🧵 /pnrg/ - point nine repeating general
Anonymous at Mon, 6 May 2024 19:04:13 UTC No. 16162687
Discuss the properties of the mysterious number 0.999... and its elusive relationship with one. Not much is known about 0.999... and I hope to encourage civil discussion to push the limits of our collective knowledge on this topic.
>What's so interesting about 0.999...?
In recent decades, the study of 0.999... has led us to a number of discoveries and applications in unexpected places, such as differential geometry and combinatorial game theory. 0.999... remains an active research topic and a treasure trove of mathematical insight, but there is still so much we have to learn about 0.999....
Central to the study of the 0.999... is perhaps one of the most important unanswered questions of mathematics: the unity conjecture. The unity conjecture states that 0.999... = 1. As of now the unity conjecture remains to be proven or disproven in any definitive way.
>Isn't 0.999... already proven to be equal to 1?
There are many naive proofs circulating online that 0.999... = 1. But there is still a lot of debate on this topic, so the issue is not settled. Many of these naive proofs have been debunked.
>You can obviously tell 0.999... is less than 1, just by the way it's written.
Numbers are not necessarily as they seem. 1.0 is equal to 1, even though they are written differently. There are easy ways to show that 1.0 = 1, but the case of 0.999... is not so trivial.
>I have proven/disproven the unity conjecture!
Feel free to share your results here, and your peers will debate the validity of your argument.
Anonymous at Mon, 6 May 2024 19:29:03 UTC No. 16162744
Claim: [math] 0.999_{\dots} \neq 1[/math]
Proof: We use induction. The base case is trivial: [math] 0.9 \neq 1[/math]. Next we introduce the notation that [math]0.9_n = \underbrace{0.9999999}_{n-\text{man
Now the inductive step: we assume [math]0.9_n \neq 1[/math]. Then trivially [math]0.9_{n+1} \neq 1 [/math]. It might help to notice that [math] 1 - 0.9_{n+1} \neq 0[/math].
This implies that [math]0.9_n \neq 1 \qquad \forall n\in \mathbb{N}[/math]
Finally, we define [math] 0.999_{\dots} := \lim_{n\to\infty} 0.9_n[/math].
[math]\therefore 0.999_{\dots} \neq 1 \qquad \square [/math]
Anonymous at Mon, 6 May 2024 20:05:19 UTC No. 16162849
>>16162687
>Not much is known about 0.999...
>But there is still a lot of debate on this topic, so the issue is not settled.
Remember anon, the fact that you don't know something doesn't mean it isn't known, and the fact that people are arguing about it doesn't mean it isn't settled among the knowledgeable.
Anonymous at Mon, 6 May 2024 20:06:30 UTC No. 16162854
3/3 = 1
3 * 1/3 = 1
1/3 = 0.333333333...
3 * 0.33333333333... = 0.999999999
..
0.99999999.. = 1
/thread
Anonymous at Mon, 6 May 2024 22:50:47 UTC No. 16163078
>>16162849
Profoundly misinformed for how confidently you say it
Anonymous at Mon, 6 May 2024 23:00:02 UTC No. 16163100
>>16162854
[eqn]\begin{align}
x &= 0.999... \\ 10x &= 9.999... \\ 10x - x &= 9.999... - 0.999... \\ 9x &= 9 \\ x &= 1
\end{align}[/eqn]
/also thread
you idiots will argue about anything.
Anonymous at Mon, 6 May 2024 23:02:59 UTC No. 16163109
>1/9 = .1111...
>2/9 = .2222...
>3/9 = .3333...
>4/9 = .4444...
>5/9 = .5555...
>6/9 = .6666...
>7/9 = .7777...
>8/9 = .8888...
>9/9 = .9999...
Garrote at Mon, 6 May 2024 23:04:28 UTC No. 16163114
>>16162744
What is true of all elements of a sequence might not be true of the number that the sequence converges to.
Anonymous at Mon, 6 May 2024 23:30:24 UTC No. 16163146
Imagine thinking about number in numeral and decimal and not geometrically
Anonymous at Tue, 7 May 2024 00:24:49 UTC No. 16163222
>>16163179
this has been refuted. see
>>16162744
Anonymous at Tue, 7 May 2024 00:29:18 UTC No. 16163227
>>16162744
induction only works for finite indices n
>but cant induction show that something is true for all (positive) integers?
yes, but integer are finite by definition. there is no index n such that [math]0.9_n = 0.999... = \lim_{n = 1}^{\infty}\frac{9}{10^n}[/math]
>>16162854
>>16163100
you have to show that the sequence converges before youre allowed to do arithmetic on it.
>>16163109
see above, and all you need to prove that 1/9 = 0.111...
Anonymous at Tue, 7 May 2024 00:36:51 UTC No. 16163248
>>16163227
>and all you need to prove that
and also*
Anonymous at Tue, 7 May 2024 00:53:17 UTC No. 16163272
>>16163227
by this argumentation, limits don't exist and therefore 0.999... is not a number.
Anonymous at Tue, 7 May 2024 01:50:31 UTC No. 16163356
>>16163272
i dont think you understand what a limit is
Anonymous at Tue, 7 May 2024 01:53:12 UTC No. 16163361
>>16163356
Nah that's you. Limits are NOT equalities.
Anonymous at Tue, 7 May 2024 01:54:53 UTC No. 16163363
>>16163361
they are if you write an equal sign next to them. what do you even think the purpose of limits are?
Anonymous at Tue, 7 May 2024 02:06:27 UTC No. 16163380
>>16163363
You = Faggot.
QED
Anonymous at Tue, 7 May 2024 06:02:02 UTC No. 16163617
1/9 = 0.111...
+
8/9 = 0.888...
=
9/9 = 0.999...
Anonymous at Tue, 7 May 2024 06:21:38 UTC No. 16163627
>>16162687
Is this where the midwits congregate?
1 and 0.999... are two ways to write the same number, much like 1/2 and 2/4.
Anonymous at Tue, 7 May 2024 06:24:32 UTC No. 16163631
>>16163627
A claim that is commonly repeated without satisfactory proof. It's known that 1 and 1.0 are two ways to write the same number, but whether 0.999... and 1 are the same number remains to be shown. The unity conjecture is still an open question.
Anonymous at Tue, 7 May 2024 08:05:56 UTC No. 16163677
>>16163631
it is impossible to satisfy those that can't see the error on their ways, and as such it is futile to argue with the pigeon's known as finitists
Anonymous at Tue, 7 May 2024 09:40:46 UTC No. 16163737
[math] \displaystyle
1= \dfrac{3}{3}=3 \cdot \dfrac{1}{3}=3 \cdot 0. \bar{3}=0. \bar{9}
[/math]
Anonymous at Tue, 7 May 2024 11:13:49 UTC No. 16163802
>>16163627
>the same number, much like 1/2 and 2/4.
Those aren't the same number though. The first says in two events one hits. The second says in four events two hits.
Think of it this way. Say the denominator is the amount of women you've fucked and the numerator is the amount of women you've given an orgasm to. Clearly 1≠2 and 2≠4.
Anonymous at Tue, 7 May 2024 11:21:44 UTC No. 16163806
What does point nine repeating mean?
Anonymous at Tue, 7 May 2024 14:31:27 UTC No. 16163997
>>16163100
9.999... - 0.999... isn't 9, it's 8.999...1
Make up a definition at Tue, 7 May 2024 15:24:23 UTC No. 16164071
>>16162687
To answer questions about infinite decimals we must first define what infinite decimals mean. We intend to have 0.000... = 0 and 0.999... = 1. If we reveal one digit of an infinite decimal at a time, at each step we should gain more precise knowledge of its value:
A number starting 1. is at least 1.000... (1) and at most 1.999... (2).
A number starting 1.4 is at least 1.4000... (1.4) and at most 1.4999... (1.5).
A number starting 1.41 is at least 1.41000... (1.41) and at most 1.41999... (1.42).
Therefore we shall define an infinite decimal to represent the real number which for each of its finite truncations (e.g. 1.414... -> 1.4) is at least as large as the truncation (e.g. 1.4) and at most as large as the truncation incremented in the final digit (e.g. 1.5). The reader may verify that this defines a sequence of nested intervals with rational endpoints whose lengths are eventually smaller than any positive rational number, and therefore defines a unique real number according to the definition of real number we are using (the nested interval construction).
Applied to 0.999..., our definition states that the number is at least 0 and at most 1, at least 0.9 and at most 1.0, at least 0.99 and at most 1.00, and so on ad infinitum. The real number 1 meets all these conditions, and thus 0.999... = 1 as desired.
If you disagree with this definition, you are free to make up your own, but be aware this is the definition we will be using on the exam.
Anonymous at Tue, 7 May 2024 15:31:43 UTC No. 16164080
Anonymous at Tue, 7 May 2024 15:33:31 UTC No. 16164084
>>16163631
>It's known that 1 and 1.0 are two ways to write the same number
They're not though. In reality, all numbers represent measurements are derived values from measurements. 1.0 and 1 have different significant figures, indicating different precisions. 1.0 for example may refer to 1.0 cm on a millimeter graduated ruler (I.e. is actually 10 mm). 1 may instead refer to 1 mL on a milliliter graduated cylinder.
Anonymous at Tue, 7 May 2024 15:36:45 UTC No. 16164089
>>16163100
This is a glowie thread for demoralization that gets spawned from time to time, that is why it is so tiring.
Anonymous at Tue, 7 May 2024 15:36:58 UTC No. 16164090
>>16164071
a self-correction:
>sequence of nested intervals
*sequence of closed nested intervals
This is important since our definition of real numbers does not admit sequences such as (0,1), (0.9, 1.0), (0.99, 1.00), ... .
>>16162744
>>16164080
That post does not justify the step [math](0.9_n \neq 1 \forall n\in \mathbb{N}) \implies (\lim_{n\to\infty} 0.9_n) \neq 1[/math].
Anonymous at Tue, 7 May 2024 15:39:15 UTC No. 16164094
>>16164089
It's called trolling, and it's a art. That said, posting copycat threads is not great art.
Anonymous at Tue, 7 May 2024 16:10:09 UTC No. 16164126
>>16164071
You haven't proven uniqueness of such a definition.
Anonymous at Tue, 7 May 2024 16:34:09 UTC No. 16164151
>>16164090
Sure it does. Lim just means for arbitrarily large N, which is allowed per previous step.
Anonymous at Tue, 7 May 2024 16:50:10 UTC No. 16164164
>>16164126
I did leave some small details for the reader to avoid getting bogged down in trivialities. What part are you having trouble with, the proof that the intervals are nested or the proof that their lengths are eventually smaller than any positive rational number?
🗑️ Anonymous at Tue, 7 May 2024 17:14:34 UTC No. 16164196
>>16164071
>we must first define what infinite decimals mean
do you even know what "positional notation" means?, because if you do then it is so straight forward that it is practically down hill
Anonymous at Tue, 7 May 2024 17:16:27 UTC No. 16164202
>>16164094
>posting copycat threads is not great art.
i'd honestly call it AI trolling
Anonymous at Tue, 7 May 2024 17:23:49 UTC No. 16164214
>>16164196
Defining a finite decimal as a finite sum is straightforward. If you want to define an infinite decimal as an infinite sum, then you must define an infinite sum. That can be done in a manner that agrees with the definition I gave in >>16164071, but I prefer the simpler, more concrete definition in that post.
Anonymous at Tue, 7 May 2024 20:17:03 UTC No. 16164551
>>16162687
the first 2 digits arent the same so they arent equal.
simple as
Anonymous at Tue, 7 May 2024 23:29:33 UTC No. 16164916
>>16163227
go fuck you you and your stupid retard argument I dont understand, anon who clearly knows more about maths than me
hah what a nerd
Anonymous at Wed, 8 May 2024 01:12:00 UTC No. 16165040
>>16163179
>A simple proof by induction
That's not a proof by induction, because there is neither a base case nor an inductive case. It's just a proof by arithmetic, and
> you have to show that the sequence converges before youre allowed to do arithmetic on it.
>>16163227
Anonymous at Wed, 8 May 2024 04:08:04 UTC No. 16165213
Anonymous at Wed, 8 May 2024 04:12:59 UTC No. 16165215
Just merge this with /mg/.
Anonymous at Wed, 8 May 2024 04:16:52 UTC No. 16165219
This board has the largest concentration of Dunning-Krugers I've ever seen. How did this place get to this point?