So does 0.999...=1 or what?
In the standard axiomatic system in which the decimal expansion is formalized, this is a theorem, yes.
/thread
x = 0.999...
10x = 9.999...
10x-x = 9x = 9
x = 9/9
x = 1
0.999... = 1
>>7882286
stop posting this. you're using the thing you want to prove in line 2.
>>7882292
No he's not. What is 10*9.999...?
>>7882292
no he doesn't
>>7882254
Infinity is a strange concept and almost all equations turn into a mess when they involve it. Just take a look at all this -1/12 nonsense blabbery.
>>7882254
tell me a number that fits in between the two and it doesn't
>>7882254
by the axiom of choice, there is a number between 0.99999... and 1.
Which means the two numbers are different.
>>7882454
I didn't think it was possible to pull something like that out of one's ass.
>>7882286
now i understand
>>7882706
What if you had an infinite set of asses?
>>7884279
Might be possible.
>>7882254
0.999... not equal to 1
0.999... gets rounded to 1
Imperfect real-life implementations of math in machines may sometimes do errors in some corner cases similar to this.
>>7884878
Your comment about machines is true but that doesn't make the statement false.
>9/10*(1 + 1/10 + 1/100 + ...) = 9/10[1/(1-1/10)] = (9/10)(10/9) = 1
Decimal expansions in general are not unique.
>>7884896
Perhaps I should add 0.999... = 9/10*(1 + 1/10 + 1/100 + ...)
>>7884896
9/10*(1 + 1/10 + 1/100 + ...) = 9/10[1/(1-1/10)]
These are not equal
It is the same thing as 0.999... = 1
It is simply not true.
>>7884910
what are geometric series
>>7882358
> have nothing to say
> manage to stretch out my meaningless babble over a couple paragraphs
> try to get seem credible by referencing a meme number thats on numberphile
0.9999......, otherwise known as the wildberger constant, was discovered by none other than nj himself. It has been shown that there are finitely many rational numbers, but a lot of numbers nonetheless, that are less than 1 but greater than the wildberger constant.
>>7885106
As per one proof:
0.111... = 1/9
9*1/9 = 0.999...
0.999... = 9/9 = 1
But it would have implications as follow:
0.999... + 0.111... = 1/9 + 9/9 = 1
And moreover:
1 = 1/9 + 9/9
1 = 10/9
0 = 10/9 - 1
0 = 10/9 - 9/9
0 = 1/9
Logic breaks. Much better way would be saying that 0.999... ≠ 1 and it sometimes only seems so, because of limitations of the language.
>>7885106
>>7886440
Rather then saying that 0.999... = 1 which should be intuitively incoherent. I would acknowledge shortcomings of developed tools in mathematics and say that they fell short.
Anyway, I'm not really competent to argue over this. However, I acknowledge that it is much deeper and funnier problem that I first anticipated.
>>7886440
>0.999... + 0.111... = 1/9 + 9/9 = 1
you're retarded
>>7882254
You dumbass, you used a >my degree >any job I want >300k starting troll thread pic on a .999...!=1 troll thread.
>>7886536
You did not follow my logic.
if 0.111... + 0.999... = 1
and if 0.111... = 1/9
I believe we agree to this point.
so
0.111... = 1/9 | multiply by 9
0.999... = 9/9
This was according to proof that 0.999... = 1
Now we have:
0.111... + 0.999... = 1
and because 1/9 = 0.111...
and 9/9 = 0.999...
therefore 0.999... + 0.111... = 1/9 + 9/9 = 1
Conclusion, logic breaks.
>>7886547
Troll thread or not. This is a real dilemma.
https://en.wikipedia.org/wiki/0.999...
"The equality 0.999… = 1 has long been accepted by mathematicians and is part of general mathematical education. Nonetheless, some students find it sufficiently counterintuitive that they question or reject it. Such skepticism is common enough that the difficulty of convincing them of the validity of this identity has been the subject of several studies in mathematics education." -- Wikipedia (link above)
>>7886554
We don't agree to that point. I'll break it down for you, ass bag:
0.111... + 0.999...
= 0.0111... + 0.0999 + 0.1 + 0.9
= 0.0111... + 0.0999... + 1.0
see where this is going?
>>7882254
if [math] \displaystyle
\frac{1}{ \infty} = 0
[/math]
then yes
>>7886554
.111... + .999... Doesn't equal one
It would be more like
.000...1 (infinite zeroes and a one at the end, this is actually zero)
So
.000...1+ .999... = 1
Do the adding yourself, you're just an uncultured idiot, don't try to argue about stuff you don't know a thing about
>>7886582
Well, when you are right, you are right.
I confused my self at some point but I never claimed that I'm competent to argue over this.
>>7886594
Oh, you are too harsh to me. So I made one mistake. Granted, that it was quite simple mistake but still. We all argue here because none of us are competent to argue anywhere that actually matters. Besides this is a good place to practice and make stupid mistakes, so sue me.
>>7886554
>You did not follow my logic.
>if 0.111... + 0.999... = 1
>and if 0.111... = 1/9
>I believe we agree to this point.
10/9 = 1 ? really?
>>7886605
Yeah, yeah. Can we move pass this? So I did math in my head before my morning coffee and I overlooked.
[math] \sum\limits_{n=1}^{+\infty} \frac{9}{10^n} = 1 [/math]
>>7886593
I think people asking this question tacitly assume we are considering the real numbers, do not contain [math]\infty[/math].
sure, trade me my .99 cents to your dollar a couple times
If I had a scale and it was perfectly accurate and I put a weight that weighed less than 1kg, then the dial would read something that wasn't 1kg.
So, 0.999... cannot be 1kg now can it? I mean, 1kg is 1kg, its not >1kg, its not <1kg - its 1kg.
So, 0.999... must be <1
>>7886711
0.99 ≠ 0.999...
"The equality 0.999… = 1 has long been accepted by mathematicians and is part of general mathematical education. Nonetheless, some students find it sufficiently counterintuitive that they question or reject it. Such skepticism is common enough that the difficulty of convincing them of the validity of this identity has been the subject of several studies in mathematics education." (https://en.wikipedia.org/wiki/0.999...)
Consider this.
1/9 = 0.111... | multiple both sides by nine
9/9 = 0.999... = 1
I did not believe it either at first, but did some surfing and... just read the wiki page.
>>7886747
Common mistakes from the wiki page.
You interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length.
Seeing two manifestly different decimals representing the same number appears to be a paradox, which is amplified by the appearance of the seemingly well-understood number 1.
Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.
>>7886826
Both can be correct it literally only depends on how you define 0.99...
If it is a real number the identity is immediate by the nested sphere theorem
If it is a hyperreal number, define it as 1-h : 0<h<a for all real numbers a. That's not a metric space though
Fumbling around with ambiguous notation and those dubious dot-dot-dots as a key part of your argument and not stating your assumptions is not kosher though
>>7886905
I guess you are right. It can be defined different ways. It is interesting though, don't take the fun out of it. Luckily it is my day off.
>>7886905
0.999... < 1 so it is amount of infinitely small number smaller.
0.999… + h = 1
0.999… = 1 – h
So intuition is correct and there is infinitely small number that fits between. But it can be defined other ways too in different situations and sometimes it actually seems to be equal to 1, cool. How often one gets actually exited about math.
>>7886896
No I don't, I just know that anything that's not 1 is either >1 or <1 and that 0.999... Is lower
All this math bullshit, Christ...
>>7886951
At some point, no matter what you pick for error margin, 0.999… will get close enough to satisfy us mathematically.
With limits, if the difference between two things is smaller than any margin we can dream of, they must be the same.
>>7886949
>So intuition is correct
You're missing the point. The intuition can be *made* correct by an appropriate choice of definition. However we don't really do this because including h and its sums and differences with real numbers requires us to throw out the idea of well-defined distance, which is foundational to analysis (calculus)
There are of course alternative approaches to analysis that use the hyperreals and they're pretty interesting
>>7886984
I was referring to hyperreals when I said that intuition is correct.
>So intuition is correct and there is infinitely small number that fits between.
So that is not missing the point. I also posted the Archimedean property post.
> Our current number system assumes the long-standing Archimedean property: if a number is smaller than every other number, it must be zero. More simply, infinitely small numbers don’t exist.
Acknowledging the fact that in normal circumstances there is no infinitely small or big numbers.
So, how am I missing the point? I still feel that intuition is correct and there's infinitely small number between and you said it yourself that it can be looked other ways too.
> The intuition can be *made* correct by an appropriate choice of definition.
> There are of course alternative approaches to analysis
Where is the contradiction?
>>7886984
I still don't get why 0.333... < 0.34 if 0.999... = 1. So I guess you are at least half right by saying that I'm missing the point.
The main problem here is people writing stuff like 0.999...
As soon as you write it properly, for example 1-10^(-x) for x tending to infity, or as >>7886673, it's properties are easier to describe.
>>7887027
Thank you! That is more clear. Suddenly I get it.
>>7887093
[math]\lim_{x\to +\infty} 10^{-x} = 0[/math]
by summation,
[math]\lim_{x\to +\infty} 1 -10^{-x} = 1[/math]
keep in mind we're talking limits and not actual numbers.
>>7887025
What people aren't giving you when they "prove" 0.999...=1 is the rigorous proof. This method answers your question (which is a good one).
[math]\sum_{n=1}^{\infty} \frac {3} {10^n} = \frac {1} {3} < \frac {34} {100} [/math]
These infinite decimal expansions need infinite geometric series to be reasoned with.
>>7882254
>>7886996
>young child: "1+1=2"
>you: "Well, ACTUALLY, if you're working in the integers modulus 2, 1+1=0. So it depends on the situation, but sometimes 1+1 actually seems to be equal to 2."
Don't be pedantic. There's no reason at all to assume anything other than the real numbers when a question like this is stated. And if you are working with hyperreals or something, it becomes much less clear how you should even interpret "0.999...".
>>7882264
this
i can't believe people are retarded enough to argue over this
>>7882454
>axiom of choice
>using a meme to disprove another meme
>>7886593
Would that be a countable or uncountable infinity my good sir.
>>7882254
yes op, our system of writing down numbers is not that great, you can represent some of them in more than one way if you really insist.
you're free to try to invent a better one if this current one confuses you so much
(or don't, why ruin the weekly 0.999... threads)
>and I lied, it's actually pretty great, if you want to actually start to appreciate it just try to solve some equations using roman numerals and shit
>>7886440
Wat a faggot.
>>7882286
line 3 doesn't look right to me. Isn't this where you get a 9 + 0.0000.........1 ?
>>7889119
>x = 0.999... | *10
>10x = 9.999... | (-x)
>10x-x = 9x = 9 | /9
It seems off due the lack of markings but it is not. He subtracts with x which leaves 10x-x=9x and other side looses the decimals because x=0.999..., leaving 9x=9.
>x = 9/9
>x = 1 | x=0.999...
>0.999... = 1
>>7889119
More simple way to do the same thing however is following key part.
1/9 = 0.111... | *9
9/9 = 0.999...
1=0.999....
But like it has been said many times before, much better way to look at this is some of these:
[math]\sum\limits_{n=1}^{+\infty} \frac{9}{10^n} = 1[/math]
[math]\lim_{x\to +\infty} 1 -10^{-x} = 1[/math]
>>7887140
so why don't infinitely small numbers exist?
if numbers can expand into infinity in both negative and positive directions, why can't they expand in a direction approaching but never equaling 0?
Not being contrarian, just bad at math
A better question is does 3.999999....... equal 4?
>>7889603
They can, but real numbers contain neither infinite numbers nor infinitesimals. There are number systems that do, but they aren't what people have in mind when posting this meme.
>>7889606
Let me put it this way.
When x approaches infinity, 4-10^(-x) approaches to 4.
[math]\lim_{x\to +\infty} 4 -10^{-x} = 4[/math]
>>7889606
Can you think of a single number between 3.99... and 4? There isn't any, therefore they are equal.
>>7889606
https://latex.codecogs.com/gif.latex?3 + \sum\limits_{n=1}^{+\infty} \frac{9}{10^n} = 4
>>7882254
1/3 =0.333...
3/3 = 1
1/3 + 1/3 + 1/3 = 0.999... or 1
>>7889636
The element 4-h of R*, where R* denotes the extended reals and h is the hyperreal element
This literally proves mathematicians to be autistic
IF THE 9S GO ON FOREVER THEN THAT MEANS THEY NEVER BECOME 1
THATS LITERALLY WHAT FOREVER MEANS. INFINITE 9S
0.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
IF THOSE 9S GO FOREVER, HOW THE FUCK WILL IT BECOME 1??????????????
>>7886747
>a scale and it was perfectly accurate
that's where your logic fails, real life comes in quanta of 'stuff', you can't break up matter into infinitesimal parts.
0.999... falls infinitesimally short of 1 but since standard real number system has no infinitesimals the number equals 1.
this is why I don't think 0.999... should belong in real numbers.
>>7889906
Infinitely close is the same thing as equivalent on the real number line.
>>7889906
Yes, you forget one thing though. I didn't get it till yesterday either. What you forget is how numbers are defined in mathematics and the rule is that there is no infinitely large or small numbers. This is not necessarily easy thing to swallow, I think, but if one ponders about it enough, it starts to make sense. There are hyperreal numbers which allow one to calculate with infinities.
I found really good page yesterday for those who want to understand this better.
Page's title is: A Friendly Chat About Whether 0.999… = 1
http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/
>>7889603
That image is a little off by saying "infinitely small numbers don't exist." Instead, what it should say is that this particular number:
0.000 ... 01
doesn't exist. That notation above just can't work because what it is trying to say is "write an infinite number of zeros, then put a one at the end." You can't ever finish writing an infinite number of zeros, so you can never get to the point where you write the one. The one can't ever exist.
What that number is incorrectly trying to express is:
[math]\lim_{x \rightarrow \infty} \frac{1}{{10}^x}[/math]
And there is no getting around the fact that that limit is exactly zero.
>>7889906
Please define infinity rigorously before talking about infinity.
A decimal with a sequence of repeating digits, say of size k, is represented by the limit as some upper bound n approaches infinity of the sequence of partial sums (d/10^(k-1))/10^ki where i is the sum index and d is the sequence of digits which repeats.
So in our case, .999999... is the decimal we wish to represent, and the sequence of repeating digits is just "9". Now, we have that .9999= the limit as n approaches infinity of the sum from i=1 to n of 9/10^i. Now this clearly converges to 1 by the delta/epsilon definition. Hence, .99999...=1.
I hope reading this without latex was as painful for you as reading your post about infinity was painful to me.
>>7882358
>2016
>he thinks the Riemann conditional convergence theorem is complex
>>7886440
Why do people take such shitty bait. People have to be smart enough to realize that 10/9=|=1, right?
>>7889840
I should have said real, well done
If 0.9999... can exist
Why can't 0.00....1 exist?
Seems to me that all of your '''''proofs''''' are just autistic cherry picking
>>7890124
no, in standard number system 0.999... is just another way to write 1 or 'one' or 1/1
>>7889906
>IF THOSE 9S GO FOREVER, HOW THE FUCK WILL IT BECOME 1??????????????
Heine-Borel theorem
>>7890139
it's called infinitesimal and they're not part of standard system.
https://en.wikipedia.org/wiki/Infinitesimal
>>7890124
if 0.999... does not equal 1 then there must exist some number a such that.
1 - a = 0.999...
Using your previous reasoning that the nines go on forever then a must look something like this.
a = 0.000...
With presumably a one at the end. But since the nines go on forever so to do the zeros so we can never write a one. Therefore a cannot exist and our assumption that 1 does not equal 0.999... has been shown to be fundamentally incorrect.
>>7886580
Some middle school students don't get it, it's no dilemma.
Maybe their teachers aren't good enough
>>7890196
it's not pointless since in math we're not always talking about real world things.
but yes, the whole thread is pointless since it's not a real mystery but a gimmick to capture the attention of newb whose intuition is wrong (because they do think it real world terms).
>>7890199
doesn't that mean that math is autistic?
>>7882254
the series equivalent of 0.999 converges to 1.
>>7890196
I think that's the suspicion but I'd argue we don't and may never understand the universe so fundamentally.
>>7890200
does that mean you don't value things that this 'autistic' math has given you?
like for example, computers.
>>7886826
>1/9 = 0.111... | multiple both sides by nine
>9/9 = 0.999... = 1
That's how I show it to my pupils, also:
>0,99999... is just a dumber way to write one
Of course one has to posit that
0,1111...*9=0,999999...,
that is, the multiplication algorithm for finite many decimals also holds for infinite. 7th graders don't have issues with this, have you?
>>7886949
That h is an infinitesimal, but we're not talking hyperreals here
>>7890218
gtfo
>>7890218
>What is programming?
>>7890218
>computers are physics
Math is the language of physics you dumb nigger.
>browse sci for the first time
>this thread
fuckin wew lads
>>7890212
>>7886949
That h is an infinitesimal, but we're not talking hyperreals here
That was my sudden realization moment. I was referring to hyperreals and if you look one post deeper, you'll notice that I was answering to post that precisely told me the difference between hyperreals and real numbers.
Nevertheless this seems to be counterintuitive as many people allege (me yesterday but not today). I was just pointing out that there is some logic in that intuition and there is, it is called hyperreals.
>>7890218
You're correct, maths has not given you anything.
Otherwise you wouldn't be so moronic.
>>7889603
>so why don't infinitely small numbers exist?
that's called 0 you philistine
>>7890239
>maths
>culture
autism is like the opposite to culture
>>7890244
there is literally nothing wrong with being autistic
>>7882264
You can't /thread your own post retard
>>7890232
Get used to this thread. Someone always posts it.
>>7882286
x = 0.888...
10x = 8.888...
10x-x = 8x = 8
x = 8/8
x = 1
0.888... = 1
>>7892147
>10x-x = 8x
wew lad
>>7892147
>Notating arithmetic in plaintext
>10x-x = 8x = 8
>0.888... = 1
Decent bait, got me to reply.
>7892147
>x = 0.888... | *10
>10x = 8.888... | -x ; x=0.888...
>10x-x = 9x = 8.888... - 0.888...
>9x = 8.888... - 0.888...
>9x = 8 | /9
>x = 8/9
>x = 0.999... | *10
>10x = 9.999... | (-x) ; x=0.999...
>9x = 9 | /9
>x = 9/9
>x = 1 ; x=0.999...
>0.999... = 1
>>7890232
Every.
Fucking
Week.
welcome to my personal hell
>>7892147
>10x-x = 8x
sup brainlet!
