/sci/ - Science & Math

[Anonymous No.11761008]

I've posting 0.999... = 1 threads for almost 4 years now.

in the beginning i really didn't believe they were equal. i'd try to come up with many formulations for why they couldn't be equal and they'd all be shut down. i think its time we the 1 != 0.999... community accept we're wrong.

the meme has ran it's course.

let's find a new mathematical relation to debate for the years to come.

[Anonymous No.11761013]

>>11761008
0.999...=1 still has NEVER been formally proven, it's literally pseudoscience

[Anonymous No.11761018]

The Schopster, makin' copies.

[Anonymous No.11761022]

>>11761013
Two proofs:

Via Cauchy sequences:

0.999... = (0.9, 0.99, 0.999,...) = lim as n-> inf of 1-1/10^n = 1

Via Dedekind cuts:

Assume to contradiction that x is a rational number such that

0.999... < x < 1

0.999... is greater than any finite string of 9s so for any natural number n

1-1/10^n < 0.999... < x < 1

1-x < 1/10^n

10^n < 1/(1-x)

n < log(1/(1-x))

Let n = ceiling(log(1/(1-x)))+1

ceiling(log(1/(1-x)))+1 < log(1/(1-x))

This is a contradiction, so x does not exist and 0.999... = 1.

[Anonymous No.11761029]

>>11761013
[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]

[math] \displaystyle
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 = 0.999...\\
\dfrac{1}{3} = 3 \sum_{j=1}^ \infty 0.1^j = 0.333...
[/math]

[Anonymous No.11761038]

>>11761029
>>11761022
nice keyboard mashing, how about you post a proof next

[Anonymous No.11761040]

>>11761013
Funny how even pseudos use "pseudoscience" as an insult. How self-hating they must be.

[Anonymous No.11761042]

>>11761038
>monkey sniffs at book, isn't impressed
wanna bananna?

[Anonymous No.11761044]

>>11761008
The golden ratio in nature. There are science deniers who go through all sorts of contortions to claim that this is logarithmic, instead of the more natural explanation that it's simply the golden ratio continuously unfolding from an embryonic zone in the middle.

[Anonymous No.11761056]

>>11761044
>non-moving .gif
into the trash you go anon

[Anonymous No.11761073]

>>11761056
ok zoomer

[Anonymous No.11761154]

>>11761044
LOL the spirals don't match after only two rotations.

[Anonymous No.11761163]

>>11761013
1/3 = 0.33333...
0.3333... x 3 = 0.99999...
1/3 *3 = 1
ergo 0.9999...=1

[Anonymous No.11761166]

>>11761163
>1/3 = 0.33333...
Wrong

[Anonymous No.11761169]

If you accept that
1. All numbers can be represented by infinite decimal expansions in the form of a function f: N->{0,1,2,3,4,5,6,7,8,9, ".", "-"}, (for example, 0.99999... is the function f(0)=0, f(1)=. , f(2)=9, f(3)=9, f(n)=9 for all n>3 . 1 can be represented by g(0)=1, g(1)=., g(2)=g(3)=... = g(n) = 0). A sequence is valid if it has at most one "-" in front and exactly one "." somewhere in the sequence. We assume that every valid representation determines a unique number but not necessarily that every number has a unique decimal representation (for this is what we will want to disprove).
2. You can do arithmetic on numbers in expected ways.
Then you are forced to have
0.9999....=1
Proof:
By comparing each digit, we see that 1 is not less than 0.999....
Now subtract the two numbers (which you can do by assumption 2)
1-0.999... = x >= 0
As every number can be represented by an infinite decimal expansion (assumption 1), there is a sequence of digits for x.
If there are any nonzero digits of x after the dot ".", there is a first nonzero digit and it has a well-defined natural number position N. But clearly that cannot be, since then it would mean that x is bigger than some fraction 1/10^N. Thus every digit of x must be zero and so x itself is zero.
So 1-0.999... = 0
now again, by assumption 2., you can add 0.999... to both sides to get
0.999... = 1
QED

[Anonymous No.11761172]

>>11761169
Possible objections you might have:
1. Not all numbers have a sequential decimal expansion of the form f:N->{digits}. Evidently, an expression like 0.0...1 cannot be made into such a sequence because for every position in a natural sequence there are finitely many terms coming before it.
To make sense of an expression like 0.0...1 rigorously, you could instead define it as an ordinal sequence f:(w+1)->digits. w+1 is ordinal obtained by taking the natural numbers and adding one element to the end that is larger than every other element. (Look up ordinals here: https://en.wikipedia.org/wiki/Ordinal_number))
So 0.0....1 could be interpreted as f(0)=0, f(1)=".", f(2)=...=f(n)=0 and f(w)=1.
Now the problem still arises because not only you've gained a lot more numbers in this way, they would not be considered to be numbers by mathematicians because you cannot add, subtract them.
To illustrate this:
What's 10* 0.0...1? Intuitively, we shift the decimal point by 1 (do you have other suggestions?). But that would result in exactly the same representation, hence the same number 0.0...1! And we cannot have that because if 10*0.0...1 = 0.0...1, subtracting 0.0...1 we find 9*0.0...1=0 and so if we assume 0.0...1!=0, we can divide by it to find a blatant contradiction 9=0. So we see that even allowing nonstandard decimal sequences doesn't solve the problem: we need to be able to do arithmetic on numbers in expected ways and assuming 0.99...!=1 always leads us to a contradiction!
2. Not all numbers have decimal representations. In that case, there's not a lot to say here on my part except to ask what do you mean by a number then? Because in all these discussions a prevaling implicit assumption has been that numbers mostly ARE their decimal representations. "What's 0.999..? Obviously it's the number you get by writing 0 and 9999 repeating": there is no notion that it's just a notation that represents some number: it's a number itself.
cont.

[Anonymous No.11761176]

>>11761166
oh here we go again..

[Anonymous No.11761177]

>>11761172
<cont.>
So if you believe not all numbers have decimal representations, please provide your own definition of a number. I assume by numbers you don't mean equivalence classes of Cauchy sequences or Dedekind cuts, because in those formulations 0.999... = 1 is a trivial theorem.
3. You cannot do arithmetic in expected ways on all the numbers. In that case, there is nothing left to argue about, it's all just a matter of a definition of what you mean by numbers. For me and most mathematicians, numbers are something you can add, subtract, multiply (although not necessarily divide in themselves, e.g. the integers are not closed under division). Although, appealing to commons sense, in what sense are your numbers "real" if you cannot even add, subtract, multiply them?

That's it for now.

[Anonymous No.11761182]

[Anonymous No.11761196]

>>11761013
>>11761038
ya'll making it harder than it needs to be

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

[Anonymous No.11761197]

https://youtu.be/x-fUDqXlmHM

>...99999= -1
>...99999.999...= 0
Saying that 0.999...= 1 makes as much sense as the previous two statements

Mathematics is not a perfect language. Things can be "proven" mathematically, but still not be correct and we know this because there exist inherent flaws in maths itself, no matter how few. Anyone who believes that 0.999...=1 must also believe those first 2 extremely absurd statements as well.
>inb4 pseuds fight tooth and nail to somehow justify them as correct
0.999... is infinitesimally close to 1 but will never be equal to 1. Anyone who believes otherwise is insane.

[Anonymous No.11761203]

>>11761154
But they obviously converge to that limit.

[Anonymous No.11761206]

>>11761166
>>11761029

[Anonymous No.11761215]

>>11761206
>if i bang my head against my keyboard maybe they will think i'm correct

[Anonymous No.11761217]

>>11761196
>clunky

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

[Anonymous No.11761220]

>>11761215
>>11761042

[Anonymous No.11761221]

>>11761197
>Things can be "proven" mathematically, but still not be correct
what do you mean ? what exactly is "correct" here ? give examples of statements which can be proved but aren't correct please.
>because there exist inherent flaws in maths itself, no matter how few
give examples of flaws in maths please.

[Anonymous No.11761226]

>>11761197
Highly based video. I knew the result ...9999 = -1 in the 10-adics and generally in other adics but never thought of it as a counterpart for .9999... = 1.
The reason why is also top-notch.

[Anonymous No.11761232]

>>11761217
>mashing all that into 1 line with fifty = signs
i have a fit when peas and corn mix, as if im gonna write it like your abomination of a proof

[Anonymous No.11761234]

>>11761197
You completely missed the point of the video. Pretty fitting for a 0.999...=/=1 tard.
He definitively proved that as long as 0.999... has an answer, i.e. represents an actual number, then it's 1. The same with ...9999 =-1. There is absolutely no contradiction here.
What you're saying is that
>0.999... is infinitesimally close to 1 but will never be equal to 1
But in this you're assuming that 0.999... DOES have an answer. But then it must be equal exactly 1.
Basically, you're retarded.

[Anonymous No.11761235]

>>11761196
Based.
Using simple algebra to solve something driving people mad.

[Anonymous No.11761241]

>>11761197
>0.999... is infinitesimally close to 1
what does it mean for two numbers x and y to be infinitesimally closed ? give a precise definition please and prove that it holds for 0.999... and 1. no handwavy bullshit, I want math

[Anonymous No.11761248]

>>11761232
sniff some more glue, it's working

[Anonymous No.11761252]

>>11761248
huh?

[Anonymous No.11761275]

>>11761008
Holy shit retard, what is one third plus one third plus one third?
Now do the same problem with decimal notation instead of fractions.
You are literally too fucking stupid to live.

[Anonymous No.11761277]

>>11761196
>1
>1 plus an infinite string of 0s
imagine the coping mechanism you'd need to believe it's the same thing

[Anonymous No.11761287]

>>11761277

[Anonymous No.11761296]

>>11761275
>3/3
reduces to 1
>0 followed by infinite string of 9s
reduces to 1 followed by an infinite string of 0s

Lmao.

[Anonymous No.11761316]

>>11761296
dumb chucklefuck

[Anonymous No.11761321]

>>11761203
What is "they?" The shell clearly diverges from the golden spiral as you move out or inwards.

[Anonymous No.11761337]

>>11761316
what I love most about you copers is how you all have your favorite clips of nonsensical coping.
like victims of a religious cult all parroting your favorite bits of holy scripture.

>>11761217
you even argue over which bible verse is better lol

[Anonymous No.11761340]

>>11761337
>and not a line of math was seen
typical inbred

[Anonymous No.11761342]

>>11761337
>"dumb chucklefuck" is a line of math
cope harder

[Anonymous No.11761346]

>>11761340
>"dumb chucklefuck" is a line of math
cope harder

[Anonymous No.11761352]

>>11761342
>>11761346
>stammering
lol

[Anonymous No.11761357]

>>11761352
Ok ranjeesh

[Anonymous No.11761374]

>>11761352
>i have no argument but you replied to the wrong post, therefore my batshit assertion that
>> 3/3 reduces to 1
>> implies
>> 0 + infinite string of 9s reduces to 1 + infinite string of 0s
>is true
>smuggie.jpg

Lmao the absolute state of your coping mechanism

[Anonymous No.11761401]

>>11761374
>>> implies
your whole argument was that 0.999... is not 1.000... because they look different. so 3/3 = 1 doesn't imply 0.999... = 1, but it does imply that your logic is retarded.

[Anonymous No.11761428]

>>11761401
>because they look different
lolwut
how much cognitive dissonance must you be in right now? that in order to cling to your batshit 3/3 analogy, you have to invent nonsense i didn't say and attribute it to me

[Anonymous No.11761524]

>>11761428

[Anonymous No.11761534]

>>11761524
yep that's exactly how cogdis works.
your coping mechanism converts that text into "they look different" and you get a false memory of nonsense

here's your argument
>(1+2) / 3 = 1
>therefore 1 = infinite string of 9s
>*coping intensifies*