/sci/ - Science & Math

It's false because 0.999... is not smaller than 1.

>>12267404
why? it seems true to me 0.xxx is always smaller than 1

>>12267406
It isn't. The confusion exists because our way of representing numbers sucks for these cases.

>>12267393
Define ...

>>12267393
>0.99999999999999999... < 1
False. Use the contrapositive of the density of reals to show this isn't true.
>1 < 1.00000000000000000....0001
True. Since you have numbers following the..., you are inadvertantly asserting the ... is a finite number if 0's, since the sequence is terminating (at a 1). So yes, that will be greater than 1.

>>12267393
0.9...=1=1.0...1

0.1=10^-1
0.01=10^-2
0.001=10^-3
:
0.0...1=10^-inf=0

>>12267435
>>12267411
>>12267413
>>12267432
CAn you explain this? 10^-inf does not seem very rigorous. I agree that if you take the limit it approaches the same, but the limit essentilly just says "these are close enough", not that they are the same, thats at least how it is with epsilon-delta and such

>>12267447
>the limit essentilly just says "these are close enough", not that they are the same
awww, it's retarded

>>12267447
schoolkids use it every day all the time
[math] \displaystyle
\lim_{x \to \infty} \dfrac{x+1}{x} =
\lim_{x \to \infty} 1+ 1/x = 1+0 = 1
[/math]

>>12267447
0.999... and 1 aren't "close enough". They're the exact same value represented differently.

what about 1.000000.....000

>>12267413
this
>>12267447
[math]\infty[/math] is not a number kek
this is why mathematician hate decimal representation

>>12267483
anything after the ellipsis = 0 and can be removed

>>12267393
>1 < 1.00000000000000000....0001
True. The difference is 0.00000000000000000....0001
>0.99999999999999999... < 1
False. The difference is 0.00000000000000000...

Prove < =/= =

You can define "numbers" as functions
[math]f:\xi \to \{0,1,2,3,4,5,6,7,8,9,.\}[/math]
Where [math]\xi[/math] is an ordinal and define
[eqn]f<f' \iff f(y)<f'(y) \\
\text{where } \ y= \min \{x: f(x)\neq f'(x)\}[/eqn]
(if the domain ordinals are different, we extend the domain to the larger one by zeros).
With this definition,
0.999...., 1, 1.000....001 are indeed well-defined objects and satisfy the order relations in the OP.
In fact you also have
1.000....1 > 1.000....01 > 1.000...001 > 1.000....0001
It's all well and dandy. The problem is that with these "numbers" you can't do arithmetic, that's why mathematicians don't take them seriously.
For example you have no good way to define
1.000...5 + 1.000...5
These two numbers are greater than 1 so we expect their sum to be greater than 1. So there must come a place where the sum is greater than 1, but if it does than it will be too large for our needs. There is no ordinal number immediately before [math]\omega[/math], we say it's a limit ordinal as opposed to a successor ordinal.

>>12267490
>difference is 0.00000000000000000....0001
[math] \displaystyle
0. \bar{0}1
= \lim_{n \to \infty} 0. \underbrace{0 \dots 0}_{n ~ \text{times}}1
= \lim_{n \to \infty}
\left [
\left (
\sum_{k=1}^n \dfrac{0}{10^k}
\right )
+ \dfrac{1}{10^{n+1}}
\right ]
=0
[/math]

>>12267488
well just imagine that there's something just a little bit higher than 0 but lower than 1 at the end

>>12267496
>1.000...5 + 1.000...5
1+1

>>12267503
0.5/inf = 0

>>12267504
1+1 is just 2
We want our arithmetic to be such that
a<b, c<d => a+c<b+d.
So we want 1.000...5 + 1.000...5 > 1+1.

>>12267510
5/inf = 0
1.000...5 = 1

>>12267512
Dude are you retarded? I am not talking about the real numbers here. Read
>>12267496

Cirno already confirmed this.

>>12267532
>not talking about the real numbers
ok, idgaf then
feel free to disappear up your own asshole

>>12267500
No, the "...." in
>1.00000000000000000....0001
can't represent an infinite string of zeros, because the string clearly ends in a one.

The only representation that makes sense is for "...." to represent an arbitrarily long, finite string of zeros. So the difference between the string in question and the number 1 is an arbitrarily long, finite string of zeros that ends in a one: 0.00000000000000000....0001

>>12267393
The thing is, that technically only only one of those 3 numbers exists.
0.999...999 is either finite, or is still missing a few 9s at the end.
Same with 1.000...001.
How much is 0/1? Because that's the difference between those 3 numbers.

>>12267496
>a limit ordinal as opposed to a successor ordinal
The reason this doesn't work for number values is actually very intuitive and visual. It makes sense to put a new digit next to a string of digits that has no end. You're not doing anything contradictory. It doesn't make sense to put a new digit at the end of a string of digits that has no end, because the place you're trying to put it in doesn't exist by definition.

>>12267638
>It makes sense to put a new digit next to a string of digits that has no end
>It doesn't make sense to put a new digit at the end of a string of digits that has no end
Dude what lmao

>>12267638
>It doesn't make sense to put a new digit at the end of a string of digits that has no end
It also doesn't make sense to have a string of these threads that have no end, but ehh.

>>12267654
?
If somewhere doesn't exist, you can't put something there.

>>12267691
But it does. It makes sense to put the a new digit at the end of a string of digits. That's what the ordinals are for.
The new digit is the last digit, it's at the end.

>>12267393
Yes, it's less, distance is 0.

>>12267696
It can't be at the end of something with no end. You're imagining a new digit adjacent to something with no end, not at the end. That's the whole point of [math]\omega+1[/math] notation. The 1 isn't the last element of the [math]\omega[/math] string, it's the first element of a new string.

>>12267736
Ok I see what you mean. You can't put it next to the last digit because there is no last digit (no end) but you can still put it so it's at the end of the digits. In any case, it makes sense to put the digit at the end (meaning in the resulting string the new digit is at the end).
For example, (0,1) in the reals has no end but you can put 1 so it is at the end. You can say that you put 1 at the end.
Similarly you can embed countable ordinals in Q while respecting the order.
The naturals could be
1-1/n, then you can put an ordinal at the end by adding 1. Nobody would deny that 1 is the last digit, and it's not just next to the numbers but the end of the numbers. It's quite continuous.

>>12267432
I can write 1 as 1.0 if I want, I can also write it as 1.0000 or 1.0000000000000000000000000000

>>12267393
Petition to ban "..." notation and replace it with "receding off into an infinite horizon" notation just so idiots stop trying to put shit at the "end" of it or stop saying it's "always" doing something over some duration of time.

>>12267777
>Petition to
Is reddit not good enough for you? Just go back bro, nobody wants you here.

>>12267557
>infinite has an end
hurr durr

>>12267788
Oh no I used a word. Cry about it.

>>12267777
0.9... = 0.(9) = [math] 0.\bar{9} [/math]

>>12267819
I'm aware. (So what the fuck is this " 1.000 ... 001 .. .101) bullshit people keep making up and making endless threads about.

>>12267839
It's called a troll.

>>12267839
as >>12267500 shows
anything after the ellipsis can be ignored
0.9...=0.9...6544786=1

>>12267803
>>infinite has an end
>hurr durr
I said the opposite. Infinite means no end.

>>12267754
>It's quite continuous.
It's not. It's specifically not continuous. That's the point.

>>12267393
The difference between 0.999... and 1 is 0

>>12267393
The only way that 1-0.999... can differ from 0 is if one of the digits of 0.999... eventually is something other than 9. By definition, it never does, so the difference is always 0.

>>12268256
>>12268270
These.
0.999... deniers deboonked by pure logic

>>12268070
It's connected under the order topology.

>>12268070
>>12268415
It would be discontinuous, i.e. disconnected space if instead of adding one at the end you add one, then add one before the last one but after all the others and do so for all natural numbers. Then it would be disconnected, because the original natural numbers and their complement would be two open sets.

>>12268415
>>12268422
It could be "connected" as in "somehow adjacent to," but not continuous or "at the end of."

If the digit at [math]\omega+1[/math] were continuous with the string described by [math]\omega[/math], it would simply be part of [math]\omega[/math], not a new thing.

>>12268270
thats not right tho, is it? 0.9 is not 1, 0.99 is not 1 0.999999999999999999 is not one. Just adding dots afterwards does not make it anything. No matter how many 9s its not one. Its not logical in the slightest

>>12267393

>>12269090
>let's pretend finite is infinite

>50 replies
Holy fuck it’s ALWAYS the SAME FUCKING BAIT, not even the daily “show your love for Hitler” thread on /pol/ gets this attention

>>12268550
>It could be "connected" as in "somehow adjacent to," but not continuous or "at the end of."
It's literally at the end of the whole string. It's the last element.
>If the digit at ω+1 were continuous with the string described by ω, it would simply be part of ω, not a new thing.
Doesn't follow.

>>12269173
>It's literally at the end of the whole string.
ω+1 isn't a string, it's literally two strings—one of which is countably infinite, the other of which consists of a single element.
>Doesn't follow.
It's has nothing to do with following, it's the whole structural point of ω+1.

>>12269151
>Hitler
nein nein nein nein...

>>12267447
>but the limit essentilly just says "these are close enough"
t. doesn't know what limit is

>>12267393
>1.00000000000000000....0001
brainlet

People who argue about this shit are moronic.

Look, its time to put an end to this nonsense. Get one thing straight, 0.999... DOES NOT EQUAL 1.

Everyone with half a brain knows that. However its a convenient Band-aid, to cover up the inadequacy of current mathematics. So we accept it, knowing that its wrong, simply because we dont have any mathematical system that does a better job at the moment.

Think of it this way. Say some guys designed an airplane that they say relies upon pink elephant flying power. They build it, it flies. Not perfectly, but it gets the job adequately done. Some other dude comes along and points out that pink elephants dont exist. The inventors turn around and say "We know, but it works for now, so until you got some better explanation, or until you build a better airplane, kindly please do fuck off"

>>12273201
gr8 b8 m8

>>12273201
1/inf=0

>>12267393
Your brain is too small to comprehend infinity, just quit it.

>>12267393
>The year 2020
>People still dont accept MOAN as their Lord and Savior of Mathematics
>MOAN is "Mother Of All Numbers"
>Its the largest number possible. It lies at the very end of the number line.
Hurr Durr! MOAN + 1 !!!
> No, fuckhead, it is not possible to add anything to MOAN. Its the largest number possible. By definition. You do understand "definition" dont you? No? Then lets make an analogy your poor stupid little brain might actually grasp
>Any attempt to add anything to MOAN results in you sailing off the edge of existence, to be devoured by trans-dimensional sea monsters.

>>12273234
>Its the largest number possible
nope, it's larger than any real number

>>12273242
Moan is my concept, bitch, I will define it however I like. Go make your own construct.

>>12273234
I'm conceptualizing MOAN+1 right now and there's nothing you can do to stop me

>>12273253
I dont need to stop you. The trans-dimensional sea monsters will. Eat some parsley. They like their food seasoned.

>>12273244
no such thing

>>12273369
There is now, bitch. Deal with it.

>>12273376
>Deal with it.
I did >>12273369

>>12273382
Stay in denial.

>>12273390
ty, very comfy
https://www.wolframalpha.com/input/?i=largest+number+possible

Required reading for this thread:
-Baby Rudin, chapter 1 up to (including) 1.22.