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Through specific mathematical processes and...
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Through specific mathematical processes and conversions, and through these ONLY, can 0.999... = 1.
No black and white answer, just specific circumstances which dictate whether or not 0.999... can = 1.
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anyone with a decent understanding knows .99... =1
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I think the best way to resolve this question is to define what you mean by 0.999...

What do you mean by 9/10 + 9/100 + 9/1000 + ...

What does it mean to perform an infinite sum.

Once you answered these questions, the answer to "is 0.9999... exactly equal to 1" should be obvious.
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Summerfag, math was disproven long ago, it's just convenient because, like classical models of physics, it works well enough.
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9(1/10)+9(1/10)^2+. . .= (9/10)/(1-1/10)=1
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>>702462848
Then explain it, faggot. If you understand it, explain it clearly.
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>>702463042
>>702463079
disprove .9999...
I know of 3 different methods used to obtain .9999=1
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>>702462888
>>702462848
Yes, but most people see this as someone stating that a number that is shown to be smaller is equal to a bigger number.
It is initiated by people who see the statement as simply a decimal that is equal to a whole number.
This is the only reason why this should not be shown as a sort of unbelievable clickbait for the purpose of therein educating using processes.
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>>702463216
>>702463079
>>702462500
the final method is to simply have 1/3= .333...
(1/3)*3=1=.333...*3=.999...
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Its quite apparent that the way that the fact is proposed that it snags people into learning something if they can suspend thier disbelief and apply processes in thier head to the question, but only AFTER a shitstorm.
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>>702463042
lines 4 to 5 is where it goes wrong
Please define the square root operation
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>>702463735
Are you high?
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>>702463783
Exactly. The duality shows its arbitrariness.
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>>702462500
there you go. dude has other nice articles as well

mathforum.org/dr.math/faq/faq.0.9999.html
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>>702463903
Someone who hasnt read this shite before sees a thread like this.
Thread comment "0.9 recurring =1"
They think that its impossible.
Shitstorm ensues, they learn a few mathematical processes that prove the thread comment.
Rinse and repeat.
The threads normally consist of fucking drawn out maths lessons because people still see a statement that tells you a number that begins in a decimal smaller than the exact number of 1 is the same value.
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>>702463514
You introduced a fraction and a different divisor than the original. Nice smoke n' mirrors. Again, where is your clear explanation again???
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>>702462500
Imagine it as a girlfriend
just like in life you only get really really really close to it ,but yet never get one.
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>>702464313
from the same guy a way easier explanation:
mathforum.org/library/drmath/view/55746.html
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>>702464313
>>702464604
These link to viruses, asshole.
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>>702464410
1/9 = 0.111...

now multiply both sides by 9

9 * 1/9 = 10 * 0.111...
9/9 = 0.999...
1 = 0.999...

It does not get more clearer than that
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>>702464871
>9 * 1/9 = 10 * 0.111..

sorry typo

9 * 1/9 = 9 * 0.111...
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>>702464752
those link to a math forum you troll
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>>702464410
if 1/9= .11111...
2/9= .2222...
then it would make sense that 9/9 = .9999.... and 9/9=1.
Also use the convergence theorem in geometric series. see >>702463079
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>>702464410
The introduction of fractions and whatnot are the reason why processes are able to show that 0.9 recurring can = 1.
Simply looking at the statement at hand and only being able to use decimals in an attempt to prove it isnt enough.
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>>702464871
Except OP starts with x=.99999. He does not start with 1/9=.111111.
That's my point. Nobody can clearly explain OP's original problem, and why math gives arbitrary outcomes, like multiple answers, some of which are tossed and labeled "extraneous". Oh, convenient, just throw away the parts that don't work.
Just stop, your embarrassing yourself.
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>>702465185
My point exactly, smoke and mirrors. An equation that purportedly "works" should be explainable on its face. Whatever, the autism is at a fever pitch in this thread.
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>>702465173
That's called inductive reasoning. Mathematics should be proved with DEductive reasoning. Fucking charlatan.
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>>702465389
what are you talking about? you have two valid equations, OPs and mine. Manipulated in a valid way and both show that 1 = 0.999...
The approach is just another. You have to fortidude your understanding of algebra if you don't understand it instead of accusing others of making an error.
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How do we express "just under 1 then"?
Usually I do 1⁻ but in my understanding that was equal to .999...
Does 1⁻ exist or is it just a concept?
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>>702465789
theoretical mathematics isn't your thing, is it?
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>>702465389
It's not an arbitrary outcome. It proves that if x = 1 it is also equal to 0.999... because 1 = 0.999...
The other euqation comes to the same result.
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>>702465937
Here's a theory: fuck you. Come back when you understand the difference between theory and fact.
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LISTEN HERE YOU FUCKERS

I got the image without approving of it myself. Just needed a related picture.
There are processes which work, but bear in mind, ALL OF THEM USE FRACTIONS BECAUSE YOU CANT DISPLAY INFINITE NUMBERS WITH DECIMALS AND NOT CONVERT THEM. If the decimal doesnt end, there is no way to pull out a fucking 0.000......... hence the fractions
Of course the shit isnt going to be exactly feasable using only decimal logic. Some people will only see it as the tiniest of decimals below 1. I dont even fucking know anymore
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>>702465389
Do you have a method that shows that 0.999... =/= 1? I would love to see it. I won't throw it away as extrenuous.

I'm a mathematician btw. I tutor kids, and I'm studying to be a teacher.

I have never seen any proof that attempts to prove that 0.999... =/= 1. I didn't even realize it was called into question until this thread.

If you have such a proof, I can take a look at it, and formalize it with first order logic, or something.
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>>702466150
0.000..........1*
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just google it, not hard.
wikipedia.org/wiki/0.999...
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>>702466125
Seems like a suggestion and not a theory.
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>>702466245
>I didn't even realize it was called into question until this thread.

It's not called into question.
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>>702465789
Wat.

Inductive reasoning is how you know that there's no largest infinite number. You wouldn't be able to even conceive of the notion of decimal places without inductive reasoning, since the whole basis of the Arabic numeral system is to assume an infinite number of possible digit places.

You can think of decimals as the summation of a^n as the index n, begins at negative infinity and approaches positive infinity. As such, the strategy of multiplying 0.111... is completely legitimate, since it's just distribution.
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>>702466245
Uh, fuckface, you are the one positing that .9999....=1, not me. That bullshit assertion is on YOU to prove, not me. So far all I see is people saying, "oh, it follows from a pattern that works with other fractions THAT ASSUME THAT WHICH IS TO BE PROVED!!!"

Sorry, I reserve allcaps for truly stupid situations, and please don't go into teaching. The kids are dumb enough.
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Numbers like .33 are decimal approximations. Numbers like 1/3 are exact values. This is why if you actually do math you don't use both at the same time.
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>>702464410
you realize that repeating decimals are a symptom of the base you choose, right?
any time you ever come up with 0.999... in a calculation, go back a step, then convert everything into base 3, now redo the step and convert back to base 10.
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Remember, patience is key. Try not to swear like crazy at each other at the mere sight of a conflicting opinion, and instead understand why it is considered to be true from another standpoint, then go from there.
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>>702466245
0.999... = 0.999...
EZ
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>>702466937
Thanks for making /b/ a safe space. In that spirit, I will include a trigger warning with this post. You are about to be called a faggot.

Faggot.
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>>702466312
YEAH YOURE WELCOME RETARDS
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>>702464175
"the duality" of the square root operation is anything but arbitrary. That said I think the point you are attempting to get at is that mathematics is the study of the consequences of axioms. These axioms describe a system, which, obviously is not necessarily related to the world we live in. However to assert from that the meaningless statement "math was disproven" shows a deeper lack of understanding.
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>>702467177
LISTEN HERE YOU FUCKING SPIC BASTARD CUNT WANKER PRICK ASSHOLE FAGGOT CHINK NIGGER SPIC HONKEY BITCH

thats pretty mean but i appreciate the warning
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>>702466150
>YOU CANT DISPLAY INFINITE NUMBERS WITH DECIMALS AND NOT CONVERT THEM

Sure ya' can. Just use a geometric progression.
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>>702467080
I would agree with this. It's called the reflexive property. It does not apply to 1=.9999.
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10x=9.990 not 9.999

retards
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>>702466647
What's the decimal representation of 1/3? Triple it? That number is either equal to 1 or division and multiplication don't work.
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>>702462500
Except step 3 is improper, one side is minus x, the other by .99, have to choose one or the other and do both sides by it. Yes, it's equal, but at the same time, algebra is butthurt.
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>>702466125
How autistic are you?
if it weren't for abstract mathematics electrical engineering would only be a mere fraction of what it is today. Go actually learn something past elementary mathematics.
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>>702467359
I picture you saying this in a calm, British voice while wearing a top hat, monocle, and tuxedo top, but totally pantsless. You're also holding a martini.
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>>702466125
>what is an inductive proof
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>>702466647
Mmk, well there's a been a couple proofs in this thread already showing that 0.999... = 1

So if you're not smart enough to figure out math on your own, you could try looking at some of those.
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>>702467687
well you got me pegged on the pantsless
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>>702467418
The fuck? Im out of my depth, summerfag OP out.
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>>702467359
I always get excited to learn something in the style of Kurt Gödel's incompleteness theorem when these "MATH ISN'T REAL" threads pop up, but am always sorely disappointed.

Funny thing, I don't think I've ever learned a thing from a skeptic before. Have you?

I get this one a lot.

"It's not up to me to prove what I think isn't true, it's up YOU to prove to ME what I think isn't true, IS true."

But I've never once learned something from them. Thinking back, I could have spent my entire life ignoring them, and know exactly as much as I do now. Hm.
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>>702467621
> electrical engineering
And yet science can't explain why charges oscillating in AC with no net change in position still yield energy that moves in one direction.
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>>702466794
disprove the geometric series proof one then.
>if you don't know which one that is, just take a seat and keep the dumb to yourself
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>>702466937
fuck you nigger
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>>702467696
My point exactly. Use deduction, but nobody here is.
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>>702467418
>>702467874
samefag, but humorous and true
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>>702467451
how are you not the head of some mathematical research facility
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>>702468513
rest
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>>702468083
I personally find Godel (and the entire mathematical history of the time re set theory and such) absolutely fascinating. What it really hints at is the lack of our ability to model what we describe as truth using traditional axiomatic systems. Of course I always found this to be quite an unintuitive result, because I always assumed (and still hold) that truth can be modeled by a formal system in one manner or another, though this is could get more into philosophy and metaphysics than mathematics.

I also, to be frank, don't find it surprising that I have not learned any deep fundamental flaws with systems by random people on /b/ that are not already common knowledge.

In addition it has always astounds me the attitude people can have (not just concerning mathematics) that if we don't have enough evidence for to show that something is true or isn't its only natural to choose based upon some arbitrary "gut feeling" of some sort and believe that whole-heartedly instead of just accepting that you don't have enough information and trying to figure out more.

I suppose it shouldn't surprise me, but the rigor with which some people will insist upon believing something simply because there isn't a 100% proof that it is false is quite disconcerting sometimes.
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>>702468114
I've never heard of that. link it or lies.
Also, explanation: they travel through a higher dimension.
>inb4 "That's just sci-fi!" go join the crowds saying objects heavier than air can't fly. We'll never understand stars. Hell, even Einstein though it would be impossible to detect gravitational waves.
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>>702469219
Everyone "disagreeing" is trolling. Nobody actually disputes these axioms. Now go on back to redit-tumblr-wherever. The kids are playing here.
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>>702469259
Asshole, I said nobody can explain it, so what the fuck am I supposed to link to? How about YOU link to an explanation???? Man, the Saturday night fever. Fever of autism.
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>>702469219
>I suppose it shouldn't surprise me, but the rigor with which some people will insist upon believing something simply because there isn't a 100% proof that it is false is quite disconcerting sometimes.

In an information age, certain humans have decided to mirror the shape of a binary code, when establishing their belief patterns in life;

upon hearing a possibility, they are compelled to believe with absolute certainty that it is "true," 1, or to believe with absolute certainty that it is "false," 0. They cannot continue their train of thought before assigning of these two values to every idea they ever come across. If you interrupt them before they finished inputting their binary code, they actually forget the train of thought that led them belief pattern they were interrupted in the process of assigning binary values to.

I think it's neat. They're like... little computer people. They even use similar vocabulary so that ordinary humans know who they are.
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x=0.999
10x=9.990
10x-x=9.990..-0.999
9x=8.991
x=0.999

maths doesn't lie you complete fuck ups
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>>702469219
>>702469551
if that's what reddit is, why the fuck do I hang in this cesspool of fb fap and "death is inherently funny cause i'm uber edgy"
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>>702469794
My niggah. Everyone thinks I'm trolling, but this is exactly what I was trying to say. Thank you.
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Lol nerds
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0.999.... is a concept that represents that the 9s are repeating forever into infinity. It's not a number. Furthermore the equivalence between the concept .9repeating = 1 is not arbitrary in any way. If you think it is you can't comprehend calculus at all and are not qualified to question it until you spend the time to learn it.

>>702463042
(-1/1)^0.5 != i/1; the rest of the math is thus also incorrect.
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>>702467177
>I should people able to say stupid shit without people pointing it out to me.
Why do anti-SJWs hate freespeech?
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>>702470197
.9999 is not a number.
If it's not a number, then does it make sense to multiply it by 10?
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>>702469219
isn't using postulates considered a formal system?
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>>702469784
You described it perfectly, though I would argue that it far predates the information age. As a matter of fact, I think for most people the "decision" they make when they encounter a choice is based purely on which one is more consistent with whatever decisions they have already (probably arbitrarily) made.

>>702469551
I honestly think /b/ has a higher average IQ, everyone on here is retarded, but they are much more self-aware about their autism than leddit or especially fucking tumbl
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>>702469710
link to the problem
If real, it can be explained in theories, not yet proven though
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>>702470401
.9999 is a number. 0.999repeating is not a number.

Why does it make sense to multiply it by 10? because now the concept is 10 times bigger. Apple is not a number, but if you have one apple and then I ask you to multiple the amount of apples you have by ten, is it not 10 apples?
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>>702469784
I believe this has more to do with personality than external influences
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>>702470518
Wait, you're an electrical engineer and you don't realize that alternating current is separate from the flow of energy?
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>>702470403
A set of postulates, or axioms as I described them describe a formal system yes. Godel's incompleteness theorem essentially shows that any set of postulates that meet some sufficient requirements result in some statements that can neither be proven or disproven, which is a huge and somewhat counter-intuitive result that has very big implications. Not sure what your question is beyond that.
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>>702469710
It's because it's not oscillating AC per se. It's oscillating in a 3d spiral, which has a bias in direction, depending on the rotation of the spiral.

If you offset the 2d vertical AC component of the charge by half a wavelength, it will change the direction of the energy current.

The energy is produced as a result of E = Mcc, since the oscillation offsets the charge's mass due to relativistic time dilation. Understand through the lens of quantum field theory, the probably of detecting charge in that region is higher, when its rate of oscillation increases.

I'm pretty sure all this is covered in Einstein's original 1905 paper about the electrodynamics of moving bodies, and was actually the entire point of that paper, relativity just being a minor consequences in order to connect the dots.

Or else I'm completely misunderstanding what you're getting at here.
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>>702465844
it is a surreal number, not a real number. It is as conceptual as any other number
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>>702470718
An apple is a discrete thing, not a concept, so your analogy fails
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>>702469794
it's supposed to be an infinitely recurring 9.
.99 != 1
.999....=1
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>>702464410
Roughly .571, or
1-(2-(pi/2))
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>>702471020
An apple that doesn't exist in reality. Now it's no longer a discrete thing.
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>>702470197
disprove the geometric series equation then.
https://en.wikipedia.org/wiki/0.999...#Infinite_series_and_sequences
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>>702470967
No, this is helpful, and a legit thank you. I was trolling because I couldn't find a clear explanation in simple searches, and was hoping someone would take the bait and give me an answer. I could have just asked, but...well, we're on /b/, right?
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>>702471010
>How do we express "just under 1" then?

1-n, as n approaches 0.

Just isn't implicitly a number. If you want to treat "just under" as a mathematical concept, then you need to define how far away "just this far" is. Which is done by just letting n = "just this far."

That's all you can ever do with math.
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what is the number between 1 and 0,999... ? If they are different numbers, there must be one in the real set by definition
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>>702471327
For sure.

I like to rewatch this video from time to time, to make sure I'm visualizing waveforms correctly.

The dialogue mentions only specifically polarized light, but the 3d animations he uses to add wave vectors together is... pleasing to the eyes. Relevant to most geometries that deal with transverse waves in three dimensions.
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>>702471462
https://en.wikipedia.org/wiki/Surreal_number

but you can take it as a limit in reals if you want to, there is some dualism going on here
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>>702469794
fucking retard there cant be different number after 9 if 9's are infinite
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>>702470874
I'm not an electrical engineer. What I got from your post is that ac moves in one direction with no net change in position. Unless that direction is a circle I hold your statement unlikely; however, it would be extremely fascinating if true. Kind of like like electrons changing orbit fields
no, I'm not a chemist
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>>702471580
There is no difference, op is a retard. 0.999 reoccurring is just a different way of writing 1
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>>702471327
it's strange that you've got to manipulate emotions to get a non-personal response sometimes
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>>702472008
I know, I was talking about 1- , that is a total different thing
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>>702472392
What? You asked what is the number between 1 and .999 reocc. did you not?
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>>702472392
>>
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>>702471462
fine, infinite-small under 1
Math and autism go hand in hand in this concept
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the layout is cheesy, but this is a good explanation of how people can disagree about this.

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>>702471812
I really appreciate how they broke down the vector addition so slowly. I have a hard time visualizing that in my mind, and am amazed at folks who did that 100+ years ago without computers. And yes, very pretty graphics.
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>>702471580
what's the number between 1.999..(infinitely recurring 9s)..9998 and 1.9999... then?
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>>702462500
if x =1
then 10x !=9.999

x=0.9999999999999999
10x=9.999999999999999
10x-x=8.999999999999999
8.999999999999999!=9

if .999.... = 1 then 1 = 2 and 2=3 etc, etc.
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>>702472259
Again, we're on /b/.
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>>702471842
>https://en.wikipedia.org/wiki/Surreal_number
No, I dig it. Didn't realize that was a thing.

The 1-n approach is sloppy, no doubt. If there's a different way to arrive at a number, then it exists, no matter how many abstractions ya' go through to get there. I've got some reading to do.

*glasses bump*
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>>702473035
>>702472008
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>>702473079
...you're not too bright are you
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>>702473550
Nope, what lead you to that conclusion?
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>>702473278
if .99...998 = .999... and .9999=1 then 1=.999..998
by this logic we can reach 0 = 1
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>>702473157
and then you got an even bigger set, Games, where there are some fucked up numbers that are "smaller, bigger and not equal to" real numbers
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Not going to dive in to this db8. Just want to point out that any repeating number is NOT 100% accurate.Just as close as we can depict with decimal notation. That's why we use fractions in the first place.
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>>702465389
It's not arbitrary, it's a quirk of the decimal system. 0.11111... is simply 0.1 in base 9.
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>>702473950
But .999 reoccurring does not equal .998
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>>702474368
recurring decimals are abstract, not concrete
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>>702474565
he said .9999... equals 1 because there are no real numbers separating them
by that logic:
.99..(infinite 9s)..998 = .999... = 1
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>>702474748
I don't know who said that, but I said .999 reocc. is just a different way of writing 1, not because there are no numbers between, but because that's just how it is.
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>>702474552
We're uh, in a base 10 system. Count your fingers or toes to see what I mean. Then again, you probably have lost some from frostbite, as you're clearly a neckbeard who wears shorts and sandals in all seasons. Whatever, don't change the rules to fit your broken argument.
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>>702474748
what does it mean that you got an 8 (and then an end) after an infinite secuence?
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>>702475123
It means you are taking the bait.
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>>702475080
math is the same in all bases, the only thing that changes is representation
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>>702475123
two infinitesmall under 1
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>>702475258
More like MISrepresentation. This is why kids hate math. Unnecessarily convoluted, and 128 replies later, no simple, clear answer.
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>>702475479
Clear answer to what question?

I'm sorry, I'm a math tutor, studying to become a teacher. I didn't realize you had a question.

If you state the question again, I'll try to answer it clearly.
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>>702475123
>>702475262
or one infinitesmall under if .999...=1
it's not the best way to express it though
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>>702475262
infinitesimals are not part of the reals
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>>702475736
1-2ε
happy?
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>>702475479
ok, here you go:

1/3 = 0,3333...
3/3 = 0,9999...
1 = 0,9999...
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>>702475604
not him, but how are infinitesmals defined? I understand they're smaller than any real number, but if you add enough of them together they should equal a real number, right?
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>>702475893
that's the same as you said before, still an infinitesimal, not a real number.

Anyways, there is no infinitesimal inbetween 1 and 0,999... for the same reason there is no number (real or not) between 2 and 2. Doesn't matter how you approach it
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>>702476222
how do I say .999..(infinite 9s)..998? It's just as real of a number as .999..(infinite 9s)..998
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>>702476409
how do I say .999..(infinite 9s)..998? It's just as real of a number as .999..(infinite 9s)..999
fixed
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>>702476213
Yes. Integration wouldn't work, if it wasn't just adding rectilinear aligned infinitesimals with coefficients in front of them. But it does, so that's neat.
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>>702476452
I think there is no such thing as 0,999...998, because infinite secuences don't 'land' in some number after an eternity. They just keep going.
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>>702476452
You'd have to use summation notation.

Σ(f(n)*a^n) as the index of n begins at 0, and approaches ∞ . Then just define f(∞) = 8, and f of any other input is 9.

https://en.wikipedia.org/wiki/Summation

It's not an easy thing to type, without knowing all the proper alt-codes for subscript, and superscript, which I do not.
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>>702477189
There are ways to mathematically define it. I could say something like the square (1-n) as n shrinks to 0, minus nn. (1-n)(1-n) - nn = 1-2n+nn - nn = 1-2n. So this way as n shrinks down to zero, it's twice as far away from 1, as 1-n is, even though n is infinitely small in either case.
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>>702476487
By trying to understand the value of an infinitesmal, I've just realized how hard of a concept infinity is.
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>>702477556
oh, that's pretty cool actually
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>>702477573
I'm having a crisis actually
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>>702471048
if you mean the area asked by the image, it's pi/4-(1-pi/4)...pi/2+1
and not pi/2-1 as you stated...
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>>702478213
realized my error nevermind....it's the same area
i misscalculated the on the final step
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>>702477573
You can demystify it by doing all of your arithmetic with modular arithmetic first, and extending your number base as needed.

https://en.wikipedia.org/wiki/Modular_arithmetic

Many infinite sequences have simple solutions when you apply them to modular arithmetic. For instance, the famous geometric progression, 1+2+4+8+16... appears to equal -1. This is weeeeird, right? How can a sequence that gets larger become negative?

But if you do it with modular arithmetic, it makes perfect sense. Real fast, let's make a table of powers of 2, with a modular base capping the largest value out at 5.

So,
2^0 = 1
2^1 = 2
2^2 = 4
2^3 = 8, which rounds down to 3
2^4 = 16, which rounds down to 1 again

Since it repeats, you really only need to think about the first four powers.

1+2+4+3=9, which rounds down to 4, which is -1 in modular 5 arithmetic.

You can do this for every base of modular arithmetic, and find that it is true. So, if it's true for every modular base of arithmetic, why wouldn't it be true when you're counting in mod base ∞ ?

And is true when counting in mod base ∞ . So, sweet. Math makes sense, if you just understand that infinity wraps around, but it's also so big that you'll never get there by adding real numbers. It still wraps around though.
>>
hehe
now i'll drop the real f bomb:
1+2+3+4+5+...=-1/12
off to sleep
good evening trying to understand that fellows
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>>702462500
The problem here that all of you fags are just skipping is the fact that f.ex 1/3 doesn't actually equal 0.333... , thats an approximation used to make us able to convert from irrational fractiona to numbers. 1/3 = 0.333... Is by no means a precise answer, but 1/3 * 3 =1 is. It all has to do with the amount of uncertainty we accept in the final answer!
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>>702478487
excellent....could >>702478505 make sense with this logic?
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>>702478505
I don't accept that solution. If anyone wants to show me what proof they use to arrive it, I'll show, using my modular method, where there's an invalid operation. It has to do with the number of values that are left shifted. Although it is true that you can cancel out certain values in an infinite sequence by adding them to others, I maintain that the amount of values being cancelled in this way are relevant. You can't just validly right shift them out of existence, and this is demonstrable when using modular arithmetic with a base that extends to ∞ .
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>>702479046
shifting sums isn't the only proof....
there's also the analitical expansion of the zeta function (Riemman's one)...a proof that uses calculus
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>>702478966
The value I get when I use infinite base modular arithmetic to compute that infinite sequence is... well, ∞/2.

But it's definitely not 1/12. Although 1/12 is a famous result, I reject it. I assert that Σ(n) for n = (0,∞) does not converge on a real number.
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>>702463042
line 6?

sq both sides should be 1/i^2which equlas 1/-1?
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>>702479235
No. Line 4.

sqrt(1/i) =/= sqrt(i/1)

It should be sqrt(1/i) = -sqrt(i/1)

Idunno. They covered this in MY school. We were carefully explained that square roots have two possible solutions, and that this matters when rotating around the complex plane. Which is what this proof is doing--just rotations.

If you do it again with Euler's identity, you'll see that what the image does is impossible. Since 1/-1 and -1/1 are multiplicative inverses, by definition their square roots must be conjugates.

When dealing with advanced mathematics such as complex number theory, you can't afford to leave out things like how roots have two possible solutions.
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>>702479200
i answered on >>702479187
also i said -1/12...no 1/12
be aware