/sci/ - Science & Math

>>10532817
Two real numbers are not equal If and only IF there exists a third number that is strictly between them.
There does not exist any such real number that you can shove between .9999.... and 1, therefor they are equal.
Most of what you wrote is just nonsense, it is gibberish

>>10532817
>s would work in a base-11 number system
here it is in hexadecimal, or 16-base
[math] \displaystyle
\begin{align*}
1 = \left (\frac{15}{16} + \frac{1}{16} \right )
&= \text{0x0.F} + \frac{1}{16} \\
= \text{0x0.F} + \left ( \frac{15}{256} + \frac{1}{256} \right )
&= \text{0x0.FF} + \frac{1}{256}\\
= \text{0x0.FF} + \left ( \frac{15}{4096} + \frac{1}{4096} \right )
&= \text{0x0.FFF} + \frac{1}{4096} \\
= \text{0x0.FFF} +\left ( \frac{15}{65536} + \frac{1}{65536} \right )
&= \text{0x0.FFFF} + \frac{1}{65536} \\
&\vdots
\end{align*}
\\ \displaystyle
\Rightarrow \text{0x}0.\overline{\text{F}} = 1
[/math]

>>10532825
good proof but if OP is so fucking retarded that he doesn't realize base systems literally do nothing or change anything in math then there's no hope for him. Good for you to show this though

>>10532824
>two numbers are equal if there is no number between them
Simply repeating the claim isn't sufficient. This is for what I require proof. It's conceivable that they are next to each other but distinct.
Something isn't gibberish just because you're stupid.

>>10532817
>proof that .999... = 1 that doesn't just presuppose one infinity eliminates another, i.e. that necessarily, infinity - infinity = 0
real (as in correct) math says that .999... = 1,
and not only does it *not* use " infinity - infinity = 0".
it explicitly says infinity - infinity is UNDEFINED

>>10532833
Show me.

>>10532817
[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]

>>10532831
It's not "simply repeating the claim" and calling me stupid when I'm clearly smarter than you (you don't even realize that convergent series are equal to the limit of convergence) doesn't make what you're saying any less gibberish.
Also, mathematicians are the people LEAST likely to just accept something. They demand step by step proofs for every single claim, every proposition, every edge case. They are autistically devoted to sculpting out the most perfect and generalized behavior of any structure, of any people in any discipline.
>>10532837
There are now 2 proofs showing you this, you going "durr but it makes no sense gurgadurr" isn't a counter argument.

>>10532837
https://www.wolframalpha.com/input/?i=inf-inf

>>10532817
please stop

>>10532840
>you don't even realize that convergent series are equal to the limit of convergence
You've done it again, and you're apparently too stupid to realize it. This is the same thing as saying "you don't even realize that .999 = 1."
>>10532840
I agree that how something doesn't make sense isn't an argument for why it's gibberish. I agree that is equally plausible that if something seems like gibberish, I'm just stupid. What I've asked to be shown is a mathematical proof that doesn't make metaphysical presuppositions about infinity. Surprise, they all necessarily make metaphysical presuppositions (this means to suppose ahead of time in case you're inclined to call this gibberish again). They're called axioms.
>>10532842
No dude, not that infinity - infinity is defined as undefined. Read my reply to this retard ^
>>10532845
no

>>10532840
>you don't even realize that convergent series are equal to the limit of convergence
You've done it again, and you're apparently too stupid to realize it. This is the same thing as saying "you don't even realize that .999 = 1."
>>10532840
I agree that how something doesn't make sense isn't an argument for why it's gibberish. I agree that is equally plausible that if something seems like gibberish, I'm just stupid. What I've asked to be shown is a mathematical proof that doesn't make metaphysical presuppositions about infinity. Surprise, they all math necessarily makes metaphysical presuppositions (this means to suppose ahead of time in case you're inclined to call this gibberish again). They're called axioms.
>>10532842
No dude, not that infinity - infinity is defined as undefined. Read my reply to this retard ^
>>10532845
no

>>10532840
>you don't even realize that convergent series are equal to the limit of convergence
You've done it again, and you're apparently too stupid to realize it. This is the same thing as saying "you don't even realize that .999... = 1."
>>10532840
I agree that how something doesn't make sense isn't an argument for why it's gibberish. I agree that is equally plausible that if something seems like gibberish, I'm just stupid. What I've asked to be shown is a mathematical proof that doesn't make metaphysical presuppositions about infinity. Surprise, they all math necessarily makes metaphysical presuppositions (this means to suppose ahead of time in case you're inclined to call this gibberish again). They're called axioms.
>>10532842
No dude, not that infinity - infinity is defined as undefined. Read my reply to this retard ^
>>10532845
no

>>10532868
>>10532871
quite a stammer you've got there

>>10532877
ocd. You cannot delete posts this often, has me anxious.

>>10532817
Because no matter how many decimals you add to .9999, it will not equal an exact one.
But by the time you add an infinite amount of extra digits, it'll be the same as one, as there will be absolute no visible difference.

>>10532817
Because that is how the sets defining real numbers are constructed. If the difference between two cauchy sequences of real numbers has the limit 0, then both real numbers belong to the same equivalence class. If you reject this construction of the real numbers, then why should anybody accept that 0.9999... exists at all? I can see 1 on a ruler, but I cant see 0.99999....., and I have no reason to believe that any infinitely precise scientific instrument would report a measurement of 0.99999.....,the instrument could easily just report 1.

>>10532817
https://en.wikipedia.org/wiki/Geometric_series

[math] \displaystyle
1 = \frac {3}{3} = 3 \cdot \frac {1}{3} = 3 \cdot 0. \bar{3} = 0. \bar{9}
[/math]

>>10532817
Couldn't you always decrease the axis, "zoom in", so the curve always leaves a gap between 1.0 and 0.999... and follow this infinitely to show that there always will be a gap?

Like: if 1.9<x<2.1 fills all 20 points there, and make 0.8<y<1.0 (as an example), the gap would appear larger. And then you do something similar for x =3, and 4, and n, as n -> inf

>>10532892
Bullshit, 1/3 does not equal 0.333...

>>10532817
If I have an infinite number of dice that I roll an infinite number of times, will there be one that rolls a 6 every single time?

>>10532840
>you don't even realize that convergent series are equal to the limit of convergence
You've done it again, and you're apparently too stupid to realize it. This is the same thing as saying "you don't even realize that .999... = 1."
>>10532840
I agree that how something doesn't make sense isn't an argument for why it's gibberish. I agree that it's equally plausible that if something seems like gibberish, I'm just stupid. What I've asked to be shown is a mathematical proof that doesn't make metaphysical presuppositions about infinity. Surprise, all math necessarily makes metaphysical presuppositions (this means to suppose ahead of time in case you're inclined to call this gibberish again). They're called axioms.
>>10532842
No dude, not that infinity - infinity is defined as undefined. Read my reply to this retard ^
>>10532845
no
>>10532882
>But by the time you add an infinite amount of extra digits, it'll be the same as one, as there will be absolute no visible difference.
which is why in practice they are the same, but not necessarily in actuality. The same with asymptotes. Why are they not considered the number they approach?
>>10532886
If they're the same, then why are they distinct in concept such that you're able to reason about them as possibly distinct things?

>>10532910
heh.. sandwich.. tasty

>>10532912
Because that is how equivalence classes work retard
1+1 and 2 are concepually different, but they belong to the same equivalence class
That is what 1+1=2 means

If you're really truly interested in understanding how and why this is true, simply read https://en.wikipedia.org/wiki/0.999...... That article proves it irrefutably ten times over. However experience tells me the vocal idiots in threads like this are not at all interested in learning or challenging their beliefs.

>>10532912
They really are in most cases, more often than not, in serious use, the infinitely small difference is ignored.

>>10532817
>Show me one (1) proof that .999... = 1 that doesn't just presuppose one infinity eliminates another, i.e. that necessarily, infinity - infinity = 0. If you can't or won't, then show me proof that necessarily, infinity - infinity = 0.
there's plenty of proofs that demonstrate that both 0.999... and 1 are different representations of the same thing. it's an artifact of decimal representation.

>>10532902
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...

>>10532935
1/inf =/= 0

>>10532902
[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 \\
\dfrac{1}{3} = 3 \sum_{j=1}^ \infty 0.1^j = 0.333...
[/math]

>>10532936
https://www.wolframalpha.com/input/?i=1%2Finf

>>10532938
The transition between line 2 and 3 has a false implication

>>10532817
>Why can't .999... not equal 1?
What do you think .999... even MEANS, op? How do you think that notation is defined, mathematically?

If you have no answer to that, then your whole question and doubt is meaningless.

>>10532944
Wolfram alpha cant solve a bunch of problems, this means nothing

>>10532947
line #3 is independent from line #2, retard

>>10532954
No its not

>>10532953
shitposter can't solve a bunch of problems, this means nothing

>>10532921
Adding or any relationship between two or more elements being equal or equivalent to an element, is not the same thing as two elements on opposite sides of the equation being conceptually distinct and yet somehow equal, e.g. 1+1 is a mental function. They are becoming 2 by adding them. .999... and 1 are not functions. They're simply two conceptually distinct elements and yet are supposedly equivalent.
>>10532934
>are different representations of the same thing
Then how can they be analogous, i.e. a consequence of their own, separate and distinct, i.e. neither comes from or has any relation to the other that isn't artificially conceived, such that they both exist conceptually as two separate and distinct objects, yet they are somehow the same. And what about asymptotes? It's my understanding that they infinitely approach but never reach a number? Is this wrong?
>>10532948
Trying to appeal to what it means is just a rhetorical tautology. It's equivalent to: This means that because this is that because this means that. It's circular reasoning, as are all rhetorical tautologies. And no, questioning your indoctrination doesn't mean I must not understand something.

>>10532963
>Then how can they be analogous
them being analagous is exactly the point. you arrive at the same number through different methods, and their representation is different because of quirks of the way we've chosen to represent the numbers

>>10532957
#3 has nothing to do with #2.
It does nothing but plucks the first number from the series out from the sigma.
look at the value of j, retard
>kids these days, jfc

>>10532963
>Trying to appeal to what it means is just a rhetorical tautology. It's equivalent to: This means that because this is that because this means that. It's circular reasoning, as are all rhetorical tautologies. And no, questioning your indoctrination doesn't mean I must not understand something.
No, you misunderstand me. What I'm asking is: what do you think mathematicians mean when they write 0.999..., and what do you mean when you write 0.999... ? How does that notation map to the underlying logic of real numbers, in your mind?

>>10532973
Retard, line 2 references a fraction, and then line 3 implies that this infinite series is somehow related to the fraction
They are not related, so line 4 doesn’t follow, so the whole proof is wrong

>>10532982
you can erase everything else but line #3
and it is a trivially true statement
>wow, this retard really is something else

>>10532982
>the whole proof
>>10532839

>>10532988
Hurr durr, erasing line 4 and everything after makes it true
Yea, and it erases your conclusion, too

>>10532992
p^ infinity =/= 0

>>10532912
>What I've asked to be shown is a mathematical proof that doesn't make metaphysical presuppositions about infinity
>This is the same thing as saying "you don't even realize that .999... = 1
Not either of them, but you're really moving the goalpost on a request that they have have no obligation of fulfilling. It's axiomatically true by set theory. There are no infinitesimals and you can't add or subtract anything from .999... that wouldn't make it higher or lower than 1 so they must be in the same class. Math does not have to conform to reality. It doesn't matter that the conclusion triggers you because of some metaphysical implications you yourself are tying into it. The only relevant argument against it would be if it leads to an inconsistent system, which you have not shown that it does. There are other systems where infinitesimals do exist and someone making the opposite argument you are against those systems would be equally retarded.
tl;dr: Math is not philosophy, fuck off.

>>10532817
Everyone who says .9 repeating doesn't equal 1 is either a brainlet who hasn't taken calc 2, or a troll.

damn, this retarded shit AGAIN??? What's next, le epick portal problemo or los flies in le jar maymay?

>>10533009
Still better than iq threads

>>10532995
give an example

>>10532995
Do you know what a limit is?

>>10532994
so #2 and #3 are independent

>>10533020
I know that limits are just ways to take conceptually distinct mathematical objects, and to arbitrarily connect them

>>10532831
>It's conceivable that they are next to each other but distinct
Not to anyone who knows anything about math. What that guy said is an elementary property of the real numbers.

>>10533026
kek
look who's projecting

>>10533012
True kek

>>10533026
"5 + 7" and "12" are conceptually distinct expressions. Do you think that 5 + 7 = 12?

>>10533023
In truth yes, but the way the statements were written implied that they are dependent statements
That is why i said false implication, the intention of the statements was for them to be dependent, but that wasnt true

>>10533043
They are not conceptually distinct statements
The reality of 12 requires 5+7 to exist
12 is just a simpler form of 5+7, and 5+7 is a more complex statement

>>10533018
That is the example

>>10532902
What is 1/3 equal too then????

>>10533069
2/6

>>10533049
Lets get straight to the point, can you find a difference between:
[math]A=\left \{ a \in \mathbb{Q} : a <1 \right \}[/math] and [math]B=\left \{ a \in \mathbb{Q} : a < \sum_{k=1}^{\infty} \frac{9}{10^k} \right \}[/math]?
Or are you implying that [math]A\setminus B \neq \emptyset [/math]?

>>10533071
In decimal notation.
1/10 = 0.1
What is 1/3 equal to if not 0.3333...?

>>10533073
Moving goalposts: the post
>oh no, i cant prove 0.999... = 1
>lets force these definitions of real numbers again, despite the definitions not being philosophically grounded

this entire fucking thread

>>10533078
You cant write 1/3 in decimal notation
you can write something than an engineer would accept, or a scientist, but not a philosopher

>>10533081
What is physically grounded? Natural numbers? Or the empty set? Because given the empty set, you can define quite well all of the rest, so do tell what is wrong with that definitions of real numbers or how would you define them?

>>10533088
*philosophically grounded

>>10533088
You cant define the real numbers
Limits require an equivalence relation between conceptually distinct expressions
equality doesn’t exist arbitrarily, or abstractly, so it is impossible to define limits, and so its impossible to define real numbers

>>10532817
since when 0.(9) has any infinity in it, faggot?
following your bullshit reasoning
0+0+0+0+... != 0

>>10533098
Well now your'e just trying to be all philosophical and smart. YOU think that numbers work that way, while the consensus does not. So have fun defining number whatever you want them to be while the rest of the world dabs on you.

>>10533087
>but not a philosopher
And that matters why? The real numbers are constructed in a way that where either you have a coherent decimal system and .999... = 1 or you get missing representations. Mathematics chose the first option. It's not a philosophical point. It's a direct result of the axioms you chose. Even in philosophy, that's as rigorous as you can get.

>>10533087
>but not a philosopher
Of course you can. Just picture an infinity of successive 3 decimals.

>>10533112
Its fine when missing representations are for non-co-existant possibilities
maybe the zeitgeist was captured by some brainlet in the 19th century, but, in truth, the second option is what reality is
This is totally philosophical, because finding true axioms is beyond the scope of engineering or math

>>10533044
>implied
fuck off back to >>>/lit/

>>10532817
What makes a number different from another? You can fit yet another number in between them. Can you fit a number between 1 and 0.999...? No. They're thus the same.

>>10533087
[math]
0.1_3
[/math]

>>10533172
Base 3 is not decimal, brainlet

>>10533130
>what reality is
That’s a mighty big truth claim. If you want to play the mental masturbation game, you can’t just make statements like that. Regardless, it’s irrelevant to math what you think reality it is. Both types of mathematical systems, as far as we know, are consistent. If you can’t show otherwise then your criticism is vapid.

>>10532817
Either learn what a supremum is and how the real numbers are defined by them or fuck off.

>>10532817
Alright so i spent better part of an hour (which is a lot for me to work on something i didn't already know) to make this graph.

Wouldn't this "gap" continue to exist infinitely if you just continue to decrease the axies infinitely like I've very poorly done in this example?

>>10533272
The "gap" exists at any point where you cut off the infinite series. If you cut off the infinite series, you're talking about a different number.

>>10533362
Exactly, that's what I'm saying.

But technically that "disproves" every convergence of a sum i suppose

>>10533367
No, it doesn't. Again, truncating the sum means you're looking at an entirely different number. It's like saying "3=2, because 3=2+1 and I can just drop the 1". Qualities that your truncated sum possesses are not necessarily shared by the untruncated sum.

>>10533374
But the sum isn't truncated, only our perspective of the sum's graph.

In the animation we only zoom into the convergence point of the same sum. At any point of this limit where it approaches "infinitly small", we can draw a grap which makes the gap between 1-10^-x and 1 which is infinitely wide. Therefore, it never actually closes

>>10533367
no, all it proves is, that infinity isn't a number

>it's ANOTHER brainlets that are just learning about convergence think they're right when every mathematician in the world is wrong thread

>>10533384
There is no value between .999... and 1.

>>10532970
Them being analogous is a reason to think "you know what? maybe the fact that I can't comprehend how .999... could not equal 1 isn't good enough of a reason to consider them equal outside of practical application."

.999... = .999...9
.999...9 - .999...8 = .000...1 = 0
.999.9 = .999..8
.999...8 - .999...7 = .000...1 = 0
.999...8 = .999...7
.
.
.
1 = 0

>>10532976
0.999... = 0.999 to infinity, which in turn = 1 if and only if this is sufficiently proved. This is so obvious you must think 0.999... being defined as 1 is somehow proof that this is true, hence rhetorical tautology.

>>10533394
But 1 - 0.9 = 0.1, 1-0.99= 0.01, 1-0.999=0.001 and so on. When does it stop being a value between 1 and 1-10^-x?

>>10533411
you can't do 0.999...8, then it's not 0.999... repeating infinitely

>>10533413
The ..., There is no numerical value for x that satisfies 1-10^-x = .999...

>>10532997
A working system does not imply truth, you fucking retard. Obviously .999... would work as 1 in every practical model. The contention is that this doesn't necessarily mean .999... actually, in the real world, equals 1. Therefore, it doesn't necessarily equal 1, and this truth value remain so until you prove negation of the possibility that it doesn't equal 1. Therefore, the burden of proof is on you to show that .999... necessarily = 1, not on me to prove anything otherwise.
>you're really moving the goalpost on a request that they have have no obligation of fulfilling
Request for a mathematical proof that doesn't make metaphysical presuppositions about infinity is explicitly outlined in the OP. Therefore, I haven't moved any goal posts; the goal's just been too narrow (rigorous) for you to score in the first place. Even if I had, no one has ever been obligated to answer me.
>Math is not philosophy
>what is philosophy of math
All mathematical axioms are properly based on metaphysical inference, you idiot.

Also, your claim, and that's all it's been, has been something like "I can't comprehend how two numbers with no numbers between them cannot be the same, and it works in practice, therefore they are the same." What about the asymptote?

>I'm clearly smarter than you
Unironically, no one is smarter than I.

>>10533435
So the whole concept hinges on using a specific numerical system which enables it? What's the point of math if you can just remove/invent new rules which fits you depending on the argument?

>>10533452
In no system would you be able to quantify any difference between .999... and 1.

>>10533455
Only because someone invented the " ... " notation which disables the difference between 0.999... and 1. I mean, ofc it makes sense if you change the rules of math in order for it to make sense.

>>10532997
A working system does not imply truth, you fucking retard. Obviously .999... would work as 1 in every practical model. The contention is that this doesn't necessarily mean .999... actually, in the real world, equals 1. Therefore, it doesn't necessarily equal 1, and this truth value remains so until you prove negation of the possibility that it doesn't equal 1. Therefore, the burden of proof is on you to show that .999... necessarily = 1, not on me to prove anything otherwise.
>you're really moving the goalpost on a request that they have have no obligation of fulfilling
Request for a mathematical proof that doesn't make metaphysical presuppositions about infinity is explicitly outlined in the OP. Therefore, I haven't moved any goal posts; the goal's just been too narrow (rigorous) for you to score in the first place. Even if I had, no one has ever been obligated to answer me.
>Math is not philosophy
>what is philosophy of math
All mathematical axioms are properly based on metaphysical inference, you idiot.

Also, your claim, and that's all it's been, has been something like "I can't comprehend how two numbers with no numbers between them cannot be the same, and it works in practice, therefore they are the same." What about the asymptote?

>I'm clearly smarter than you
Unironically, no one is smarter than I.

>>10533452
The decimal system isn't some kind of epic ruse people came up with so they can laugh at retards who don't understand infinite series implies 0.999... = 1. We choose this representation for convenience and it happens to have these somewhat unexpected (and probably not ideal) properties.

>>10532817
>Side note: I wonder if said or similar proofs or tricks would work in a base-11 number system?
You sounded intelligent but merely misguided, until you said this

>>10533459
The "..." notation here just signifies that it continues in the same way ad infinitum. The question has nothing to do with notation, it's whether the decimal consisting of infinite nines is equal to 1, which, no matter how you write it, is true.

>>10533468
>infinite series implies
>math doesn't care if it's ideal ;_;
straw man much? No one gives a fuck about how ideal it is. The question is whether the infinite series .999... implies 1, not what an infinite series implies.
>>10533472
So, there are stupid questions ;_;?

>>10533468
>>10533475
But the concept of something approaching infinity is a man made concept, no? Infinity can't be proven, so at best it's a guess or a mutual agreement between mathmaticians where they say "yeye whatever, fine, let's just roll with this"

>>10533460
What does "0.999...", "1" and "equals" even means "in the real world"?

>>10533505
>at best it's a guess

Definition is:
An unbounded quantity that is greater than every real number.

>>10533510
it means that they are exactly the same
duh

>>10533516
That just breaks all intuitive sense from a real number line perspective. If there's no possible intuitive grasp of the mathmatics, what are you even calculating? Noone can say they "understand" infinity because it doesn't even exist. Even complex numbers makes some sort of sense, seeing that they at least are present in 3d transformations of any object.

It may exist in some sort of quantum coherence sense though, maybe. But now im just mixing my popsci understanding of stuff

>>10533411
1 = 1
1 - 1 = 0 = 0
1 = 1
1 - 1 = 0 = 0
1 = 1
:
:
duh

>>10533538
ts kiddo, that's the definition

infinity isn't a number, because by definition it would be greater than itself

>>10533546
yeah okay who's cock do i need to suck to invent my concept "shrlacluckle" functions as follow: (any number)/0 = 2.31 shrlacluckles? Why 2.31? Why not!

>>10533553
sigh
https://www.wolframalpha.com/input/?i=infinity

>>10533556
yeye im all over that, how do i make my own whimsical mathematical definitions? Seeing that they don't need to make much sense at all in order to exist

>>10533563
Well technically you don't, but you do need them to adhere to the accepted axioms (which you didn't do so far) and you need to prove your system is consistent. Do that and people will listen to your math.

And btw, the axioms were chosen because they're intuitive and because they make sense when relating them to the real world, so try to think again how why infinity does make sense the way it's defined.

>>10533563
>don't need to make much sense
you not understanding =/= doesn't make sense

open your mind a bit, try these
https://www.youtube.com/watch?v=BBp0bEczCNg
https://www.youtube.com/watch?v=FVZqPaH94qU

>>10532831
He gave a proof by contradiction, albeit a very shitty one. This is not an open question in the mathematical community, friend. I can tell you aren't one of the .999999 =\= 1 shitposters, since you actually are looking for a mathematical truth.

>>10533571
>so at best it's a guess or a mutual agreement between mathmaticians where they say "yeye whatever, fine, let's just roll with this"
>"why not? because we're mathmaticians"

She literally confirmed what i just said. If you can just throw the "rules" away and say "why not" to things, what's the use? It's borderline ethically wrong.

>>10533403
or maybe you're not a genius and your not understanding a problem is your fault, not math's

>>10533597
>"rules"
silly, there are still rules
you just don't like them for some emotional reason
that's between you and your shrink

>>10533571
Ok so what she's saying is that 1 - 0.999... = 1/K? There is an infinitely small number, which is called a monad?

>>10533481
it should be obvious that the underlying properties of numbers doesn't change just because you write it in a different format. calls into question whether you understand any of the terms you've been throwing around.

anyways you should really take a closer look at this guy's post >>10532825 , it was a quick /thread
starting from the very first equation (1 = 9/10 + 1/10), the left side of each equation is perfectly equal to the right side. keep splitting terms up and combining all but the last (by ANY proportion, it doesn't have to be 9/10 and 1/10) and substituting, and you will see it always adds up to .99999... even though we only used the concepts of addition and equality, and we never broke the rules of either

>>10533419
.999... is 9 repeating infinitely with a 9 on the end

>>10533674
there is no end of 0.999... to put a 9 on

>>10533625
infinitesimal.
not in R
from the perspective of R, there is no daylight between 1/K and zero, so 1/inf=0

remember how in school you moved into an alice-in-wonderland world using the laplace transformation and played with the s-variable
then the whole thing made sense only after you transformed back to the real t-variable.

It's a bit like that, infinitesimals are useful in analysis but they aren't real.

>>10533680
im telling you after you put infinitely many nines just put a 9 on the end and you get what im saying
you get what im saying?

>>10533705
>infinitely many nines
"infinitely" doesn't have an end,
it's like asking what's north of the north pole, no such thing.

>>10533708
just take infinitely many blocks with 9 on them and one with a 9, start arranging them and after you place one place the latter 9, remove it and place one of the infinitely many ones, this procedure i call .999...9

The whole debate is actually this:

Does everything which has a start, also come to an end?

(the correct answer is no, the universe is infinite but our temporal grasp of information is finite)

>>10533411
Let's assume these are all valid steps. At some point, you'll get to 0.999...0

What follows next? The alleged 0 digit would disappear by subtraction and carry over to the next digit, so 0.999...9, but that just wraps around to your first step without making any headway. All these numbers would be identical and a finite loop, so you still have no infinite algorithm to "work back" from infinitesimals to the naturals.

>>10533198
0.5 is weird in base 3, so it doesn't exist?
brainlet

>>10533460
>actually, in the real world, equals 1.
But you keep misunderstanding this. Whether you're that retarded that you can't comprehend what i've told you 100 times or you think you're just acting, you're still a retard. First of all, you're making a claim that you've given no justification for. Why, in reality, is it not equal to 1? You're merely claiming there's a discrepancy and telling people to disprove you. That's not how math or philosophy works.
>Therefore, the burden of proof is on you
That's not how burden of proof works.
Second of all, it's irrelevant to math how the real world operates. The real world also has no perfect circles. Why should that matter? Your entire argument is a non-sequitur. It's like claiming a guitar is useless because you can't eat it. The two things have nothing to do with each other.
>mathematical proof that doesn't make metaphysical presuppositions about infinity is explicitly outlined in the OP
You've been given them. You haven't accepted them on the grounds that they're tautological. But that's how all of math works. Everything follows from a few axioms so .999... = 1 is true by definition of the our real number system so if you ask for a mathematical proof you can't cry that the axioms of math are assumptions. It's nonsensical. It's like discussion the problem of evil in a religious context and saying "well you can't even prove God exists". Ok? Then don't have the discussion and fuck off. There are other systems that won't trigger you, see how much company you have using them.
>your claim...it works in practice
I never appealed to practicality. Don't strawman me.
>I'm clearly smarter than you
who are you quoting?

>>10533460
the burden of proof is on anyone who makes a claim. if you claim something to be true, you need to support that claim.

you're claiming that 0.999... =/= 1. that's a claim that needs proof. someone else posting the opposite, that 0.999... == 1, also is making a claim that needs proof.

>muh burden of proof
is something that very rarely actually applies

>>10532827
Notice they didn't respond to what is undoubtedly the hardest proof ITT, or to any post that quoted it. And it's the second post, so it's not like they don't see it.
It's almost like the whole thread is bait... but not quite. I think it's more like they just ignore any argument that blows them out, and focus on the ones they can "argue" with.

>give me a proof that 1=0.999... but I want you, no matter what the proof you give me, I will say it's wrong
That's fun. Good thread

Ok, OP. Lemme approach this in a non-judgmental and pedagogical way. You clearly have some kind of misunderstanding about the real numbers. Don't worry. Together we'll get to the bottom of this and walk out with a better, more grounded, knowledge of math as a whole. First, you're asking for a proof that doesn't make "metaphysical assumptions". You seem to have a background in philosophy so surely you must understand that every logical system requires axioms to work with and thus, a proof of pretty much anything without making any metaphysical assumption is impossible. Under those constraints, I can prove that 0.99...=1 only to the extent that I can prove that I am an entity independent from you. You know, the demon and all of that. So, do we agree your request is impossible by its very nature?

>>10533812
Better version.

>>10533812
>>10532825
Maybe I’m dumb but...what’s going on here? Looks like it implies its true because the limit is 1?

>>10533728
What follows after .999...0 is .999...89 and you do this infinitely many times and all the infinite digits to the left start declining until you reach the infinitith digit on the left which after infinity amount of work reaches 0

>>10534398
>infinite digits
>start declining
wew lad

>>10533520
Which they are, both in our mathematical system, AND the real world

>>10534398
0.99.....8 is the equivalent of
[eqn]9\sum_{k=1}^{\infty}(10^{-k})~~~- (10^{-\infty})[/eqn]
which is completely nonsensical.

>>10534485
1/(10^inf) = 0
so no problemo, just ignore the 8

>>10533505
>man made concept
All concepts are man made. There is a rigorous meaning of the limit as n goes to infinity.

>>10532817
>For practical calculation, the answers of which are always a finite representation of numbers, .999 does equal 1, but in reality, not really.
kek, what does "in reality" even mean?
Do you seriously mean to suggest that numbers actually exist in the real world? Then show me a molecule of [math]3[/math]. Or should I say a molecule of [math]2.999\ldots[/math]?
Numbers only exist in mathematics, and mathematics only exists in our heads. We define what mathematics is, and we defined the underlying axioms at its foundation in such a manner that, within the system (which everyone except schizophrenics and retards uses), [math]0.999\ldots = 1[/math].

>>10533098
it's because you think of limit as in calculus where you know how to compute them but don't understand what they are. limits are well defined, they're not a ''distinctive object'' arbitrarily chosen to be equal certain numbers.

>>10534796
Don't all university calculus courses teach you epsilon delta proofs for limits?

It is an accepted mathematical truth at this point, isn't it? That is, unless you start changing rules and axioms to fit your argument.
https://www.wolframalpha.com/input/?i=0.999...%3D1

>>10534539
>not knowing the beauty of the forms

>>10534811
Yes.

>>10534803
no? it's more about technique than applying the actual definition of a limit in most unis.

>>10532824
Whats the first real number that comes before .999... and first real number that comes after 1 then? It seems that axiom would be self-defeating if there is no answer. The first real number could not have another number between it or it wouldnt be the first, but if it didnt have another number between it then it would be "equal"

Saying "there is no first real number" is brainlet tier. If there is no first, there is no second, 3rd,4th etc. But you must eventually "reach" a different real number. If theres 2 numbers and no starting point between them then there can be no real number between them, hence all real numbers are equal according to the axiom, but that isnt true.

>>10535092
>If there is no first, there is no second,
yeah, the Reals are "un-listable", unlike the rational numbers (Q)
https://youtu.be/elvOZm0d4H0?t=2m
https://youtu.be/elvOZm0d4H0?t=4m

>>10535092
The real numbers are uncountable. For any list of infinite strings, you can generate a new one that cannot be in the list by taking the [math]k[/math]th digit of the [math]k[/math]th entry and changing it, and real numbers happen to be expressed as infinite strings. Let's demonstrate.
For convenience, we'll just work in binary so we only have two choices for digits, but this proof is of course valid in any base that isn't [math]0[/math] or [math]1[/math] (which themselves are not valid bases with which to express real numbers).
[math]
1.000 \ldots \\
0.101 \ldots \\
1.101 \ldots \\
0.011 \ldots \\
. \\
. \\
.
[/math]
So let's look at the [math]1[/math]st digit of the [math]1[/math]st number, the [math]2[/math]nd digit of the [math]2[/math]nd number, the [math]3[/math]rd digit of the [math]3[/math]rd number, and so on to [math]\infty[/math].
[math]
{\color{red} 1}.000 \ldots \\
0.{\color{red} 1}01 \ldots \\
1.1{\color{red} 0}1 \ldots \\
0.01{\color{red} 1} \ldots \\
. \\
. \\
.
[/math]
Let's put them all into a number.
[math]
{\color{red} 1}.{\color{red} 101} \ldots
[/math]
And then let's switch the digits. [math]0[/math] becomes [math]1[/math], et vice versa.
[math]
{\color{red} 0}.{\color{red} 010} \ldots
[/math]
This number, by construction, cannot be in the original list. If it were, then this process would result in a different number which would also not be in the original list.
So there is no first or second or third or fourth or whatever real number because they are literally uncountable.
Fun fact, though. Separately, we can prove that there is a rational number between every real. Somehow.

>>10535092
The real numbers are uncountable. For any list of infinite strings, you can generate a new one that cannot be in the list by taking the [math]k[/math]th digit of the [math]k[/math]th entry and changing it, and real numbers happen to be expressed as infinite strings. Let's demonstrate.
For convenience, we'll just work in binary so we only have two choices for digits, but this proof is of course valid in any base that isn't [math]0[/math] or [math]1[/math] (which themselves are not valid bases with which to express real numbers).
[math]
1.000 \ldots \\
0.101 \ldots \\
1.101 \ldots \\
0.011 \ldots \\
. \\
. \\
.
[/math]
So let's look at the [math]1[/math]st digit of the [math]1[/math]st number, the [math]2[/math]nd digit of the [math]2[/math]nd number, the [math]3[/math]rd digit of the [math]3[/math]rd number, and so on to [math]\infty[/math].
[math]
{\color{red} 1}.000 \ldots \\
0.{\color{red} 1}01 \ldots \\
1.1{\color{red} 0}1 \ldots \\
0.01{\color{red} 1} \ldots \\
. \\
. \\
.
[/math]
Let's put them all into a number.
[math]
{\color{red} 1}.{\color{red} {101}} \ldots
[/math]
And then let's switch the digits. [math]0[/math] becomes [math]1[/math], et vice versa.
[math]
{\color{red} 0}.{\color{red} {010}} \ldots
[/math]
This number, by construction, cannot be in the original list. If it were, then this process would result in a different number which would also not be in the original list.
So there is no first or second or third or fourth or whatever real number because they are literally uncountable.
Fun fact, though. Separately, we can prove that there is a rational number between every real. Somehow.

>>10532817
The uncomfortable truth is that maths, being axiomatic, depends on physics for validation. Maclaurin expansions and infinitesimal limits work physically., so they have weight in maths. I wish people would stop taling about maths and physics as if they were separate things.

>>10532817
>Why can't 1 equal 2
You should consider a career in licking windows.

>>10535243
so the limit of approaching a value then hits up against the smallest piece of matter in it's path?
so observing a final limit is impossible since it could potentially change the value
so a limit also contains a probability
SNARF

https://www.youtube.com/watch?v=E3vtfHoZi0k

>>10535243
engineers say the darndest things
kek

>>10532817

>>10535218
you put all that work in to TeX something you could have just screencapped from wikipedia. why?

>>10532817
The next thing they'll say is that 0.4999… is 0.5 or 0.3999… is 0.4

>>10535218
>>10535150

I know they're uncountable/unlistable. There's an infinite number of values between 0 and 1 etc.

But the axiom in question is self defeating if they're uncountable/unlistable and that's the point. If every single real number in the entire infinite set between 0 and 1 does not have a first number before it or first number after it then they are all the same number according to the axiom.

The difference between two real numbers can't arbitrarily begin existing because, again, there'd b no number between them and, again, according to the axiom they'd be the same. You'd have an infinite number of "same numbers" between 0 and 1, ie all of them.

>>10535966
>the same number
when there isn't anything in between

infinite amount of stuff between --> not the same number

wtf are you rambling about?

>>10532831
>Simply repeating the claim isn't sufficient
It's not a claim. It's an axiom.
Give it a rest.

>>10532817
>It's like your tiny brains just can't grasp the concept of infinitely-repeating digits,

I know you are, but what am I?

>>10536484
150+ posts and THIS is your contribution..?

>>10532817
WHO GIVES A FUCK???
THIS IS NOT AN INTERESTING QUESTION ITS JUST DEFINITIONAL BULLSHIT THAT RETARDS ARGUE ABOUT TO FEEL INTELLECTUAL

>>10535759
Because linking the numberphile video would have been an admission that he didn't come up with that himself, and he wanted to look really smart in a 0.999 = 1 thread

>>10532831
Ma boi, numbers can't just be next to each other. If you go at a smaller increment, you can always fit another number between the two

>>10535759
Because I'm an addict and [math]\LaTeX[/math] is my drug.
>>10537262
Where in my post did I imply that I came up with that myself?
Wait. Do you actually think Numberphile came up with Cantor's diagonal argument? Is this peak retardation?

>>10537973
>Is this peak retardation?
Hey now I'm not the one trying to educate people in a troll thread

>>10537973
well that's the lamest use of latex ever seen,
no worry about withdrawal symptoms

>>10532831
Formal proof: 0.9999... can be defined as the limit of the series
[math]\sum_{j=0}^{n} 0.9 * 0.1^j[/math] as n goes to infinity.
Now, we say K is the limit of this series if [math]\forall \epsilon > 0, \exists N \in \mathbb{N} : \forall n \geq N, \left|K - S_n\right| < \epsilon [/math], where [math]S_n[/math] is the partial sum [math]\sum_{j=0}^{n} 0.9 * 0.1^j[/math].
If we set K = 1, given any [math]\epsilon > 0[/math] we can take any N > [math]log_{0.1}(\epsilon)[/math], and then we have [math]\left|1 - S_n\right| = \left|0.1^n \right| \leq 0.1^N < 0.1^{ log_{0.1}(\epsilon)} = \epsilon, \mathbb{QED} [/math].
If you didn't understand the epsilon-delta notation used here, then you shouldn't have asked the question you fucking retard. Get some formal education. Popmath morons must be cleansed from this board.

>>10532817
Here you retard:

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

0.999... and 1 are different symbols for the same number.

What's the Tooker approved approach to this question?

>>10532817
Easy
1/9 = .111...
(1/9)9=(.111...)9
9/9=.999...
1=.999...

>>10533412
all mathematics is tautalogy.
In mathematics, conventionally 0.999... is short-hand notation for the limit of a particular sequence.
A limit is rigorously defined concept in mathematics, and it turns that the the limit of the sequence in question is indeed 1.
If, for some reason, you use the notation to mean something other than aforementioned sequence, then it may or may not be equal to 1 (depending on how you choose to define 0.999...).

>>10533570
I would say that another significant reason for choosing the set theortic axioms that underly modern mathematics is for the emergent theory to contain all previously known mathematics; geometry, algebra modular arithmetic etc.

Rational and Irrational numbers are a different set of information. Like the difference between Hex and Binary code.

When you take a whole number(rational), and divide or multiply by an real number like pi(irrational)... you get an irrational number as a result. Because you are switching from rational numbers to irrational numbers via the equation. The only way you can get away from an irrational results in an equations, is if the irrational number is nullified. An irrational number is a measurable ratio of a rational number. They're two different formats of numerical information. example, (1*(1/3)) is equal to (1*(0.33333~), but, they have classifications of format. (0.99999~) as a real number is the same number as 1, a whole number.

>>10533272
oh great another fetish

>>10532817
An infinitesimally small unit does not equal zero, else calculus would mean fuck all. If .999... is equal to 1, then 1=0.

>>10543139

>>10534803
Didn't read Bolzano or Weierstrass once in this thread.

>>10532825
based and hexpilled

Can we just euthanize all the people who try to disprove 1=0.999... by trying to end infinitely repeating digit?

0.999... is the infinite series 9/(10^1) + 9/(10^2) + ... + 9/(10^n)

It entails everything between 0.999... & 1 so it converges to 1

Guys, 0.999...=1 look at a math textbook

>>10543139
>>10533689

>>10533005
Calc 2? You should be able to understand the fraction-based proof literally in elementary school.

>>10543139
infinitesimals are not used in calculus

>>10544424
And by the time you realize that that proof is technically not rigorous because you can't manipulate infinite series like that unless they are absolutely convergent, you should be able to trivially prove that [math]0.9\sum_{k=0}^{\infty} 0.1^{k}[/math] is—to [math]1[/math].