/sci/ - Science & Math

>>12320752
Yeah.

>>12320752
Yes.

>>12320755
>>12320758
Are you going to expand on that? Please I feel like I'm going crazy.

>>12320763
Yup.

>>12320793
What the fuckk?? Just fucking tell me why

So, what's the difference between 0.9999... and 1? 0.000...1? But that one never comes, and 0.0000... is equal to 0. Surely that one isn't hard to swallow.

>>12320752
I think the confusing thing may be notation for you?

0.9999.... is just [math] \lim_{j\rightarrow\infty}\sum^j_{n=1}\frac{9}{10^n} [\math]

i.e. the limit of the sequence of partial sums.

Now you can use the definition of the limit to show that this is just one.

>>12320853
It still doesn't look right. How can a decimal number also be a whole number? There must be something wrong with your argument. Maybe there are infinite small numbers?
>>12320886
I don't know what limits are, sorry anon.

>>12320886
It's part of a board flood, you are replying to pasta.

>>12320899
Fuck thanks for letting me know.

>>12320853
The one comes immediately after a countable infinity of zeros.

>>12320763
[math]
\begin{align*}
n &= 0.999... \ (1)\\
10n &= 9.999... \ (2) \\ \\
\text{eqn(2) - eqn(1)} \\ \\
9n &= 9 \\
n &= 1
\end{align*}
[/math]

>>12320886
no, 0.99999.... is the sum, not the limit of the partial sums.
the limit of something is not necessarily equal to what it is.
if you can prove that that sum is equal to the limit of its partial sums, then i can believe you

>>12320954
prove that 9.9999....0 - 0.99999... = 9
protip: you can't

>>12320955
I know you’re fucking with me. Why do I bite?

There is no such thing in standard analysis as an infinite sum outside of the notion of limit.

>>12320752
sure but the awnser is tio

1/9 = 0.11111111...
0.1111...*9 = 0.9999...
1/9*9 = 1 = 0.9999999...
decimals are niggers.

>>12320964
He doesn't have to prove anything since it's correct by axiomatic definition. It's your task to disprove it.

>>12320980
yep, can confirm

axioms of real numbers:

commutative + *
associative + *
identity + *
inverse + *
distributive of * over +
<= transitive
<= antisymmetric
<= connex
compatibility of <= with + *
least upper bounds
9.9999....0 - 0.99999... = 9

>>12321001
>not constructing the reals as the completion of the rationals and then proving LUB plus the axiom of 9.99...0-.9999=9

Disgusting

>>12320752
Well it’s an axiom or some shit like that but it seems off

>>12320752
Idk dude in my class I was the only person to understand why .999=1 and also why the distance between 1 and i is 0 or something I don't even remember. But I was the smartest kid. For some reason y'all just can't grasp it. Like think about it there is no ...1 that just makes even less sense. If you subtract .999... from 1 you're not gonna get .000...1 bc it doesn't exist. It may seem as if it is fundamentally less than one because duh but it's just not. Imagine if you will an infinite list of items you could select. You could select each one at a time forever (.999...) or you could "select all" (1). Either way you are selecting all items

>>12320752
no op it 1.000...

>>12320752
The real problem is that most of the proofs you've seen are bunk, they don't involve the definition of the real numbers at all. One popular construction of them is to consider them as infinite sequences of rationals that you would expect to converge (formally, Cauchy sequences), but you also need to add the condition that if the difference of two sequences approaches 0 (formally, the limit of the sequence is 0), then those sequences define the same real number. We add this so that we can work with them as just a number without worrying about the sequence that defines them (For example, {1, 1.2, 1.25, 1.25...} and {1.25, 1.25, 1.25...} ought to define the same number). If you consider the number 1 as {1, 1, 1...} and 0.999... as {0.9, 0.99, 0.999...}, then the difference between these two sequences is {0.1, 0.01, 0.001...}, whose limit is 0. By the definition above, these two sequences describe the same number.

>>12320752
1/inf=0

>>12321244
1/inf = undef

>>12321254
1/inf = 0
1 + inf = inf
1 - inf = -inf
inf + inf = inf
inf/inf undefined
inf - inf undefined
1^inf undefined

you can't do everything with inf as with a number, doesn't mean you can do nothing tho

>>12320979
no, that only means fractions are retarded and false. if you have 1/9*9 it will never equal 1, same thing with 1/3*3 it will always equal to 0.999....

>come up with concept "the number line"...which goes on for...forever?
>so come up with concept called "infinity", which has no basis in reality, doesn't matter, we are still doing maths, right?
>0.999.... OH FUCK!
>I know! Let's define it as "1", we are still doing maths, right?
>0.999... = 1 because... because... because I SAID SO!
>I call myself a mathematician
>smug smile jpeg

And there we have it in a nutshell.

>>12321049
>If you subtract .999... from 1 you're not gonna get .000...1 bc it doesn't exist
Opinion discarded

>>12321267
9×(1/9) is 1 by definition lmao

>>12321267
>1/9*9 isn't 1
topkek, you're worse at math than kids at school

>>12321269
>I've never read a definition of the real numbers
>there must not be any

t. retard

>>12320763
No.

Basically mathematicians think there's a little gap just below 1 where there are no numbers. You can get numbers up to around 0.999 but much closer to 1 than that would put you in the gap and the value has to jump up to 1.

>>12321332
Theres no gap, adding the infinite sum (9/10 + 9/100 + 9/1000...) gives you 1

>>12321321
>The definition I subscribe to is the only one.
>my mind is smaller than a pea
>but i am too stupid to understand that
>I jerk it to Jesus

>>12321406
https://en.m.wikipedia.org/wiki/Real_number

The real numbers refer to a specific thing, dude, theres no ambiguity in what that phrase means

>>12320913
You can start a new string, but nothing can come at the end of an infinite string, because an infinite string doesn't end.

>>12321434
Then why does -e^i*pi equal exactly 1?

>>12320964
Correct.
We know that "..." can't represent an infinite string here, because the string ends with a 0.
So we can assume "..." represents an arbitrarily long finite string of 9s.
So the answer can't be 9, it must be 8 followed by an arbitrarily long finite string of 9s, followed by a 0.

9.999...0 - 0.999... = 8.999...0

Nobody's ever seen infinity, so we can assume it doesn't exist.

0.99999.... is equal to 1 because eventually the universe runs out of storage space for all the 9s and rounds it up to 1.

>>12321444
non sequitur

>>12321452
Infinity is much simpler, much more intuitive, and much easier to imagine than a very large random number that fluctuates wildly according to data storage technology.

>>12321454
No, its three numbers with three infinite strings that converge to a non infinite product.

>>12321466
What does that have to do with the post you replied to?

>>12320955
sum is literally defined to be the limit of partial sums

>>12321480
He said infinite strings can not ever terminate, but if you take three infinite strings together, they quickly terminate.

>>12320964
Where did that 0 come from?
9.9 - 0.9 = 9
9.99 - 0.99 = 9
9.999 - 0.999 = 9
9.9999 - 0.9999 = 9

brb going to calculate this to infinite precision. I'm God, so I can do that.

...

Alright I'm back. It was 9.

>>12321483
That was my post, I'm the same guy.
The infinite strings of digits representing the numbers e and π don't terminate in any integer base.
Why do you think it's relevant that an arithmetic manipulation of those strings can result in a number that can be represented by a finite string of digits?

You don't even need to complicate things with irrational values.
For example: 0.999... + 0.999... = 2

>>12321496
>Why do you think it's relevant that an arithmetic manipulation of those strings can result in a number that can be represented by a finite string of digits?
Arithmetic manipulation of those strings would be impossible with your faulty assumptions, why do you think it is irrelevant that non terminating multipliers can converge to a terminating product?

>>12321503
How is any of what you're saying relevant to the post you replied to? (my post)
>>12321434
>You can start a new string, but nothing can come at the end of an infinite string, because an infinite string doesn't end.

>>12321512
Because we do math with infinite strings, such as e and pi, all the time, so is not impossible like you claim.

>>12321434
what definition of "string" are you using?

>>12321529
What does doing math with values that can only be represented digitally by infinite strings have to do with whether or not something can come at the end of an infinite string?

>>12321533
ur mom

>>12321377
yeah try a number just below 1 though

>>12321016
>completion of the rationals
In a constructive setting there are various non-equivalent ways to complete a space. If you want to convince most people you need to be careful.

>>12321533
>values that can only be represented digitally by infinite strings
Even infinite strings can be represented as finite ratios.

>>12321530
Let's say a sequence of digits such that every digit after the first has a successor and a predecessor.

>>12321548
Some, yes. So?
This is like the twilight zone of non sequiturs this morning.

>>12321556
Let's clarify that, if finite, the last digit will of course have no successor.

>>12321562
So you are wrong and it is not impossible to use infinite strings to produce finite results, infinite strings have plenty of utility in finite mathematics.

>>12321569
>So you are wrong and it is not impossible to use infinite strings to produce finite results
Lol what
Where did you imagine I claimed anything like that?

>>12321444
because
[math] \displaystyle
e^{ix}= \cos(x)+ i \sin(x) \\


[/math]

>>12321541
no such thing
a finite, long enough line of 9s will get between it
an infinite line of 9s will reach 1

>>12321580
Those are both continuous functions.
1 is discrete.
Something can be continuous and discrete.

>>12321459
There's no fluctuation according to data storage technology because the theoretical limit to storage is not a function of technology.

can anyone prove that 1/3*3 is 1? saying that 3/3 is 1 therefore 1/3*3 is circular logic

>>12321591
just use
x=pi

>>12321592
How so? If everything's already baked into the cake, then what's the fixed value of this theoretical limit to storage?

>>12321595
1/r * r = 1

>>12321595
Try using 1+1+1 and the definition of addition.

>>12321595
what do you think the symbol 1/3 is? the very definition of 1/3 is the number which gives 1 when multiplied by 3

>>12321601
https://www.nature.com/articles/35023282

The universe has data processing and storage equivalent to a black hole of the same mass and has ultimate memory space I = S/kBln2. Compute the entropy of the observable universe and you've bounded the information content of any computable number.

>>12321605
>Try using 1+1+1
You mean 0! + .999... - e^i*pi?
How does that help? It just makes things more confusing.

3/3 is false because 1/3 is 0.333... and if you were to add 0.3333... 3 times it would add up to 0.99.... so you can't use fractions as an example to prove that 0.999... is equal to 1 in decimals

>>12321609
>3/3 is false
hahahaha

>>12321275
9/9 != 1
based retard.

>>12321612
(9-9)! = 1
fixed your equation for you

>>12321612
found the home schooled

1/1, 2/2, 4/4, etc could all equal 1 because in decimals they would add up to 1 if you do 0.5+0.5 or 0.25+0.25+0.25+0.25

however, if you were to 3/3 it would not equal 1 because you would have 0.333...+0.333...+0.333...=0.999...
there would always be a 0.00...1 left

same thing with 1/9

>>12321608
Using different symbolic representations of 1 only makes things more confusing to people who don't know what the symbols means.

It's like using arcane inquotidian latinate verbiage to describe something, instead of the punchier day-to-day anglo saxon words we hear more commonly.

>>12321621
this may be shocking, but 3/3 is not something to be done. it already is a number.

>>12321607
The mass of the universe isn't constant.
Try to "compute the entropy of the observable universe" and let me know what fixed value you get.

>>12321633
How can you even know anything about what 1 means if you refuse to acknowledge its various identities?

>>12321639
The mass of the universe doesn't have to be constant. If it changes, then our largest computable number will also change. That's just how it is.

However the mass-energy of the observable universe (that is, within our light cone) is constant.

>>12321641
Where did I refuse to acknowledge any of the symbolic representations of 1 you listed?

>>12321644
>If it changes, then our largest computable number will also change.
Which was my point to begin with. There's no possible fixed value—it fluctuates wildly.

>>12321621
>base 10 is holy

>>12321652
It doesn't fluctuate wildly, but we can understand that the computability of numbers depends on where we are in the history of the universe.

If we were in the far future when the universe is nearly extinct and all the stars have gone, then we just wouldn't have the same computation resources that we do today. It would be retarded to say that our maximum possible computable number is the same as theirs.

>>12321648
When you said it was too confusing to acknowledge all of its identities.

>>12321639
>Try to "compute the entropy of the observable universe" and let me know what fixed value you get.
Try counting to infinity. Why is it more reasonable to claim that numbers go on forever than to say that we can only speak about what we can actually, at least in theory, compute?

>>12321667
you can compute with infinity just fine, lol

>>12321673
Infinity doesn't exist. You can program a constant called "infinity" into a computer and attach it to some degenerate cases to treat it as though you could count forever, but that's just a larp.

>>12321673
I want to compute with U where U is an integer chosen in the random uniform interval between 0 and TREE(3).

Now U is a lot smaller than infinity. How do you compute with U?

>>12321679
>How do you compute with U?
in the exact same way as you would with the number 2 for example

>>12321676
While (1){x++;}

>>12321667
Here's my post that started this tangent
>Infinity is much simpler, much more intuitive, and much easier to imagine than a very large random number that fluctuates wildly according to data storage technology.
It's very easy to imagine that something doesn't end or stays the same forever.
It's much harder—in fact, it's impossible—to imagine a large finite value that changes all the time and can't even be calculated temporarily.

>>12321676
>Infinity doesn't exist
neither do numbers, anon

>>12321682
How? There's no computer that could possibly be constructed to do it. Your belief in U, or in TREE(3), or in infinity, is a religious belief. You'll never see any of them, and nobody who ever will be alive ever will see any of them. They do not exist.

>>12321666
When did I say that? You're the one who said it was confusing >>12321608
I said that it would only be confusing to people who were unfamiliar with those symbols>>12321633

How can it equal one? If 0.999... equals one that means that 0.000...1 is of no value because one minus 0.999... has no effect. 0.999... is less than one because 0.9 is less than one, so what's 0.999...? Should be less than one, but we're told it's one, and that cannot be.

>>12321655
It certainly does. Even this equation you wrote
>>12321607
>I = S/kBln2
seems to be predicated on bits being the most compact storage vehicle, which we already know isn't true.

>>12321685
God is intuitive, but fails basic logical tests for being a valid and useful concept, just like infinity.

We can calculate the computing storage of the observable universe, and it doesn't change on any timescale that would worry a human being. For all intents and purposes, it's constant.

The entropy of the observable universe is roughly 10^104 k. That means that the largest number can occupy about 10^104 bits. If you write 0.9999.... with about 10^100 9s, you run out of memory and it turns into a 1.

https://iopscience.iop.org/article/10.1088/0004-637X/710/2/1825

>>12321692
> seems to be predicated on bits being the most compact storage vehicle, which we already know isn't true.
Take it up with the theory of computation retard.

>>12321691
Nah that just means you don't know what the symbol ... means.

>>12321691
0.000...1 is either a finite string, or two strings.
It has nothing to do with 0.999...

1 - 0.999... = 0.000...

>>12321697
>retard
Anon's Law: you lose.

>>12321683
I'll agree with this post when your algorithm actually gets to infinity.

>>12321693
>about 10^104 bits
About isn't a value.

>>12321705
What is the Sorites fallacy

>>12321687
you won't see 5 either

>>12321707
It's a non sequitur to my post.

>>12320752
>I'm fundamentally misunderstanding something about mathematics.
At least you're considering the possibility

If you truly wanted to understand this, you wouldn't post this on /sci/.
Instead you would look up one of the gazillion proofs of this (preferably the easy algebraic one), make sure you understand every step, and if you don't get one, you write that one down and in turn look up, why it works.
Reapeat the above, until you reach understanding.

>>12321708
You can see 5 the exact same way you can discern any macroscopic object.

>>12321710
You need to have minimum IQ 135 to be in this conversation

>>12321715
>IQ
Anon's Law: you lose.

>>12321715
can show me the number 5?

>>12321691
>0.000...1 is of no value
exactly, it's zero

>>12321702
Infinity is a type of limit not a type of number.
The fact that the infinite loop never converges to a final state is what makes it infinite.

>>12321722
Can you show me anything? How do you know things exist?

>>12321726
If you can't compute it, it doesn't exist and your belief in it is identical to a religious belief.

>>12321733
>If you can't compute it, it doesn't exist
What is the computed value of yourself?

>>12321733
>Infinity
Has no end. Easy and useful.
>""""Maximum storage capacity of the universe""""
Unimaginable and arbitrary. Can never be computed. Has no functional or theoretical utility whatsoever.

>>12321733
>Can you show me anything? How do you know things exist?
I don't. it's you who started making claims about existence and non-existence, namely "infinity doesn't exist".

>>12320752
infinity is just an abstract concept. The question doesnt make actual sense, as 9 cant reoccur forever.
I say I am 1.9m tall, but in reality i am 1.9436711....

>>12321759
>9 cant reoccur forever.
prove it

>>12321762
i just said infinity is not real and does not exist in the real world. And before you ask: I dont believe in pi

>>12321762
>infinity is not real
That does not mean a 9 can not recur forever, since forever is not the same as infinity.

>>12321784
math isn't physics, retard

>>12321784
>I dont believe in pi
Do you believe in integers?

>>12321784
If you don't believe in infinity, how do you measure the depth of your own stupidity?

>>12321805
math is physics. there's no point in discussing numbers that you can't ever compute.

>>12320898
>How can a decimal number also be a whole number?
1.0000... is a decimal and whole number.

infinity deniers blown the fuck out of the thread
it's like generals v globetrotters at this point

>>12321269
Infinite exists in reality

ok so 0.333... is 1/3
0.666... is 2/3
so 0.999... is 3/3, but 3/3 has to be equal to 1
so it's 1

>>12322259
you can only have one right

either 3x0.333 is 0.99... or 3/3 is 1

its literally a contradiction to say both are correct

>>12322299
0.999... represents the exact same number value as 1

>>12321826
well iq is from 0 and upwards.
I think you sit around room temperature xD

>>12321798
>forever is not the same as infinity

>>12321271
Bro it DOESN'T EXIST maybe if I type in all caps you'll get it. How can .000...1 exist? Even me writing it doesn't make sense. Why don't you try to prove this exists rather than argue about the actual question since you clearly don't understand it. You probably thought .333... Was a joke when you converted fractions to decimals. You probably got messed up cuz .666... rounds to .667 (as if rounding matters in real math)

You've simply been thinking wrong your whole life

>>12322501
Look, numbers exist because they are an abstraction of what we can observe.

You can have 3 cups, 3 stars, 3 chairs, and so on. They have a common property in that we can group objects in group A and objects in group B in 1 to 1 correspondence if there are the same number in both group.

What the fuck does it mean to say that infinity exists though? There is no such thing as infinity anything in real life. There are no sets in real life that have an infinite number of members. And the concept of infinity breaks the set correspondence property of numbers.

All the paradoxes are easily resolved when you notice that infinity does not exist.

>>12321556
>>12321567
Why prevent infinite strings from having a last digit?
Just take 999... and ...999 and concatenate them together; now every digit has a predecessor and successor except the first and last.

>>12320752
yeah, if you think they aren't equal try to name a number between 0.99999... and 1

>>12322547
>Why prevent infinite strings from having a last digit?
It's not a matter of preventing something, it's what the word means—infinite, doesn't end.
>Just take 999... and ...999 and concatenate them together; now every digit has a predecessor and successor except the first and last.
If "..." means "infinite" then you can't concatenate anything to the right side of 999... or to the left side of ...999 because infinity means that there's no end point.
If "..." means "a very large finite number that I can calculate" then the result is simply another very large finite number that you can't calculate. It has nothing to do with infinity.

>>12322547
>Why prevent infinite strings from having a last digit?
It's not a matter of preventing something, it's what the word means—infinite, doesn't end.
>Just take 999... and ...999 and concatenate them together; now every digit has a predecessor and successor except the first and last.
If "..." means "infinite" then you can't concatenate anything to the right side of 999... or to the left side of ...999 because infinity means that there's no end point.
If "..." means "a very large finite number that I can't calculate" then the result is simply another very large finite number that you can't calculate. It has nothing to do with infinity.

>>12320752
The true question which you potato heads completely miss is this: Does the concept of infinity have any place in mathematics? Currently it does. Some think it shouldn't.

Personally I am for leaving it in and for also introducing flying pink elephants ( FPE ) as well. Consider the beauty of this statement.
FPE = infinity / 0.
Cant be wrong. It's what I defined FPE as being. And if I defined it as such then that's what it is. Just like infinity. Neat eh? Anyone who disagrees is a moron and sucks pig's penises.

>>12322661
Infinity simply means that something has no end or bound.
What does FPE mean?

>>12320752 and all other anti-infinity people in this thread
Are we talking about numbers that belong to the set of Real Numbers? Then 0.999... is equal to 1. There are numerous proofs of that; it was established a long time ago. Go back to class.

A number with infinitely repeating 9s after zero and a decimal point that isn't 1 cannot be part of the Real Numbers set.

If you invented a set of numbers that allows having numbers with an infinite decimal expansion "next" to each other then it's fine. Share your work and the consequences of it.

>>12322674
Flying Pink Elephants

>>12322674
And God simply means an omnipotent being who created the universe. Therefore, you should believe in God.

>>12320752

Numbers aren't even real we literally just made them up they are logical to a point but can't represent what is true with 100% accuracy
Fractions or decimals are only accurate so far as it's convenient to the situation it just happens that decimals are convenient in more practical situations
So yes 0.999...=1 for all intents and purposes because that tiny indiscernible difference apparently isn't relevant in any real world applications of numbers I guess

>>12322679
What exactly is a real number?

Numbers cannot just go on forever. Where would you write them down?

>>12322684
How does that relate to math in your view?
>>12322688
Who said you should or shouldn't believe in anything? I'm just explaining what the word means.
Whether you """"believe"""" it or not, it's very simple and very intuitive. Would you prefer it if math books used the word God instead?
A God sequence. A God series. A God decimal.
Does that make you happier?

>>12322700
What does writing have to do with anything?

>>12322697
>tiny indiscernible difference
There is no difference at all, not even a tiny difference.
1 - 0.999... = 0.000...
Always zeros forever.

>>12322726
It would be the same. There's as much proof of God as there is of infinity.

>>12322736
How can you claim that a number exists if you can't write it down somewhere?

>>12322700
We have symbols for that. You don't need to write an infinite decimal expansion down.

>>12322646
Prove that I cannot concatenate to a string that has no last digit.
Every number in the list has a predecessor and successor except the first and last. There is no need for a last digit in the first component because there is no first digit in the last component missing a predecessor.

>>12322746
Both God and Infinity exist.

>>12322746
>It would be the same. There's as much proof of God as there is of infinity.
So why do you care and why are you talking about God?
>How can you claim that a number exists if you can't write it down somewhere?
You can't write any numbers down, all you can write down are symbols that represent numbers.

>>12322746
I define Pi as the ratio of the diameter of a circle x^2 + y^2 = r^2 and its circumference. Pi exists by definition. Pi isn't a "physical" object nor does it need to be. That's true for any number.

>>12322760
Do you know what concatenate means?
You're describing the second option
>If "..." means "a very large finite number that I can't calculate" then the result is simply another very large finite number that you can't calculate. It has nothing to do with infinity.

It's all well and good to talk about axioms and constructions, but ultimately the real numbers were created to measure real world quantities such as length, volume, mass, and time. Where is the proof there is no length between 0.9m, 0.99m, 0.999m, ... and 1m?

>>12322776
Real numbers were discovered

>>12322776
0.999...m is not a physically meaningful quantity

>>12322773
>Do you know what concatenate means?
I'm not sure what it means in your head, but all I need to do is put all the digits in ...999 after all the digits in 999...
There is no need for 999... to have a last digit or ...999 to have a first.
>You're describing the second option
No, I'm not. Both strings being concatenated are infinite.

>>12322787
Why not?

>>12322776
No, the real numbers were "created" to fill in arithmetic gaps in the rational numbers.
>proof there is no length between
Just subtract it and see for yourself.
1 - 0.999... = 0.000...
The only way you can get anything other than a 0 is if the 9s end.

>>12322789
>your head
My head? Lol what
>after all the digits in 999...
There's no "after" unless the digits end. If they end, it's not infinite.
>Both strings being concatenated are infinite.
Then they're not being concatenated because there's no end point where they can connect.

Math is discovered, not constructed.
Materialism and Physicalism are false, Platonism is true.
God exists.
Stop seething about it.

Failed python nutjobs using c++ for the first time that crashed their computer trying to store a float as a char want to know your location

>>12322795
The origin of the positive reals was in the work of Greek mathematician Eudoxus, who generalized the theory of proportions to incommensurate quantities such as the legs and hypotenuse of an isosceles right triangle. Over time people began to understand these ratios as numbers. The abstract formalized versions of the real numbers came much later.

>>12322802
>There's no "after" unless the digits end.
Why not?
>Then they're not being concatenated because there's no end point where they can connect.
Call it what you like.

>>12322810
>incommensurate quantities
To be clear, you're agreeing with me completely that the real numbers were "created" to fill in arithmetic gaps in the rational numbers, and you're providing extra historical background?

>>12322814
>Why not?
Because that's what infinity means. If it's ends, it's not infinite.
>Call it what you like.
I did. What you're describing are two finite strings of unknown length.

>>12322814
>Why not?
Because that's what infinity means. If it ends, it's not infinite.
>Call it what you like.
I did. What you're describing are two finite strings of unknown length.

>>12320752
not all "..." is the same anon. consider that PI is usually represented as 3.14... or something similar even though pi has no repeating digits.
you can't represent 1/3 in base 10 with just the characters 0-9, so 0.333... is used instead, where ... stands for whatever numbers you'd need to represent 1/3 (from the 4th place after the decimal onward in the case of 0.333).
3*0.333... = 3*0.333+3*... = 1
you can think of 0.999... = 1 as 0.999 + ... = 1, where the ... stands for whatever numbers you need to represent 3*1/3 from the 4th place after the decimal.

>>12320752
0.999999999... is INFINITE
0.9 + 0.09 + 0.009 ... doesn't make any sense, because it never ends, so technically, we say the LIMIT OF 0.999999999... is 1

>>12322768
There's no such number. If there was you could write it down. Pi can be bounded by two high precision decimals, but not given exactly.

>>12322847
>Because that's what infinity means. If it ends, it's not infinite.
The digits do not end. I merely place more digits after the unending list.
>What you're describing are two finite strings of unknown length.
Both strings are infinite.

>>12322883
TECHNICALLY 0.999... already is the limit. so 0.999... = 1

>>12322895
strings of digits are not numbers, and numbers are not strings of digits

>>12322895
Pi is defined as such. Pi defines a very specific number. You don't have to "believe" in Pi. You could reject the definition of Pi (no idea why would you) and suffer the consequence of having no Pi to make things simpler.

>>12322896
>I merely place more digits after the unending list.
Say that again but slowly.

>>12322896
>after the unending
If something doesn't end, it doesn't end.
>Both strings are infinite.
Then there's no end point where they can be connected.

>>12322925
the interval (0,1) is an unending sequence of numbers. and yet 1 comes after it. how weird

>>12322913
It's all symbols that represent numbers.
1 and 0.999... represent the exact same number.

>>12322960
That's a different poster, but now you're describing two strings—one infinite string and one finite string consisting of a single digit.

>>12322960
The difference is that 1 is not placed after any other real number. Real numbers are not a sequence of numbers. There are no 2 real numbers that have no numbers in between that aren't equal.

>>12322830
Sure. The extra historical background is that the reason those gaps needed filling was to model real-life quantities such as length, volume, mass, and time. So it is a worthwhile question to ask not just whether 0.999... = 1 holds for the abstract model, but whether it also holds for the real-life quantities.

>>12322983
Even granting that poster an infinite string of rational numbers with a limit of 1—such as 9/10, 99/100, 999/1000...—the number 1 itself could only be combined with that string (in numerical order) as a separate string.

>>12322972
Please show that it is two strings from your definition of string.
Note (0, 1) ∪ {1} is neither a string nor two strings by the definition because there are numbers neither first nor last which have neither a predecessor nor a successor. But {-1/n, 1/n | n ∈ ℕ1} would be.

>>12323022
Well yeah

>>12323004
0.999... and 1 are symbolic representations of the same real-life value. There is no difference. If real-life length of a line segment is 1, the real-life length of that same line segment is also 0.999...

>>12323033
Can you prove there is no real-life value between 0.9m, 0.99m, 0.999m, and so on and 1m?

>>12323023
As I said here>>12323022
the "1 comes after" idea you proposed makes 1 a separate string even if you're talking about the rational numbers in (0,1).
If you're talking about an uncountable set of digits, then your idea makes even less sense. Is that what you meant?

>>12323081
>makes 1 a separate string
Why? Prove it from your definition of string.

>>12323038
That's a non sequitur.
If you believe that fractions exist in real life, then there are in fact too many real-life values in that sequence for anyone to count in real life.
But that has nothing to do with the fact that 0.999... and 1 represent the exact same real-life value.

>>12323086
It has no successor.

>>12323086
It has no predecessor.

>>12323101
Fair enough. But {-1/n, 1/n | n ∈ ℕ} is still a string.

>>12323104
No argument there.

>>12323092
Certainly if you take 0.999... and 1 from the abstract real numbers, they will map to the same real-life value since they were identical to begin with. But what if you add the infinite series 0.9m + 0.09m + 0.009m + ... ? Is the limit 1m?

>>12323104
Well, I'll add the caveat that it depends on how we list the numbers. But we certainly can make a single string out of them.

>>12323114
>>12323131
Then we can replace all the items by 0 except the last, which we make 1, while preserving the order, to form the infinite string of digits 0.000...001.

>>12323118
How do you suggest we translate that into real-life values?

>>12323135
There is no last item, the string is infinite.

>>12323145
False. There is a last item in {-1/n, 1/n | n ∈ ℕ}, which is 1.

>>12323153
In what order are you counting the numbers?

>>12323138
Consider a length of 1m, oriented left-to-right. It is composed of 10 equal parts, each 1/10 of a meter. The nine leftmost parts measure 0.9m. The rightmost part is composed of 10 equal parts, each 1/100 of a meter. The nine leftmost of those parts measure 0.09m. Since the 0.9m part and 0.09m part are back-to-back, their combined length is 0.9m + 0.09m. The question is whether there is any part of the 1m length which is shorter than the full 1m but longer than 0.9m, 0.9m + 0.09m, 0.9m + 0.09m + 0.009m, and so on.

>>12323169
For {-1/n, 1/n | n ∈ ℕ}, they are ordered by their value. That is, -1, -1/2, -1/3, ..., 1/3, 1/2, 1.

>>12323186
Depends on what the length is made of.

yeah bro, 0.999...=1 looks weird intuitively so i reject it despite whatever some stranger on the internet tries to tell me or teacher

before you know it, they're gonna say 9=10

>>12323191
It need not be made of anything. Even "empty" (not really empty because of fields / virtual particles) space has length.

>>12323189
Then you're not ordering it as one infinite string, you're ordering it as two infinite strings.
If you replaced every number except 1 with the digit 0, you would get the strings 000... and 100...

>>12323208
What are you using to measure the empty space?

>>12323209
what's your definition of string?

>>12323209
>Then you're not ordering it as one infinite string, you're ordering it as two infinite strings.
Can you prove that? Again, every number except the first and last has a predecessor and successor.

>>12323216

>>12321556
>>12321567

>>12323224
Then there must exist a number for which the predecessor is in the first half and a successor from the second half. There is no such number in -1, -1/2, -1/3, ..., 1/3, 1/2, 1

>>12323224
At what number are you suggesting that your negative infinite string and your positive infinite string could possibly be connected, such that that number would have both a predecessor and a successor?

>>12323213
Well, I'm not measuring them obviously, because it would take forever, but I should establish what I mean when I say that the parts involved are equal. Being equal means their lengths would be the same if you were to measure them.

Light could be a way to do it. You could use an EM wave whose wavelength was 2/10^n of the meter. Then the division points can be the places where the expectation value of the electric field is zero. For timing we could use another wave with perpendicular polarization going the other direction. Thinking about the measurement process in detail could get really complicated, but I don't see any reason why it would be impossible to measure arbitrarily small fractions of a meter.

>>12323248
Is "connected" a property a string has to have, in your view? What does it mean?

>>12320752
in analysis you say two numbers [math]a,b \in \mathbb{R} [/math] are equal [math] a = b [/math] whenever
[eqn]|a - b| < \varepsilon[/eqn]
For all [math]\varepsilon > 0[/math]
So indeed 0.999... = 1 where equality is in the sense of the above definition.
You may argue that they should not be equal. but that (in)equality will be of a different kind than the one discussed above.

>>12323305
Let epsilon = 0.000...001

>>12323293
There's no need to add anything to the original definition.
You've described one string that starts at -1 and gets closer to 0 forever while staying negative forever, and a second string that starts at 1 and gets closer to 0 forever while staying positive forever. Both of those strings fit the definition fine.

>>12323336
That's a finite string.

>>12323367
As does the combined string.
>>12323372
No it isn't. Read the thread.

>>12323376
At what point do you suggest combining your two strings?
>No it isn't. Read the thread.
That was me chiming in.
0.000...001 is a finite string starting with a 0, a decimal point, an arbitrarily long finite string of 0s, and ending in 1.

>>12323376
>the combined string
At what point do you suggest combining your two strings?
>No it isn't. Read the thread.
That was me chiming in.
0.000...001 is a finite string starting with a 0, a decimal point, an arbitrarily long finite string of 0s, and ending in 1.

>>12323398
>At what point do you suggest combining your two strings?
Prove there must be such a point. I simply combine the strings by putting one after the other.
By 0.000...001 I mean the string defined here. >>12323135 It is infinite.

>>12323415
>the string defined here
You defined two infinite strings there.
>I simply combine the strings by putting one after the other.
The negative string doesn't end (nor does the positive string). There's no "after."

>>12323448
>You defined two infinite strings there.
I defined one infinite string. It has two parts, each of which are infinite strings.
>There's no "after."
"After" here simply means that all the digits in the second string come later than all the digits in the first string. Do you claim 1 is not greater than -1/2?

>>12323489
>each of which are infinite strings.
Yes, you defined two infinite strings. Not one.
>all the digits in the second string come later than all the digits in the first string
Not in the order you listed. In your order, you've listed both the 1 and the -1 as starting points, then defined two separate sequences of numbers that each continue forever.

>>12323555
>In your order, you've listed both the 1 and the -1 as starting points
Incorrect. 1 is the ending point. The second string has no starting point.

>>12323489
>Do you claim 1 is not greater than -1/2?
Your negative string is ordered from least to greatest, and your positive string from greatest to least.

>>12323573
>your positive string from greatest to least
False. Both are ordered from least to greatest. Please pay attention.

>>12323562
>The second string has no starting point.
Then it fails the definition, which defines a string as having a first digit.

>>12323582
If having a first digit is a requirement, then it is indeed not a string. But the combination of the two remains a string.

>>12323578
If you're trying to order your positive string from least to greatest, you need to start with a least number. What number is that?

>>12323601
>you need to start with a least number
Why?

>>12323600
>the combination of the two
At what point do you suggest combining your two strings?

>>12323608
There is no point of combination. One is simply placed after the other. I've explained this before.

>>12323607
Because if you don't, then your string isn't ordered from least to greatest.

>>12323614
There's no "after." Something that is infinite doesn't end.

>>12323616
If you're quibbling about the word "least," you can consider the ordering "lesser to greater" instead.

>>12323620
It does not need to have an last element for things to come after it.

>>12323627
What's your starting point then?

>>12323637
You can't add something to the end of something that has no end.

>>12323644
Of the positive numbers ..., 1/3, 1/2, 1? There is no starting point.
Of the numbers -1, -1/2, -1/3, ..., 1/3, 1/2, 1? The starting point is -1.

Be back later.

>>12323656
It's not added to the end. It's simply added after.

>>12323656
read something about ordinal numbers and stop making yourself look like a total brainlet
t. not the one you're arguing with

>>12323662
>not at the end
>added after
Do you seriously not get the problem here

>>12323767
There is no problem. Again, something does not need to have a last element for things to come after all its elements.

>>12323774
They don't but that would violate the logical conclusion of the definition of a string.

>>12323779
What part of the definition does it violate?

Anyway going back to the -1, -1/2, -1/3, ..., 1/3, 1/2, 1 "string".

Let's extend the definition of a string.
>a sequence of digits such that every digit after the first has a successor and a predecessor
From that, it follows that you can divide a string at any point to get substrings of any length. All substrings are strings themselves.
That also means that strings *have* to have a first and last element. Which means that all of their substrings also do.
Let's assume -1, -1/2, -1/3, ..., 1/3, 1/2, 1 is a string. Because it's a string you can divide it in half into a string -1, -1/2, -1/3, ... and a string ..., 1/3, 1/2, 1. Neither of them has both a first and a last element meaning they cannot be strings. If they aren't strings they cannot be substrings of the original "string". The conclusion is that -1, -1/2, -1/3, ..., 1/3, 1/2, 1 is in fact not a string.

A follow up from >>12323790
It all follows from the fact that there is no digit that connects both parts of the supposed string.

It may or may not mean that strings have to be finite. I'm not sure about that yet.

>>12323790
>a sequence of digits such that every digit after the first has a successor and a predecessor
By this definition 12345 is not a string because 5 does not have a successor.

>>12323804
Sorry, it was said before that >apart from first and last element

>>12323790
>>a sequence of digits such that every digit after the first has a successor and a predecessor
>>12323824
>Sorry, it was said before that >apart from first and last element
000
000...
...000
...000...
000...000
000...000...000...000...
all are strings

>>12323736
>brainlet
Anon's Law: you lose.
>ordinal numbers
Yes, that poster is describing ω+ω, not ω.
Perhaps you're the one who need to brush up on ordinals.

>>12323658
>-1, -1/2, -1/3, ..., 1/3, 1/2, 1
You're describing two strings.
The one on the left can never be positive and the one on the right can never be negative.

>>12323662
>It's not added to the end. It's simply added after.
Correct, it's a new string.

Brb later.

>>12323855
>You're describing two strings.
The combination is a string. If not, explain why using a definition of string.

>>12320964
>9.9999....0

>>12320752
0.999... does not equal 1 because 0.999 isn't a real number. infinite is a mumbo jumbo abstract concept. the entire premise relies on the plausibility of manipulating infinite strings.

>>12321492
what is 0.99999...*7.2?

protip: it's 7.199999...28. an infinite string, ending in 28. multiplying infinite values is paradoxical and doesn't work.

>>12323888
>what is 0.99999...*7.2?
[eqn]0.999\dots*7.2 = \sum_{n=1}^{\infty}\frac{9}{10^n}\cdot7.2 = \sum_{n=1}^{\infty}\frac{64.8}{10^n} = \sum_{n=1}^{\infty}\frac{6}{10^{n-1}}+\frac{4}{10^n}+\frac{8}{10^{n+1}} =\\= 6+\frac{6}{10}+\frac{4}{10}+\sum_{n=2}^{\infty}\frac{6+4+8}{10^n} = 7 + \sum_{n=2}^{\infty}\frac{18}{10^n} = 7 + \sum_{n=2}^{\infty}\frac{1}{10^{n-1}}+\frac{8}{10^n} = \\ =7 + \frac{1}{10}+\sum_{n=2}^{\infty}\frac{1+8}{10^n} = 7.1 + \sum_{n=2}^{\infty}\frac{9}{10^n} = 7.1 + 0.0999\dots = 7.1999\dots [/eqn]
it works just fine man

>>12323954
you're missing the 28 at the end friend.

>>12323973
Why would there be 28 at the end?

>>12323988
all 0.999etc. number strings, when multiplied by 7.2, end in 28, regardless of size. an infinite version of that doesn't magically lose it. it's a bottomless hole with a bottom, an absurdity.

>>12324008
>an infinite version of that doesn't magically lose it.
actually it does

>>12324008
To be honest I don't get this. All this talk about what's real and what's not, what can be represented in reality and what can't... But then you claim that you can put digits after an infinite decimal expansion. But as the number of terms of an infinite series grows, any part of the sum added "after" the decimal expansion tends towards 0. So what's the actual point? What are you people arguing for?

>>12324045
I'm saying you CAN'T put digits at the end of an infinite string. multiplying an infinite string is fundamentally broken.

>>12324056
Multiplying infinite strings is fine. It works just as expected.

>>12324071
the strings you're multiplying aren't actually infinite, friend, just a rudimentary representation.

>>12323863
You defined the order you want as two strings.
12323189
{-1/n, 1/n | n ∈ ℕ},
Do you want to redefine the order you want as one string? If so, what’s the new definition?

Brb tomorrow morning. None of the rest of this is me except for the two brb posts.

>>12324095
That doesn't really matter if we know how the whole string will multiply. We don't need to literally write down the whole infinite decimal expansion.

>>12321698
>>12321725
>>12321699
Fucking newfags.

>>12324140
Again, if you think it is not a string, prove why.

>>12324173
the point is you can't, genius. how do you know the end result multiplication of a fundamentally untranscribable concept? you fucking don't. it's all assumptions, usually paradoxical ones.

>>12320763
Let me actually try to explain it.

Basically what people say is that they are "infinitely close together". The difference is .00....1.
Also:
a = .9999...
10a = 9.999...
9a = 10a - a = 9
a = 1
Thats a proof my teacher showed me.

>>12320752
mathematics doesn't about what you "believe" this isn't like religion or even natural sciences

>>12321699
0.999... is almost 1 but not one. the logic used to say 0.999... = 1 is borderline troll science comic tier

>>12321580
why does cos(2pi)=1? cos(x) is an infinite sum and an infinite sum can't be a whole number (.999.... =1)

>>12321595
>define 1/3 as the unique number such that 1/3*3=1
>therefore 1/3*3=1

>>12321595
1/3 = 0.333... is a failure of the number system, not a failure of mathematics. 1/3 works perfectly fine in base 12.

>>12320898
> I don't know what limits are
The fuck, are you in precalc?

>>12324250
They aren't infinitely close together and .00....1 is a nonsensical notation that doesn't mean anything.

There isn't really such thing as 2 different real numbers that are "infinitely close together". By definition, such numbers have to be the same number.

>>12320752
Give a math autist 99 pennies and a dollar bill and watch them squirm

>>12324267
Let k be a real number equal to whatever 0.999... is.
I believe we can both agree that k is smaller than any number x where x > 1.
For any y such that y < 1 there exists a number z with n 9s after "0." such that y < z.
k > z
Therefore k is larger than any number smaller than 1 and smaller than any number larger than one. The only real number for which those are true is 1. So k = 1.
k = 0.999... = 1

There's nothing trollish about this and I don't understand why is it so difficult to accept it.

>>12324374
Meant to write
y > z
k > y
k > z
instead of
k > z

>>12320752
You are fundamentally misunderstanding something about math. Here it is:

The definition of a number is NOT "a string of digits."

That's it. Grade school math/high school math doesn't address this, so you go through school thinking that a number is a string of digits and a string of digits is a number.

So if you got two DIFFERENT strings of digits, that's gotta be two different numbers, right?

A number is an element of a set. The digits are just a way of making the number easy for humans to deal with it. There's no reason for it to be unique.

That'd be like saying that 10 can't be equal to 7 + 3 because we already know it's equal to 8 + 2.

>>12321244
Don't stop there, dude.
1/inf = 0
2 * 1/inf = 2/inf = 2*0 = 0
1/inf = 2/inf
multiply both sides by inf
1 = 2
Yay! Where's my Fields Medal?

>>12324337
>0.999...
>real number

>>12320846
1/3=0.3...
By Axiom.

>>12324407
>multiply both sides by inf
inf/inf is undefined

>>12324374
so you denote a planck length in numbers and decide that means both numbers are the same.

When two numbers are very close to one another they are declared to be equal by mathematicians.

>>12324568
What, where? Wtf are you talking about

Yes. It has been proven time and time again.