
prove it.

Can we make a compromise? For example 0.(9)=1, but 1.(9)<2?

>>12053975
I agree OP. The only people who say that 0.999... = 1 are trolls

>>12054765
Problem with this "no u" post is that mathematical operations used in OP can not be applicable to this example. They just don't make sense here.

>>12057064
worst and cringiest bait I have ever seen

>>12057084
Stop baiting
Why do retards like to pretend 0.999...=1 so much? Do you have anything else going on in your life??

>>12053975
Based post.

If 0.999... were equal to 1, pseudorandom number generators would have a chance of outputting 1 and potentially causing system to crash as it depends on the fact that all numbers are generated in range of [0; 1), or in other words, [0; 0.999...].

>>12053975
1/3 = 0.333...
(1/3) * 3 = 0.333... * 3
3/3 = 0.999...
1 = 0.999...

>>12057111
but 1/3 is not 0.3333....

>>12057114
The absolute state of /sci/

>>12057114
When these faggots will understand the difference between approximation and real value?

It's so difficult to distinguish who's a moron and who's baiting for shits and giggles.

>>12057130
Everyone who assumes 1/3 = 0.(3) for their argument is baiting.

1/3 is finite, 0.(3) by definition is not, the latter approaches 1/3 when amount of digits is going towards infinity, but interesting property of limits is that end value is purely theoretical and requires perfect conditions.

You can be a retard and say 1/x = inf, because that's the limit of x approaching 0, but everyone will laugh at you.

Say that 1/3 = 0.(3) and you're a genius...


Go back to fundamenrals, retards.

>>12057090
I was wrong. this is the worst and cringiest bait I have ever seen.

0.999... is an algorithm that keeps appending nines.
It is not equal to one since it will never reach 1.

However, the LIMIT of 0.999... is 1.

>>12053975
Against my better judgement (and also because I'm bored) I will answer these. The first one, (0.999...+1)/2 would be great if you could actually produce such a number. As it stands, no decimal can conceivably be imagined that represents this. The argument also goes more like this: 1-0.999... is less then every single rational number-since there exists a rational less then a real[simply take a new number by a termination of that reals decimal sequence], then any real number which 1-0.999... supposedly is, would mean I can find a rational less then it-in turn, 1-0.999... is not less then every rational, a contradiction. I would like to see you worm your way out of the properly stated form of this proof rather then the strawman.

The second one is to ask what it means for a number to correspond to it's decimal expansion: the answer is the limit of the partial representations of the decimal and in that case 1/3 certainly equals 0.333... To ask such a question without even being clear how the decimal sequence (0,3,3,3,...) relates to the number 1/3 is a pointless exercise. 0.333... isn't a decimal approximation-it is the true to goodness calculated value of 1/3-at any given termination at point n,

The third one, from a formal perspective, is a bit iffy [namely, on the multiplication by 10-this needs to be better elaborated on[. However, to claim 9.999... - 0.999... =/= 9 is to reach mount stupid, since this is how decimals have always worked. Also, your DIVERGENT series is a red herring since these series are convergent, absolutely so, in fact. The sum of two convergent series is the series of the summed terms-this is a basic proof.

Basically, fuck off and learn math. All your tripe objections are easily, easily answerable if you actually care to learn.

>>12057139
1/3 and 0.(3) are both finite you fucking idiot, since they are both less then 1. The totality of the sequence of digits to represent them in decimal format are both infinite.

The decimal expansion isn't a real number but a sequence-there is an assignment to a real number by its infinite series as a decimal, which is defined as a limit.

This isn't even comparable (if you write it out in full you lazy bastard) to your 1/x case, and you still get it wrong since lim x-> 0 1/x isn't inf, only if you approach it from the positive.

Pretty rich for you to tell someone to 'go back to fundamentals'. This is real analysis 101.

>>12054395
1/inf isn’t a number, and it never becomes 0

>>12058998
I still think he's winding you up, anon.

Incidentally, does no one write it [math]0.\dot3[/math] anymore? That's how we learnt it in school.

>>12059026
https://xkcd.com/406/

>>12053975
What a brainlet thread.

>>12058980
1-0.999...=0.000...1 which is less than every positive rational

0.333... is simply a decimal approximation of 1/3 and no amount of whining and feet stomping will change that, child

0.iii...

>>12053975
this is funny OP, thanks for making it
>1
this one is funny because its circular, but then claims the counterargument is circular
>2
this one is not really an argument, but its funny, i like it
>3
this one is sneaky, but the key is the difference between convergent and divergent series

>>12059060
What a brainlet post

is there a mathematical application for which it is necessary to distinguish between 0.999... and 1? does it have anything to do with "surreal numbers"?

>>12053975
here's my proof:
2 - (2 x 0.9999...) = 1 - (1 x 0.9999...)
+ (2 x 0.9999...) + (2 x 0.9999...)
2 = 1 + 0.9999...
-1 -1
1 = 0.9999...

>>12061187
>approximation
no, it's exact
>>12054395

>>12059007
0 is a number, virgin
1/inf = 0

>>12061380
1 is less than 100
2 is less than 100
...
100 is less than 100
101 is less than 100

>>12061385
He never said 0 is not a number. Learn to read kiddo

>>12053976
Comic sans is a good font, begone redditor.

>>12061187
"1-0.999...=0.000...1 which is less than every positive rational"

Which would be a direct violation of the archimedean property. I will simply point you to a proof that for every two real numbers, there is a rational inbetween: http://fulviofrisone.com/attachments/article/453/Principles%20Of%20Mathematical%20Analysis%20-%20W.Rudin.pdf#page=18.

Alternatively, another way to view it is that the rationals are dense in the reals[simple fact of topology]-if 0.000...1 and 0 don't have a rational inbetween, then clearly the interval (0,0.000...1) consists of points in which are not rationals nor are limit points of rationals, a contradiction.

.

>>12061580
Also, it should be clear that '0.000...1' means a bunch of 0, and THEN a 1.[the sequence is of order type w+1] '0.999...' means a bunch [countably infinite] number of 9's, there is no 'after'. [sequence of order type w] Summing these in the only meaningful way would give 0.999... 1 [again, this is assuming 0.000...1 means anything which there is no system I am aware of that it does]. As you might have noticed, this does not produce 1.

...And to that, it should be noted that '0.000...1' has no meaningful definition. For a positive real number r, let a_0 be largest integer not 0 such that. r-a_0>0, then a_1 largest integer such that r-a_0-a_1/10>0, and so forth. This defines a sequence (a_1,a_2,...) called the decimal expansion. So, you have the 'decimal' (0.000...1), for what value of n in the sequence do you get this 1? If you were to actually carry this out for 1-0.999..., you would get (0,0,0,0,...), which, as you may have noticed, is just a string of 0's. And, the infinite sum a_0+a_1/10+a_2/100+... is equal to r, as the limit of the partial sums (the definition of infinite series) is r by the above algorithm. So the real number defines the decimal and vice versa-so since we do this for 1-0.999... and get decimal (0,0,0,...), what real number do you think we'll get? I am putting my money on... 0.

Also, 0.333... is 1/3 and it always has been and always will be. It's you whining against pretty much the most obvious concepts of mathematics. The left is infinite series with terms 3/10^n, which sums to... 1/3. What do you think it sums to, it is a geometric series so should converge to some number, if it isn't 1/3 done by basic plug and chug, what is it? Again, learn real analysis

Anyway you cut it, you are wrong from practically every angle imaginable

>>12061413
lrn2math boomer

>>12061582
1/3 is finite, 0.333... approaches 1/3 but never quite gets there because neither infinity nor a limit is a number and you cannot plug either of those into a sum you faggot.

The countably infinite number of 9s in the series is equal to the cardinality of the series.

x=.9999...
10x=9.999...
10x-x=9
9x=9
x=1
1=.9999...

>>12061596
>never quite gets there
0.999... is static, the length is aleph_0 from the get go.
Your naive cartoon vision of a diesel engine chugging along is ridiculous. Embarrassing even.

>>12061752
Write down the value of aleph_0 for us.

>>12061797
aleph_0

>>12061799
So you can't tell me because you aleph_0 is not a real number?

Is this faggot so fucking retarded that he will use English definition of "real" to dismiss mathematical fact that aleph_0 is not a real number and cannot be used in mathematical operations? Stay tuned to find out.

>>12061596
I am a faggot indeed but that is besides the point.

Let us make this as clear as I possibly can. Grab paper and pencil and try doing this and let me know exactly where I mess up because I think I've made it stupidly clear. Please be very specific, not some vague stuff about 'you set sum equal to limit', point to a specific logical fallacy.

Forget about decimal notation. Based on the above process, call the decimal of that number the sequence (a_0,a_1,a_2,...). It has, as of yet, no connection to a specific real number. There is a real number r, and an algorithm that produces the above sequence.

You now DEFINE a_0.a_1 a_2 a_3... as this real number by what I did above-as the limit of the given series. Still following? 0.333... is a NUMBER, and 0<0.333...<1-it is less then a FINITE number and therefore is finite. It isn't 'oo'-now, the amount of non-zero numbers in the sequence IS infinite, but that isn't what we mean when we say a number is infinite. And, the algorithm for numbers with terminating decimals still is infinite, just with redundant zeroes.

0.333... is therefore DEFINED to be the series 3/10+3/100+... which is DEFINED to be its limit of the partial sums. The limit of its partials sums is the number 1/3, so 0.333... is by definition 1/3.

>>12061807
'Real numbers' is just a term based its contrast to 'complex numbers'. If you want to say 'these aren't number', fine-I will call them a mathematical structure called the real and complex field respectively. It doesn't fucking matter.

Is aleph_0 a number? Well, if you don't want it to be a number, then so be it. I will consider it as a cardinal in the 'monoid of cardinals'. Whatever I call it, who cares-it acts like a number, doing arithmetic operations so we call it a number.

To illustrate the absurdity of your objection... write down the value of 1. Stupid question, it's 1. Exactly-the value of aleph_0 is aleph_0.

>>12061807
1/inf = 0
1 + inf = inf
1- inf = -inf
inf + inf = inf
inf/inf undefined
inf-inf undefined
1^inf undefined

you can't do everything with inf as with a number, doesn't mean you can do nothing tho

> 1
What is the N, such that 10^-N plus 0.9999... is below 1 then?
> 2
Absolute brainlet tier
> 3
An equation is not an infinite, divergent sequence.

>>12053975
By what do you mean equal? As symbols? As sequences? as real numbers? The true interpretation of the real numbers is as equivalence classes of Cauchy sequences. The two sequences in question are
[eqn] 0.999\ldots \sim (0, 0.9, 0.99, \ldots) [/eqn]
and
[eqn] 1 \sim (1, 1, 1, \ldots).[/eqn]
As sequences, we see that they are not literally equal. However, both belong to the same class of Cauchy sequences since for any [math] N \in \mathbb{N}[/math], there exists some [math] K \in \mathbb{N} [/math] such that for all terms beyond the [math]K^{th}[/math] term, any values from either sequence are withing [math]1/N[/math] of each other. So as sequences, no, but as real numbers (equivalent sequences), yes.

>>12062035
>1 + inf = inf
1+inf = -1(-1 -inf)

since (-inf) = (+inf)
1+inf = 1-inf

>>12062961
>inf = -inf
congratz, largest error possible

>>12063072
how is this still not common knowledge?

>>12063138
because it's bullshit

>>12063138
There is no -oo in the real projective line. There is only a single point at infinity, oo. Now, -oo can refer to two things... it can refer to an element -oo such that -oo+oo=0 [an inverse element], which does not exist in this number system. Or, it can be (-1)*oo by formal arithmetic properties of this system, which is just oo and so really doesn't need a new symbol.

>>12063149
>real world applications
>bullshit
>>12063160
So there is no number at that point, just a point used as a number. got it.

>>12053975
I know the image is bait because argument 1 relies on circular logic while calling out circular logic.

>>12063138
So that's why they call (R,*,+) a ring.......

>>12053975
Yep, truth.
http://aynrandlexicon.com/lexicon/infinity.html

>>12063761
But it isn't the real number line, it is the projective real number. The real number line is a ring whereas the projective one isn't.

>>12054765

what you wrote is a nonsense since 0.9999... is not equal 1 so it not possible to replace one by another

>>12065146
>>12061752

>>12064370
>aynrandl
lol

>>12065146
>>12065296
When will infinitesimals be taken seriously?

Even faggot wolfram alpha says 0.999... = 1 because that engine is too retarded to represent infinitely small difference between two numbers and assumes equality because of it.

It's not even such hard concept to grasp, all you need to know is limits. The only reason anyone could proudly say that 0.999... = 1 is limits, because yes, the former is approaching 1, but it will never be 1, no matter how many 9's there are, even aleph_0, it will still be less than 1 because of it, because if we could know value of aleph_0 + 1, it would still get 0.999... closer to 1, but also aleph_0 + 2... what about if we added 4? 5? 100...?

We came up with limits exactly to approximate these values despite them never actually being what the limit is, and yet people misuse undergrad level math to come up with retarded shit like this, what the fuck?

>>12065329
They're like virtual particles in physics, a nice lubricant for equations.
In the end however you return to real numbers. So 1/inf=0 to all non-autists.

>>12065329
>0.999... = 1 is limits
no you retard, "limits" is when you see "lim" written.
"0.9..." is just as static as "1" is

>>12065427
So is 0.000...1

>>12065329
>We came up with limits exactly to approximate these values despite them never actually being what the limit is,
Keep dreaming

>>12065427
You niggers always argue that 0.999... = 1 because 1 - 0.999... = 0.000...1

In other words, because apparently you can just plug in value
[eqn]\lim_{x \to \infty}\frac{1}{10^x}=0[/eqn]
like a retard and be right.

>>12065454
>You niggers always argue that 0.999... = 1 because 1 - 0.999... = 0.000...1
Actually we argue 0.999... = 1 because it follows from the definition of 0.999... as a sum of an infinite series

>>12065459
And the series depend on limits, yet somehow manage to introduce this retarded inconsistency. There's no real x where equatiin I mentioned becomes absolute 0, meaning that 1 - 0.999... > 0, faggot.

Come up with something less retarded.

>>12065463
So your point is that you dont accept 0.999... to be well defined as a number?

>>12065475
I accept the fact that 0.999... is a well defined number that is infinitely close to 1 but is not exactly the same as 1 because of infinitely small difference.

Its irrelevant to me how small the difference is, it isn't 0, because infinitely small difference is approaching 0, but is not 0, and I will not ignore it.

Suppose you have three equal spheres A, B, C with infinite points. Suppose you are able to remove a point from sphere B and two points from sphere C. How would you define the integer values of sphere B and C assuming A is 1.

>>12065437
" 0.000...1" is just as static as "0" is

>>12065463
>real x
right, inf isn't a number

>>12065477
>infinitely close to 1 but is not exactly the same
let's pretend finite is infinite

>>12065545
Just a reminder that all real numbers between 0 and 1 make up a bigger set than all countable integers.

0.999..., ironically, is not a part of countable integer set.

>>12065554
nope,
1 is part of the countable integer set

>>12065562
if 0.999... = 1, then 0.999... is a part of countable integer set.

This is everything that's wrong with niggers trying to do anything related to science.

>>12065572
yeah, that's what I just said
lrn2read

>>12065591
And its objectively wrong, you nigger.

>>12065463
in other words, you don't understand the definition of 0.999... as a sum of an infinite series

>>12065695
Infinite series rely on limits themselves and what you really can't grasp yourself is inconsistencies your retarded concept introduces.

>>12065703
>inconsistencies
name one

>>12065706
Your brain, you fucking faggot

>Infinite series are defined as the limit of the infinite sequence of partial sums.

Do you know what `limit` means?

You faggots literally continue repeatedly telling us that 0.999... = 1, only because

1 - 0.999... = 0 eventually, because that's the approximated limit.

How fucking retarded do you have to be to not comprehend such trivial concept?

>>12065603
ok sweetie

>>12065711
>that's the approximated
nope, exact

>>12065711
>>12065545

>>12065717
>>12065718

[eqn]1 - 10^{-x} = 0[/eqn]
And the value of x is?

>>12065727
oh honey, it's right there under the "lim"

>>12065729
By definition,[eqn]\infty[/eqn]is not a number you nigger.

I don't even have to try, you faggots out yourselves as retards on your own.

>>12065731
so?
1/inf=0

>>12065735
For [eqn]\frac{1}{\infty}=0[/eqn] to hold, [eqn]\frac{1}{0}=\infty[/eqn] needs to hold, also what the fuck does [eqn]1=0\cdot\infty[/eqn] even mean?

You're a fucking retard.

>>12065731
>>12062035

>>12065742
no, because that's manipulating inf like it is a number
you just said that it isn't a number, wtf is wrong with you

>>12065746
>>12065750

Infinity is not a number so you can't do anything with infinity as a number.

[eqn]\color{red}{\infty~is~not~a~number}[/eqn]

Get that into your thick skull, retard.

I'm showing you that it makes no sense to manipulate it like a number.

If you treat it as a number but avoid inconsistencies by saying "doen't matter", you probably should realize that this isn't >>>/x/chizo board.

>>12065754
>you can't do anything with infinity
sure you can
>>12062035

>>12065711
in other words, you don't understand what limits are. okay

>>12065754
>can't do anything
lol, schoolkids do it all the time
[math] \displaystyle
\lim_{x \to \infty} \dfrac{x+1}{x} =
\lim_{x \to \infty} 1+ 1/x = 1+0 = 1
[/math]

>>12065758
>>12065762

>1^infinity=undefined

Then you look at math and realize that
[eqn]\lim_{x \to \infty}1^x=1[/eqn]

Care to explain why you blindly quote someone as retarded as you and try to pass it as a fact?

>>12065766
>only in context of limits

Are you seriously trolling or just that fucking braindead?

>>12065711
0.999... is not an infinite series, it's a real number.

You want to know what's an infinite series? The retarded "proof" in number three of the OP.

>>12065782
So you agree that it's a real number that's not in a set of integrals and that it doesn't equal to 1.

Why the fuck are you even replying to me if you agree?

>>12065777
yes, it's an example where the lim is different from the value
feed it into wolfram alpha or go read a math book

>>12065785
It's a real number that equals 1.
Just another way of writing one.
The same way 20/20 is another way of writing one.

I'm replying to you because you are spouting retarded bullshit. 0.999... is not an infinite series.

>>12065777
>only in context

>>12065799
>wolfram alpha isn't retarded like me and knows that inf isn't a number
>go read a math book
yeah go read up on what a number is and isn't and what limits are, you nigger.
>>12065805
>>>12065807
>>>/x/

>>12065816
WA vs random 4chan schizo
lol

>>12065754
>Infinity is not a number so you can't do anything with infinity as a number.
non sequitur

>>12065821
You absolute nigger, WA tells you that
[eqn]1^\infty=undefined[/eqn] exactly because its not a number, and you can't raise 1 by something that isn't even a number.

>>12065831
how exactly is this related to your retarded opinion that 0.999... is not 1 ?

>>12065831
1/inf = 0
1 + inf = inf
1- inf = -inf
inf + inf = inf
inf/inf undefined
inf-inf undefined
1^inf undefined

feed them all in WA

>>12065754
It is possible to consider infinity as as number. One needs only to compactify the complex plane with a point added at infinity. This is called the Riemann sphere.

However, this ends up not having the nice algebraic structure we would like it to have.

But i agree, the arbitrary usage of infinity as a number is super retarded.

>>12065827
>pretentious words his philosopher professor mentioned eventually
This is /sci/, not >>>/his/

>>12065841
>i have no argument

>>12065840
what "arbitrary usage" do you have in mind? for example 1/inf = 0 is completely fine.

>>12065840
It is also possible to consider that 1 = 2 and be a >>>/x/ chizoposter and I don't give a fuck to humor that idea.

>>12065842
Then reply when you will have one, nigger.

>>12065841
>Infinity is not a number so you can't do anything with infinity as a number.
doesn't follow

>>12065848
>can't use non-number as a number doesn't follow because my schizo says otherwise

>>12065847
>Then reply when you will have one, nigger.
go back if you don't even know how to greentext

>>12065853
You literally greentexted yourself, what is there to know?

>>12065844
In the context of riemann sphere, yes it is true. However, we lose the field structure. 0/infinity is no longer defined, for example. So one can no longer do arbitrary operations that would be valid in a field wihtout being careful.

>>12065847
Compactification is no pseudo-mathematics, retard. It has a real valid point when talking about topology.

>>12065849
feed them all in WA

>>12065816
>He thinks numbers with repeating decimals are infinite series
>He doesn't realize we wouldn't be able to operate on fractions without using limits if this was the case
>He uses basic Math terms wrong
>He sends *me* to /x/
0.999... = 1
Seethe harder.

>>12065856
lol, so clueless
typical schizo

>>12065862
>clueless
Do you always talk in 3rd person or only on 4chan?

>>12065859
At this point you may aswell tell me to feed 0.999... into WA.

Talk to me when WA will respect infinitesmals.

>>12065849
X is not a number => arithmetic operations are not necessarily defined for X (but they might) and if they are, they don't necessarily obey all usual rules (but they might)
X is not a number => arithmetic operations are automatically not defined for X

one is true, one is false

>>12065888
Your brain is false.

[eqn]1^\infty = 1[/eqn] could very easily be defined with no issues but some faggot arbitrarily decided that nah, that just won't do it.

And why won't it do? Nobody knows because the whole thing is retarded.

>>12065881
>i am better than WA in math
topkek

>>12065900
there's a perfectly sane reason for why it's not defined this way

>>12065900
your Fields medal is in the mail

>>12065900
Exponentiation would lose continuity.

Suppose x^infinity is defined.

Then for all x < 1, x^ infinity = 0
However, 1 ^ infinity = 1.

One desires exponentiation to be continuous. Literally as simple as that

>>12065905
>one desires consistency
>rounds 0.999... up to 1

>>12065908
Imagine literally being this retarded.
I'm surprised you can solve captchas.

>>12065905
Wait, hold on...
[eqn]2\cdot2\cdot...\cdot2=0[/eqn]

??? Guess I'm just being trolled here.

>>12065910
my indian friend solves them for me

>>12065919
>infinitesmals don't exist
Someone told me that I'm not smarter than WA, so I went to WA and found out that they do https://mathworld.wolfram.com/Infinitesimal.html

>>12065924
more like they're not real

>>12065935
And? They're extension of real numbers, part of hyperreals
[eqn]^*R[/eqn]

>>12057645
it's not an algorithm idiot. CS fags gtfo

>>12065939
>use WA to see ur wrong
>shits on CS
Okay so can you make an argument that doesn't depend on "look it up on WA"?

>>12057645
>0.999... is an algorithm that keeps appending nines.
don't make stuff up, anon

>>12065942
read a book

>>12065969
I've read Nietzsche.

>>12056105
reee all rings are integral domains

>>12056105
[eqn]\lim_{x \to \infty}10^{-x}=0.000...1=0[/eqn]
Do I need to explain or will you come to your own conclusions?

>>12066007
youve not contradicted anything i said

>>12067573
Everything you wrote is complete bullshit.

Whatever retarded shite you wrote literally has nothing to do with infinitesmals.

>>12067577
thats exactly my point you dumb retard
I was showing that .000...1 ISNT an infinitesimal
can you read?