1=.999999...?

[TheMeInTeam]

It's an infinitesimally small number.



https://en.m.wikipedia.org/wiki/Infinitesimals

My notation is bound to be wrong ofc.

You contradict your own definition of infinity.

 

[Borachio]

OK. I've phrased that wrong. I really mean you can't do calculus without the concept of infinitesimals, because that's what you're using in effect when you take limits as delta x turns into dx. At least, that's how it appears to me - having studied infinitesimals in depth for at least 5 or 6 seconds.

 

[brennan]

I wasn't aware that I had defined infinity anywhere...?

 

[Hygro]

What?
First of all, your war-comparison is about different things with one name, while 1=0,999... is about the same thing with more names, so to say.
Then the second paragraph doesn't make sense. it's neither awkward or inelegant, really.
Also, my post 646 is important, okay

I understand your confusion. We get to a war on drugs by saying that drugs are the enemy, here is a political force designed to literally combat that which is aligned with drugs, therefore, war on drugs is war. It's also simply propaganda and nonsense, you can't declare war on drugs. Drugs are not a political entity represented by an aligned population. Drugs are a criminalized enterprise. Political entities are not enterprises. By a logical understanding of war, this is ridiculous.

We can get to 0.999... by 1/3*3 = 0.333...*3 = 0.999... = 1

By a reasoned read of the notation this is ridiculous. But the logical rules get there.

Here's where it gets interesting:

The "drug war", despite being "0.999..." (aka not being "1" aka "war" because drugs can't be a political agents; it's very clear by the contradicting logic) is in fact a war. It is an armed political apparatus designed to fight a political enemy (leftists, poor people/black people etc. but now expanded to include para-governmental gangs in Mexico and elsewhere) that results in out and out gun battles with soldiers and para-military police and their enemies, defying the rule of law (schedule 1 drugs by law must have no medical properties, yet basically all schedule 1 drugs have demonstrated medical value as defined by the legal non-militarized authorities of the government).

A reasonable understanding of what is actually a political body and what constitutes fighting a war shows us, in fact, drug war(0.999....) = war war(1).

Simultaneously the drug war is and is not a real war, depending on non trivial differences.

0.999... = 1 is true and false. True because of the math, false because the notation sucks because a literal read of the notation shows you there is a thing that by the (long division) process of (never actually) getting there shows you'll always be smaller than 1..... during the process of writing it out forever. And more absurd, if this was a base 5 system, 0.444... = 1 and the distance between the (non-existant) marginal 0.444...4 and 1 is twice as great as the difference between 9.999...9 and 1, except of course that they are all the same because the math is such, and reasonably, logically so.

That's the point. It's trivial and non-trivial, and Mise picking the drug war was a very good example.

You entered the thread saying:



In posts that followed you presented all kinds of fallacious and nonsensical things like:



If someone has mistaken that you weren't saying only that there are possible systems with 0.999... != 1, then the blame is on you.

All of this was to demonstrate that logic, which underlies math, can lead to other conclusions, ergo the equality requires some higher-order premises that aren't essential to have a working mathematics. The tone of devil's advocacy seemed clear to me, as valid (if perhaps suboptimal) social response to counter the tone of those who... well, I get to it below:

I hope you noticed that those who say 0.999... < 1 aren't pointing out inelegance in the decimal notation. Those who say that 0.999... = 1 however are saying that there's two different notations for the same number, and thus are pointing out an inelegance. It looks like we have been dogmatically, scornfully and ragefully been attacked for that too.

Well, different people are arguing different things.

What's interesting about this thread is not if 0.999... = 1, which it does. What's interesting is in how folks respond.

Terx was correct when he said that understanding math at a formal high level was proving a hindrance to many in this thread. There is an inability to step back and look at the causes of the assumptions, because damn those assumptions are so good/useful.

Though not perfectly analogous, I find this a lot in my discipline. Those who only studied economics, on issues of economy, can get very, very stuck even when they think they have an answer. Those (of us) who study political economy--one layer meta--can get very frustrated with the inability of economists to see position zero of their logic. Economists will tell you certain economic laws are inherently true and driving agents. They often aren't, but rather describe the logic of an overarching social super-system (generally capitalism) that have become truer and truer the more this system supplants other social mediations. They apply transcendentalism to historicity.¹

There are those in this thread who are so into Being The Expert they cannot actually engage the discussion. These are the people keeping the thread going for at least 2/3s of us. If akka said, "brennan, you're right, it is somewhat arbitrary that they equal each other but it's a damn good rule," brennan would go, "you're right, it is damn good and I agree that it is at least as of now the best position so I must concur the equality holds". But at this point akka is too busy telling everyone how dumb for not agreeing with him about how dumb they are.

1. I remember coming across a concept Keynes was using making up an ad-hoc example to demonstrate a small point, defering to the arbitrary classification of Britain's tax collection bureau that his instructor once used to make a related point. Total back-of-the-napkin stuff. But when I came across it I was like "bluhbluhbluh REALLY?!" because in my macroecon textbooks, that side point requiring the specific legal classification decided by one country's bureaucracy was presented as an economic law. Many arguments presented as inherent theory of monetary economics are literally backward on account those theories could only describe the specific conditions imposed by the precise gold standard of the late 19th and early 20th century.

 

[TheMeInTeam]

I wasn't aware that I had defined infinity anywhere...?

I realize I made a mistake, in that I was thinking of someone else's prior posts.

Do you accept the definition of infinity as continuing without limit? The assertion of the notation of .999... is that the 9's continue indefinitely. The assertion of .000...1 rejects the notion of infinity outright.

In making the assertion of .000...1 you are misusing the notation. The can't be a one at the end. There is no end, or it isn't infinite.

.999... < x < 1 having an answer of .000...1 is absurd. To make that answer, you *have* to reject the notion of .999... outright, to reject the concept of an infinite string of numbers.

You are, in essence, refusing to admit that an infinite string of digits is possible. If you're also willing to reject 1/9 = .111... then we have a different issue, but at least you're consistent. If you're not willing to do that, then the notion of .000...1 is absurd, like someone claiming "infinity plus one".

 

[Lohrenswald]

nonsense

if Brennan has really does what you think he does, Hygro, he's done an exeptionally bad job
It's not rediculous that 1=0,999...
The 9-series thing would have been an approximation if it ended at some point, but it doesn't, so it equals the limit

And people have repetedly said this is based on axioms, but if they were in a way that 1=/=0,999..., it would have sucked

EDIT: I was what I think is refered to as x-posted

 

[brennan]

This seems naive.

If I take a piece of string and stretch it out to an infinite distance from the middle then it can have two ends and still be infinite in length. Nothing about the concept of infinity precludes the possibility of having something on the end.

 

[Akka]

0.999... = 1 is true and false. True because of the math, false because the notation sucks because a literal read of the notation shows you there is a thing that by the (long division) process of (never actually) getting there shows you'll always be smaller than 1..... during the process of writing it out forever. And more absurd, if this was a base 5 system, 0.444... = 1 and the distance between the (non-existant) marginal 0.444...4 and 1 is twice as great as the difference between 9.999...9 and 1, except of course that they are all the same because the math is such, and reasonably, logically so.

You showed that it's COUNTER-INTUITIVE, which is pretty obvious and universally accepted - it's even the very reason this subject even exists.
I fail to see how it's "wrong". Counter-intuitive things aren't inherently "wrong".

What's interesting about this thread is not if 0.999... = 1, which it does.

You might notice that several people actually claim that it doesn't.

Though not perfectly analogous, I find this a lot in my discipline. Those who only studied economics, on issues of economy, can get very, very stuck even when they think they have an answer. Those (of us) who study political economy--one layer meta--can get very frustrated with the inability of economists to see position zero of their logic. Economists will tell you certain economic laws are inherently true and driving agents. They often aren't, but rather describe the logic of an overarching social super-system (generally capitalism) that have become truer and truer the more this system supplants other social mediations. They apply transcendentalism to historicity.¹

The difference is that economics is a "soft" science, without universal and absolute truth and based on reality with all the unknown and real facts it entails. Mathematics, on the other hand, is the opposite, the ultimate absolute science, based entirely on the absolute consequences of axioms which are determined.
Also, I'd like to point that numerous proofs of the subject of the thread don't require anything more than a grade school knowledge in mathematics. Saying that people are too far into their high sphere of knowledge to understand the point of view of laymen is pretty disingenuous.

There are those in this thread who are so into Being The Expert they cannot actually engage the discussion. These are the people keeping the thread going for at least 2/3s of us. If akka said, "brennan, you're right, it is somewhat arbitrary that they equal each other but it's a damn good rule," brennan would go, "you're right, it is damn good and I agree that it is at least as of now the best position so I must concur the equality holds". But at this point akka is too busy telling everyone how dumb for not agreeing with him about how dumb they are.

Three answers to these points :

- I'm not saying they are dumb to disagree with "me", as I'm just as much a nobody as they are and I've no deep knowledge about math (the fact that the subject of this thread actually DOESN'T require deep knowledge in math is in itself an evidence that someone who dispute the fact is unable to grasp the most basic mathematics BTW). What I'm saying them they are dumb because they disagree with the entire mass of information available about math and think they know more than the world-renowed mathematicians who invented and applied the rules which absolutely prove the truth of such a statement. Being open-minded is nice but there is a point, especially in math (which, again, is perhaps the single one science where you can get an absolutely true or false answer), where someone's errors start to be undefensible.

- There is nothing arbitrary that they equal each other. It's just a consequence of the basic axioms. THESE axioms might be somewhat arbitrary, but then math's entire principle is to take axioms and see what are the consequences. So it's kind of a Captain Obvious trying to be contrarian for the sake of it.

- I'd say the people playing the Expert are the ones who claim they know better than the whole world, not those who try to put them back in their place as regular Joe with inflated ego.

 

[Lohrenswald]

You can't pull a piece of string infinitely far. Thinking so is naive

 

[TheMeInTeam]

This seems naive.

If I take a piece of string and stretch it out to an infinite distance from the middle then it can have two ends and still be infinite in length. Nothing about the concept of infinity precludes the possibility of having something on the end.

Infinity is defined as having no end, and implying the existence of one rejects the concept of infinity. The naïve assertion is talking about an infinitely long string having ends.

You can't have something on the end of something with no end. If you reject infinity as a concept then its utility as a word is meaningless.

 

[Borachio]

We still don't know, none of us, how long a piece of string is. Despite centuries, nay millenia, of intensive research into this very pertinent question.

Neither have I received any response worth my time reading on the question of how short a piece of string can become before it's considered too short to be of any use.

 

[Hygro]

nonsense

if Brennan has really does what you think he does, Hygro, he's done an exeptionally bad job
It's not rediculous that 1=0,999...
The 9-series thing would have been an approximation if it ended at some point, but it doesn't, so it equals the limit

And people have repetedly said this is based on axioms, but if they were in a way that 1=/=0,999..., it would have sucked

EDIT: I was what I think is refered to as x-posted

There comes a point in a discussion where this happens:
Person 1: A
Person 2: I see A; B -> ~A
Person 1: I see B; C -> ~B -> A
Person 2: I see C; D -> ~C -> B > ~A
Person 1: Well B; C -> ~B -> A; and besides, A

Person 2 goes off muttering in frustration because Person 1 reached their limit and started repeating themselves because they didn't handle D.

You showed that it's COUNTER-INTUITIVE, which is pretty obvious and universally accepted - it's even the very reason this subject even exists.
I fail to see how it's "wrong". Counter-intuitive things aren't inherently "wrong".

Yes you fail to see it, but if you saw it, it wouldn't hurt your current understanding and it might make a lot of frustrating posts a lot more enjoyable, without making nonsense.

You might notice that several people actually claim that it doesn't.

Well, with OJH you have to *(-1) to what he says it comes out right. Terx is off whacking vines in the jungle, and I don't know if anyone else is still at it.

The difference is that economics is a "soft" science, without universal and absolute truth and based on reality with all the unknown and real facts it entails. Mathematics, on the other hand, is the opposite, the ultimate absolute science, based entirely on the absolute consequences of axioms which are determined.

Also, I'd like to point that numerous proofs of the subject of the thread don't require anything more than a grade school knowledge in mathematics. Saying that people are too far into their high sphere of knowledge to understand the point of view of laymen is pretty disingenuous.

Sure, math is the ultimate absolute science... starting at the point of math. Starting at the point of neoclassical and new-classical economics, their rules are the rules of economizing, unbreakable rules of what is economic. There is no room for disagreeing with anything that logically follows, because the axioms are the axioms. It's not as different as you might think. Knowing those rules of economizing means if something contradicts, it is "scientifically" not economical. This is, of course, empirically questionable. Then again, infinity is empirically not-at-ease with the math it lives in.

Obviously it's a much removed science from the mathematical source, so we call it a soft science. And good economics, which is highly historical in nature and has the theories used according to ideal-type principles, is clearly and deliberately a soft-science.

Three answers to these points :

- I'm not saying they are dumb to disagree with "me", as I'm just as much a nobody as they are and I've no deep knowledge about math (the fact that the subject of this thread actually DOESN'T require deep knowledge in math is in itself an evidence that someone who dispute the fact is unable to grasp the most basic mathematics BTW). What I'm saying them they are dumb because they disagree with the entire mass of information available about math and think they know more than the world-renowed mathematicians who invented and applied the rules which absolutely prove the truth of such a statement. Being open-minded is nice but there is a point, especially in math (which, again, is perhaps the single one science where you can get an absolutely true or false answer), where someone's errors start to be undefensible.

- There is nothing arbitrary that they equal each other. It's just a consequence of the basic axioms. THESE axioms might be somewhat arbitrary, but then math's entire principle is to take axioms and see what are the consequences. So it's kind of a Captain Obvious trying to be contrarian for the sake of it.

- I'd say the people playing the Expert are the ones who claim they know better than the whole world, not those who try to put them back in their place as regular Joe with inflated ego.

Right and so it doesn't make sense to assume those axioms when engaging someone who wants to talk outside those axioms. After all, there's no thread otherwise. Captain Obvious says, "if you can't talk outside those axioms, you aren't contributing to the thread"

Infinity is defined as having no end, and implying the existence of one rejects the concept of infinity. The naïve assertion is talking about an infinitely long string having ends.

You can't have something on the end of something with no end. If you reject infinity as a concept then its utility as a word is meaningless.

There's no conceptual reason you can't have an infinite (never-ending) middle. Nobody is rejecting infinity, but some of us are objecting to the idea that the limits of infinity are not arbitrarily constrained.

What numerical measure would a never-ending middle's length be?

 

[Lohrenswald]

Hygro, if there was a 0,999...998, what fraction 8/10^n is it?

 

[Hygro]

Oh snap! Except we're talking about conceptualization not equations.

 

[El_Machinae]

But, conceptually, 0.999...8 is a fundamentally different number than 0.999... or 1, given that it's right-side bounded and they're not. Why would you be able to do math between these two different framings?

 

[onejayhawk]

But, conceptually, 0.999...8 is a fundamentally different number than 0.999... or 1, given that it's right-side bounded and they're not. Why would you be able to do math between these two different framings?

It is no more right bounded than 0.999... Whether both/neither are right bounded is an interesting question.

J

 

[onejayhawk]

There comes a point in a discussion where this happens:
Person 1: A
Person 2: I see A; B -> ~A
Person 1: I see B; C -> ~B -> A
Person 2: I see C; D -> ~C -> B > ~A
Person 1: Well B; C -> ~B -> A; and besides, A

I like Boolean Algebra. Aced both courses.

J

 

[Birdjaguar]

Did this get posted here yet? I'm almost certain it has at some point but I'm not digging through 29.999...-some odd pages going back 1.9999... years to check.


Link to video.

Excellent! 676 posts and the answer was on youtube all the time. I now know that 0.999... = 1.

Thank you thank you.

 

[Agent327]

The proof was already posted a while ago though. (And by proof I don't mean youtube video.)

 

[sophie]

The proof was already posted a while ago though. (And by proof I don't mean youtube video.)

Yeah most of the explanations she does were made in the same (or similar) form within the first 15 posts or so. Still a good video, and quite relevant though.