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Hidden 10 yrs ago Post by Halo
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It's pretty simple: if there's anything you're curious about relating to maths or physics, ask here and I'll do my best to answer.

I ran this idea by some other folk earlier and they liked the thought, so I decided to go ahead with it. Over my years here I've answered questions regarding these topics a few times, and it seems there's a vague general interest in them. They're pretty awesome subjects and they're my passion, so just as (for example) Sherlock shares her experience in artwork I'd like to try to do the same. Now I've just got to hope this thread doesn't go without any activity and dies a lonely, sad, and embarrassing death.

As a disclaimer: anything I say should be taken with a grain of salt. I'm not even in university for these subjects yet, so if you ask me something particularly advanced I may misunderstand something in my research. I'll do my best to verify with more knowledgeable friends of mine, but yeah, do not necessarily take my word as gospel!
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Why is 0.999 the same thing as 1?
Hidden 10 yrs ago Post by Raxacoricofallapatorius
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How can I make my physics essay another (single-spaced) page in length?

Btw it's super overdue.
Hidden 10 yrs ago Post by Butter
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I was going to ask a question, then I read: " Over my years ... I'm not even in university for these subjects yet" and just kinda giggled.
Hidden 10 yrs ago Post by Halo
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8 said
Why is 0.999 the same thing as 1?


Okay, so this is actually a pretty interesting (and bloody complex) question, so thanks for kicking me off with a difficult one. I hope the wall of text doesn't scare everyone away - not all questions are as fundamental to the core of a subject as this one!

So, firstly, what you wrote there is definitely not equal to 1, but 0.999..., where the "..." implies that the "9" is infinitely recurring, is.
There are different ways to prove this, the following, for reasons of elegant simplicity, being my favourite:

x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9
x = 1

My other favourite proof involves limits and infinite sequences, but that's slightly more complicated for me to explain (particularly considering the notation doesn't work in this sort of text format) and, I feel, doesn't actually address your question. These proofs don't answer your question of why, which is the important part, they simply demonstrate that it is. The answer to your question lies in the nature of infinity itself. We tend to view infinity as a number - a number further away than we can possibly ever reach, beyond any other number, but still at the theoretical "end" of the number line. Infinity, however, is a concept. It is entirely separate from numbers and the number line. Phrases like "nearly infinite" often annoy mathematicians - something cannot be nearly infinite, either it is infinite or it is not. In this case, it is.
If the "9" digit in "0.999..." is infinitely recurring, then considering 1 - 0.999... = x, there is naught x can be but 0 (implying that 1 = 0.999...). There is no real number that exists between 0.999... and 1, no space between them, and thus they must be one and the same. There is no "0.000...0001" left over when you make the subtraction, because those recurring nines continue for eternity. If the difference between two numbers is zero, they are the same. A more mathematical way of representing this is here.

Of course, we can go far, far deeper. In the end, whether two numbers are equal or not is entirely dependent on how we define those numbers. The reason 0.999... = 1 seems so counter-intuitive is simply because of the standard way we are taught to define and interpret numbers. In reality, numbers do not simply exist; we created and defined every number in our number system. But in a standard mathematics education, we never really receive a rigorous definition of what a real number actually is - what rules and axioms are in place to define them. In other words: the question is not really "why is 0.999... equal to 1", but rather to query what our definition of the numbers involved, or real numbers as a whole, is. The argument above, about there being no space between the two numbers 0.999... and 1, depends on the fact that we do not allow infinitesimally small numbers - "if a number is smaller than every other number, it is zero", also known as the Archimedean property. We say that if the distance between two numbers is infinitesimally small, the distance between them is zero, and thus they are the same. But, well, who decided that infinitesimally small numbers don't exist? Who made the rules, and what are they? How do we define numbers? More specifically: "Assuming 0.999... and 1 are the same by our definitions of those numbers, what are those definitions; how are those numbers defined such that they are the same?"
The question to start with us: who says that two numbers having zero-difference makes them equal? That's something we pretty much just decided arbitrarily, because in almost all cases any two numbers with zero-difference are functionally identical. Two-thirds and ten-fifteenths are different numbers, but they are functionally identical in terms of their value and they have a difference of zero, so we declare that they are equal. We are doing the same in the case of 0.999.... and 1. They are different numbers but have zero-difference between them, so we say they are equal. All these "we say"s are basically one way of defining the real numbers - the "we say"s are the rules in place that decide what is or is not a real number and how they all interact. So, in essence, 0.999... and 1 are equal just because we say they are, and the reason we just say that they are is because they function identically, have a difference of zero, and we can prove it with methods that fit our rules (such as the ones mentioned near the start of this post.)
However.... while this is distinctively different from "because we say so" for multiple reasons, it still feels a little unsatisfactory. There are obviously other ways of constructing what we identifty as the real numbers, such as Dedekind Cuts and Cauchy Sequences, but those are way above my paygrade, and as it is currently almost half past midnight I will do my research into them another time and return here to explain in depth how 0.999... = 1 by these definitions of what real numbers are on another day.
My current limited understanding (beware, I've not simplified this much, I'll do that when I return to these, so you may want to skip this paragraph) is that using Dedekind Cuts, a real number is defined as a set (a group/list) of all rational numbers (numbers expressable as a fraction a/b) lower than itself, and that when you use this method to define 0.999... and to define 1, you find the sets are identical and so the numbers themselves are identical. Defining real numbers by using Cauchy sequences, however... a Cauchy sequence is one in which the numbers in the sequence become arbitrarily closer together as the sequence goes on, like (0.1, 0.11, 0.111, 0.1111...) - the difference between each successive term gets smaller and smaller as the sequence goes on. You can define every real number as the limit of a Cauchy sequence of rational numbers). Therefore, if the limit of the difference between two sequences is 0, then the two numbers defined by those sequences are the same. You can express 1 as a Cauchy sequence (1, 1, 1, 1.......), and 0.999... as the Cauchy sequence (0, 0.9, 0.99, 0.999.....). Subtracting these sequences, you end up with (1, 0.1, 0.01, 0.001.....), and the limit of this sequence is obviously 0. As the limit of the difference between the two sequences is 0, the two numbers represented by those sequences must be the same - namely, 0.999... and 1.

And, as a final note.... here is where it gets sort of trippy: depending on how you define real numbers (or, rather, what number systems you use...), 0.999... might not equal 1. They are equal in the number system we use in our daily lives and that most mathematicians use most of the time, but there are ways of defining numbers that mean they are not equal. There are entire number systems, like hyperreal numbers, that assume infinitesimal numbers exist and because of that, 0.999... does not equal 1. As ever, it all depends on your definition. But that is a topic for another day!
Hidden 10 yrs ago Post by Halo
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Raxacoricofallapatorius said
How can I make my physics essay another (single-spaced) page in length? Btw it's super overdue.


Depends what it's on, doesn't it? . I don't want to do people's homework for them, but I'm happy to highlight relevant or interesting topics and link to online resources that you can use to flesh out your work.

Butter said
I was going to ask a question, then I read: " Over my years ... I'm not even in university for these subjects yet" and just kinda giggled.


I've been on this site since 2010, and active since 2012. Multiple years in which I've answered questions on these topics. Though I admit the phrasing isn't much helping my sorely lacking credentials . And while it's true that I won't be attending university until October, I have enough understanding of these topics to answer the sorts of questions I expect most people here to have to a relatively satisfactory extent. You may as well ask your question, and take my answer with as much salt as you like.
Hidden 10 yrs ago Post by Cpt Toellner
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ITT: We feed Halo's superiority complex.
Hidden 10 yrs ago Post by Butter
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Cpt Toellner said
ITT: We feed Halo's superiority complex.


No he's certified, he even said so himself: he's been doing this on RPG for years.
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Well, we all know RPG is the most highly respected higher education institution in all the world. But hey, if the perceived objective is boosting my ego rather than fostering genuine interest and knowledge and conversation in the barren wasteland everyone is saying Spam has become of late, I'll take the hit for the team and Butter can stroke my e-peen rather than damn near choking themselves on Jorick's like last time. Please, shower me with praise, I feed on it.

Petty insults and bullshit aside, I'm not trying to claim I'm certified. I specifically added a disclaimer saying I'm unqualified as fuck and will probably get things wrong, which is pretty much the opposite of the narcissism and overconfidence you're accusing me of. This is just a topic I'm interested in and like talking about, and something I probably do know more about than the average person. Toellner, I expect pointless non-contributions like this from you, but I'd rather you pointed them somewhere else. Butter, I get that as of your return to Spam you want to make a, uh, different impression than you did last time, but being a dick for no reason isn't really going to help much with that these days - save the scathing comments for detractors to the community, rather than those actively trying to engage with it, or you'll just become a detractor yourself.
Are we done here? I doubt any of us have much interest in a stupid pissing contest.
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Thank you for that explanation. I like that you told me "why" instead of just that it "is". I've asked my last few mathematics teachers this and never really understood it, but I sorta do now.
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8 said
Thank you for that explanation. I like that you told me "why" instead of just that it "is". I've asked my last few mathematics teachers this and never really understood it, but I sorta do now.


I'm glad ^_^.
Hidden 10 yrs ago Post by mdk
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Another way to approach it is, 1 - 0.9999999..... = X. In this case, X must equal 0.0000.......1, where the 1 does not occur until after an inifinitely long sequence of zeroes. It's a null infinity, which can never be expressed; thus, there is no difference between 1 and 0.999... Where there is no difference, the values are equivalent.
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mdk said
Another way to approach it is, 1 - 0.9999999..... = X. In this case, X must equal 0.0000.......1, where the 1 does not occur until after an inifinitely long sequence of zeroes. It's a null infinity, which can never be expressed; thus, there is no difference between 1 and 0.999... Where there is no difference, the values are equivalent.

Halo said
If the "9" digit in "0.999..." is recurring, then considering 1 - 0.999... = x, there is naught x can be but 0 (implying that 1 = 0.999...). There is no real number that exists between 0.999... and 1, no space between them, and thus they must be one and the same. There is no "0.000...0001" left over when you make the subtraction, because those recurring nines continue for eternity. If the difference between two numbers is zero, they are the same.


;D
Though, unlike you, I completely fuckin' forgot to clearly clarify that the reason there's no "0.00..001" is because that requires there to be a finite number of nines, which there isn't. Whoops. ^^;;
Hidden 10 yrs ago Post by Lucian
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Hah, was this spurred because of that thread I started about the electron? On that subject, by the way, I really only meant what I said in the most rudimentary way. We've observed it's effects obviously, but not it itself and, in fact, it could be infinitesimally too small to ever observe. I wasn't saying it didn't exist, merely remarking on how it's impossible to see it.
Hidden 10 yrs ago Post by K-97
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Can anyone explain why 1 + 2 + 3 + 4 .... and so on to infinity = -1/12?
Hidden 10 yrs ago Post by mdk
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Halo said
;DThough, unlike you, I completely fuckin' forgot to clearly clarify that the reason there's no "0.00..001" is because that requires there to be a finite number of nines, which there isn't. Whoops. ^^;;


Next I'll need someone to explain to me the mathematical difference between reading 1/3 of a post, and reading the full value...... lol. I clearly skipped this part.
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mdk said
Another way to approach it is, 1 - 0.9999999..... = X. In this case, X must equal 0.0000.......1, where the 1 does not occur until after an inifinitely long sequence of zeroes. It's a null infinity, which can never be expressed; thus, there is no difference between 1 and 0.999... Where there is no difference, the values are equivalent.


But the easiest way to approach it is with some simple fractions. Start with 1 = 1, split one side into thirds, then turn them into decimals.

1 = 1
1 = (1/3) + (1/3) + (1/3)
1 = 0.3333... + 0.3333... + 0.3333...
1 = 0.9999...

Even a middle school kid can understand it with this explanation, no wall of text or even knowledge of algebra needed.
Hidden 10 yrs ago Post by TheMadAsshatter
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Given that you can make loopholes like these in math, is it possible that math is complete bullshit and the smart guys are wrong? If not, is it not true that math is flawed?
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Lucian said
Hah, was this spurred because of that thread I started about the electron? On that subject, by the way, I really only meant what I said in the most rudimentary way. We've observed it's effects obviously, but not it and, in fact, it could be infinitesimally too small to ever observe. I wasn't saying it didn't exist, merely remarking on how it's impossible to see it.


It was, yes. I've had the idea for a while, but your thread reminded me of it. And I know that you meant we haven't visibly seen it - I was just pointing out that visibly seeing it is, in the scheme of things, a very arbitrary way of "observing" something, and that detecting, measuring, observing other properties is exactly the same thing.

K-97 said
Can anyone explain why 1 + 2 + 3 + 4 .... and so on to infinity = -1/12?


Numberphile did an excellent video demonstrating this, far better than I ever could. As for why , the answer will likely lie again in the nature of infinity. I'll get back to you on this one!

mdk said
Next I'll need someone to explain to me the mathematical difference between reading 1/3 of a post, and reading the full value...... lol. I clearly skipped this part.


It is a ridiculously dense wall of text, I don't blame you for missing a part.... I might go back and restructure some of it, thinking of it.

Jorick said
But the easiest way to approach it is with some simple fractions. Start with 1 = 1, split one side into thirds, then turn them into decimals.1 = 11 = (1/3) + (1/3) + (1/3)1 = 0.3333... + 0.3333... + 0.3333...1 = 0.9999...Even a middle school kid can understand it with this explanation, no wall of text or even knowledge of algebra needed.


But it does absolutely nothing to explain why.

TheMadAsshatter said
Given that you can make loopholes like these in math, is it possible that math is complete bullshit and the smart guys are wrong? If not, is it not true that math is flawed?


It isn't a loophole. That implies this is like some sort of accidental glitch in the system, than can be taken advantage of to prove untruths or some such. It isn't - it is factually, demonstrably true as per our definition of numbers. Saying that this is a loophole is like saying 1+1=2 is a loophole. 2 is defined by the fact that it is equal to 1+1. Similarly, the way we construct the real numbers, or "how we define numbers", means we have the same definition for both 0.999... and 1, and so they are equal.
Can mathematics be flawed? I don't think so, but it's more a philosophical question - mathematical philosophy, that is. Some believe numbers and relationships in maths already exist and are "discovered", others that they are all "created" by the person who first utilises them. I can go more into mathematical philosophy if you want, just ask. ^_^
What I feel can be flawed is, say, a number system. See, the way we think of numbers on a daily basis is pretty loose and undefined - as I noted in the original post, most people have never had what a real number is formally defined for them. If you took how we think of numbers on a daily basis and codified it, made a number system out of it, there'd probably be tonnes of flaws and issues. In the study of mathematics, what things are has to be rigidly defined by rules and axioms of the system you're using, whether that be a number system or a co-ordinate system or whatever. We don't bother with that most of through time though, which can lead to flaws. Maths isn't broken - in my opinion, it sort of can't be, not in itself - but the systems we use can be, though mathematicians are careful to define and prove things, and never just "assume", to avoid that.
Brief note: everything above is purely my own opinion, I haven't done any research for this answer as I'm on my phone. If I read something when I look it up later that contradicts me, I'll correct myself.
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Halo said



Right so, why is the answer to everything 42?
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