this post was submitted on 27 Jun 2024
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Sure, when you start decoupling the numbers from their actual values. The only thing this proves is that the fraction-to-decimal conversion is inaccurate. Your floating points (and for that matter, our mathematical model) don't have enough precision to appropriately model what the value of 7/9 actually is. The variation is negligible though, and that's the core of this, is the variation off what it actually is is so small as to be insignificant and, really undefinable to us - but that doesn't actually matter in practice, so we just ignore it or convert it. But at the end of the day 0.999... does not equal 1. A number which is not 1 is not equal to 1. That would be absurd. We're just bad at converting fractions in our current mathematical understanding.
Edit: wow, this has proven HIGHLY unpopular, probably because it's apparently incorrect. See below for about a dozen people educating me on math I've never heard of. The "intuitive" explanation on the Wikipedia page for this makes zero sense to me largely because I don't understand how and why a repeating decimal can be considered a real number. But I'll leave that to the math nerds and shut my mouth on the subject.
You are just wrong.
The rigorous explanation for why 0.999...=1 is that 0.999... represents a geometric series of the form 9/10+9/10^2+... by definition, i.e. this is what that notation literally means. The sum of this series follows by taking the limit of the corresponding partial sums of this series (see here) which happens to evaluate to 1 in the particular case of 0.999... this step is by definition of a convergent infinite series.
No number is getting converted, it's the same number in both cases but written in a different representation. 4 is also the same number as IV, no conversion going on it's still the natural number elsewhere written
S(S(S(S(Z)))). Also decimal representation isn't inaccurate, it just happens to have multiple valid representations for the same number.Good then that 0.999... and 1 are not numbers, but representations.
Lol I fucking love that successor of zero
It still equals 1, you can prove it without using fractions:
x = 0.999...
10x = 9.999...
10x = 9 + 0.999...
10x = 9 + x
9x = 9
x = 1
There's even a Wikipedia page on the subject
I hate this because you have to subtract .99999..... from 10. Which is just the same as saying 10 - .99999.... = 9
Which is the whole controversy but you made it complicated.
It would be better just to have them do the long subtraction
If they don't get it and keep trying to show you how you are wrong they will at least be out of your hair until forever.
You don't subtract from 10, but from 10x0.999.... I mean your statement is also true but it just proves the point further.
No, you do subtract from 9.999999....
Do that same math, but use .5555... instead of .9999...
Have you tried it? You get 0.555... which kinda proves the point does it not?
???
Not sure what you're aiming for. It proves that the setup works, I suppose.
x = 0.555...
10x = 5.555...
10x = 5 + 0.555...
10x = 5+x
9x = 5
x = 5/9
5/9 = 0.555...
So it shows that this approach will indeed provide a result for x that matches what x is supposed to be.
Hopefully it helped?
If they aren't equal, there should be a number in between that separates them. Between 0.1 and 0.2 i can come up with 0.15. Between 0.1 and 0.15 is 0.125. You can keep going, but if the numbers are equal, there is nothing in between. There's no gap between 0.1 and 0.1, so they are equal.
What number comes between 0.999... and 1?
(I used to think it was imprecise representations too, but this is how it made sense to me :)
My brother. You are scared of infinities. Look up the infinite hotel problem. I will lay it out for you if you are interested.
Image you are incharge of a hotel and it has infinite rooms. Currently your hotel is at full capacity... Meaning all rooms are occupied. A new guest arrives. What do you do? Surely your hotel is full and you can't take him in... Right? WRONG!!! You tell the resident of room 1 to move to room 2, you tell the resident of room 2 to move to room 3 and so on.... You tell the resident of room n to move to room n+1. Now you have room 1 empty
But sir... How did I create an extra room? You didn't. The question is the same as asking yourself that is there a number for which n+1 doesn't exist. The answer is no... I can always add 1.
Infinity doesn't behave like other numbers since it isn't technically a number.
So when you write 0.99999.... You are playing with things that aren't normal. Maths has come with fuckall ways to deal with stuff like this.
Well you may say, this is absurd... There is nothing in reality that behaves this way. Well yes and no. You know how the building blocks of our universe obey quantum mechanics? The equations contain lots of infinities but only at intermediate steps. You have to "renormalise" them to make them go away. Nature apparently has infinities but likes to hide the from us.
The infinity problem is so fucked up. You know the reason physics people are unable to quantize gravity? Surely they can do the same thing to gravity as they did to say electromagnetic force? NOPE. Gravitation doesn't normalise. You get left with infinities in your final answer.
Anyways. Keep on learning, the world has a lot of information and it's amazing. And the only thing that makes us human is the ability to learn and grow from it. I wish you all the very best.
When Hilbert runs the hotel, sure, ok. Once he sells the whole thing to an ultrafinitist however you suddenly notice that there's a factory there and all the rooms are on rails and infinity means "we have a method to construct arbitrarily more rooms", but they don't exist before a guest arrives to occupy them.
It's a correct proof.
One way to think about this is that we represent numbers in different ways. For example, 1 can be 1.0, or a single hash mark, or a dot, or 1/1, or 10/10. All of them point to some platonic ideal world version of the concept of the number 1.
What we have here is two different representations of the same number that are in a similar representation. 1 and 0.999... both point to the same concept.
I strongly agree with you, and while the people replying aren't wrong, they're arguing for something that I don't think you said.
1/3 ≈ 0.333... in the same way that approximating a circle with polygons of increasing side number has a limit of a circle, but will never yeild a circle with just geometry.
0.999... ≈ 1 in the same way that shuffling infinite people around an infinite hotel leaves infinite free rooms, but if you try to do the paperwork, no one will ever get anywhere.
Decimals require you to check the end of the number to see if you can round up, but there never will be an end. Thus we need higher mathematics to avoid the halting problem. People get taught how decimals work, find this bug, and then instead of being told how decimals are broken, get told how they're wrong for using the tools they've been taught.
If we just accept that decimals fail with infinite steps, the transition to new tools would be so much easier, and reflect the same transition into new tools in other sciences. Like Bohr's Atom, Newton's Gravity, Linnaean Taxonomy, or Comte's Positivism.
The character sequence "0.999..." is finite and you know you can round up because you've got those three dots at the end. I agree that decimals are a shit representation to formalise rational numbers in but it's not like using them causes infinite loops. Unless you insist on writing them, that is. You can compute with infinities just fine as long as you keep them symbolic.
That only breaks down with the reals where equality is fundamentally incomputable. Equality of the rationals and approximate equality of reals is perfectly computable though, the latter meaning that you can get equality to arbitrary, but not actually infinite, precision. You can specify a number of digits you want, you can say "don't take longer than ten seconds to compute", any kind of bound. Once the precision goes down to plank lengths I think any reasonable engineer would build a bridge with it.
...sometimes I do think that all those formalists with all those fancy rules about fancy limits are actually way more confused about infinity than freshman CS students.
Eh, if you need special rules for 0.999... because the special rules for all other repeating decimals failed, I think we should just accept that the system doesn't work here. We can keep using the workaround, but stop telling people they're wrong for using the system correctly.
The deeper understanding of numbers where 0.999... = 1 is obvious needs a foundation of much more advanced math than just decimals, at which point decimals stop being a system and are just a quirky representation.
Saying decimals are a perfect system is the issue I have here, and I don't think this will go away any time soon. Mathematicians like to speak in absolutely terms where everything is either perfect or discarded, yet decimals seem to be too simple and basal to get that treatment. No one seems to be willing to admit the limitations of the system.
Noone in the right state of mind uses decimals as a formalisation of numbers, or as a representation when doing arithmetic.
But the way I learned decimal division and multiplication in primary school actually supported periods. Spotting whether the thing will repeat forever can be done in finite time. Constant time, actually.
No. If you can accept that 1/3 is 0.333... then you can multiply both sides by three and accept that 1 is 0.99999.... Primary school kids understand that. It's a bit odd but a necessary consequence if you restrict your notation from supporting an arbitrary division to only divisions by ten. And that doesn't make decimal notation worse than rational notation, or better, it makes it different, rational notation has its own issues like also not having unique forms (2/6 = 1/3) and comparisons (larger/smaller) not being obvious. Various arithmetic on them is also more complicated.
The real take-away is that depending on what you do, one is more convenient than the other. And that's literally all that notation is judged by in maths: Is it convenient, or not.
I never commented on the convenience or usefulness of any method, just tried to explain why so many people get stuck on 0.999... = 1 and are so recalcitrant about it.
This is a workaround of the decimal flaw using algebraic logic. Trying to hold both systems as fully correct leads to a conflic, and reiterating the algebraic logic (or any other proof) is just restating the problem.
The problem goes away easily once we understand the limits of the decimal system, but we need to state that the system is limited! Otherwise we get conflicting answers and nothing makes sense.
But the system is not limited: It has a representation for any rational number. Subjectively you may consider it inelegant, you may consider its use in some area inconvenient, but it is formally correct and complete.
I bet there's systems where rational numbers have unique representations (never looked into it), and I also bet that they're awkward AF to use in practice.
The representation has to reflect algebraic logic, otherwise it would indeed be flawed. It's the algebraic relationships that are primary to numbers, not the way in which you happen to put numbers onto paper.
And, honestly, if you can accept that 1/3 == 2/6, what's so surprising about decimal notation having more than one valid representation for one and the same number? If we want our results to look "clean" with rational notation we have to normalise the fraction from 2/6 to 1/3, and if we want them to look "clean" with decimal notation we, well, have to normalise the notation, from 0.999... to 1. Exact same issue in a different system, and noone complains about.
Decimals work fine to represent numbers, it's the decimal system of computing numbers that is flawed. The "carry the 1" system if you prefer. It's how we're taught to add/subtract/multiply/divide numbers first, before we learn algebra and limits.
This is the flawed system, there is no method by which 0.999... can become 1 in here. All the logic for that is algebraic or better.
My issue isn't with 0.999... = 1, nor is it with the inelegance of having multiple represetations of some numbers. My issue lies entirely with people who use algebraic or better logic to fight an elementary arithmetic issue.
People are using the systems they were taught, and those systems are giving an incorrect answer. Instead of telling those people they're wrong, focus on the flaws of the tools they're using.
Of course there is a method. You might not have been taught in school but you should blame your teachers for that, and noone else. The rule is simple: If you have a nine as repeating decimal, replace it with a zero and increment the digit before that.
That's it. That's literally all there is to it.
It's not any more of an arithmetic issue than 2/6 == 1/3: As I already said, you need an additional normalisation step. The fundamental issue is that rational numbers do not have unique representations in the systems we're using.
And, in fact, normalisation in decimal representation is way easier, as the only case to worry about is indeed the repeating nine. All other representations are unique while in the fractional system, all numbers have infinitely many representations.
Maths teachers are constantly wrong about everything. Especially in the US which single-handedly gave us the abomination that is PEMDAS.
Instead of blaming mathematicians for talking axiomatically, you should blame teachers for not teaching axiomatic thinking, of teaching procedure instead of laws and why particular sets of laws make sense.
That method I described to get rid of the nines is not mathematical insight. It teaches you nothing. You're not an ALU, you're capable of so much more than that, capable of deeper understanding that rote rule application. Don't sell yourself short.
EDIT: Bijective base-10 might be something you want to look at. Also, I was wrong, there's way more non-unique representations: 0002 is the same as 2. Damn obvious, that's why it's so easy to overlook. Dunno whether it easily extends to fractions can't be bothered to think right now.
Very rarely wrong actually.
The only people who think there's something wrong with PEMDAS are people who have forgotten one or more rules of Maths.
https://www.youtube.com/watch?v=lLCDca6dYpA
...oh wait I remember that Unicody user name. It's you. Didn't I already explain to you the difference between syntax and semantics until you gave up. I suggest we don't do it again but instead, you review the thread.
Well, you seem to have forgotten that the woman in that video isn't a Maths teacher, which would explain why she's forgotten the rules of The Distributive Law and Terms.
I didn't give up, you did.
I suggest you check some Maths textbooks, instead of listening to a Physics major.
P.S. you proved my point
There! Syntax. We went over this. Seriously, we did, and, no, I got the last word.
I can check any textbook from any discipline. You know what? I could even ask my school teachers. Because I'm not American and I wasn't taught shit that doesn't match up with what professionals are doing.
You're just another yank drunk on jingoism, "We do it like that, therefore, it is right".
BWAHAHAHA! I see you still didn't learn to check facts first. 😂😂😂
P.S.
Yep, Maths teachers do it right. :-)
I don't really care how many representations a number has, so long as those representations make sense. 2 = 02 = 2.0 = 1+1 = -1+3 = 8/4 = 2x/x. That's all fine, we can use the basic rules of decimal notation to understand the first three, basic arithmetic to understand the next three, and basic algebra for the last one.
0.999... = 1 requires more advanced algebra in a pointed argument, or limits and infinite series to resolve, as well as disagreeing with the result of basic decimal notation. It's steeped in misdirection and illusion like a magic trick or a phishing email.
I'm not blaming mathematicians for this, I am blaming teachers (and popular culture) for teaching that tools are inflexible, instead of the limits of those systems.
In this whole thread, I have never disagreed with the math, only it's systematic perception, yet I have several people auguing about the math with me. It's as if all math must be regarded as infinitely perfect, and any unbelievers must be cast out to the pyre of harsh correction. It's the dogmatic rejection I take issue with.
You're used to one but not the other. You convinced yourself that because one is new or unacquainted it is hard, while the rest is not. The rule I mentioned Is certainly easier that 2x/x that's actual algebra right there.
Why, yes. I totally can see your point about decimal notation being awkward in places though I doubt there's a notation that isn't, in some area or the other, awkward, and decimal is good enough. We're also used to it, that plays a big role in whether something is judged convenient.
On the other hand 0.9999... must be equal to 1. Because otherwise the system would be wrong: For the system to be acceptable, for it to be infinitely perfect in its consistency with everything else, it must work like that.
And that's what everyone's saying when they're throwing "1/3 = 0.333.... now multiply both by three" at you: That 1 = 0.9999... is necessary. That it must be that way. And because it must be like that, it is like that. Because the integrity of the system trumps your own understanding of what the rules of decimal notation are, it trumps your maths teacher, it trumps all the Fields medallists. That integrity is primal, it's always semantics first, then figure out some syntax to support it (unless you're into substructural logics, different topic). It's why you see mathematicians use the term "abuse of notation" but never "abuse of semantics".
Again, I don't disagree with the math. This has never been about the math. I get that ever model is wrong, but some are useful. Math isn't taught like that though, and that's why people get hung up things like this.
Basic decimal notation doesn't work well with some things, and insinuates incorrect answers. People use the tools they were taught to use. People get told they're doing it wrong. People give up on math, stop trying to learn, and just go with what they can understand.
If instead we focus on the limitations of some tools and stop hammering people's faces in with bigger equations and dogma, the world might have more capable people willing to learn.
There is nothing wrong about decimal notation. It is correct. There's also nothing wrong about Roman numerals... they're just awkward AF.
You could just as well argue that fractional notation "insinuates" that 1/3 + 1/3 = 2/6. You could argue that 8 + 8 is four because that's four holes there. Lots of things that people can consider more intuitive than the intended meaning. Don't get me started on English spelling.
Neither of those examples use the rules of those system though.
Basic arithmetic on decimap notation is performed by adding/subtracting each digit in each place, or multiplying each digit by each digit then adding those sub totals together, or the yet more complicated long division.
Adding (and by extension multiplying) requires the carry operation, because digits only go up to 9. A string of 9s requires starting at the smallest digit. 0.999... has no smallest digit, thus the carry operation fails to roll it over to 1. It's a bug that requires more comprehensive methods to understand.
Someone using only basic arithmetic on decimal notation will conclude that 0.999... is not 1. Another person using only geocentrism will conclude that some planets follow spiral orbits. Both conclusions are wrong, but the fault lies with the tools, not the people using them.
That's where limits get involved, snatching the carry from the brink of infinity. You could, OTOH, also ignore that and simply accept that it has to be the case because 0.333... * 3. And let me emphasise this doubly and triply: That is a correct mathematical understanding. You don't need to get limits involved. It doesn't make it any more correct, or detailed, or anything. Glancing at Occam's razor, it's even the preferable explanation: There's a gazillion overcomplicated and egg-headed ways to write 1 + 1 = 2 (just have a look at the Principia Mathematica), that doesn't mean that a kindergarten student doesn't understand the concept correctly. Begone, superfluous sophistication!
(I just noticed that sophistication actually shares a root with sophistry. What a coincidence)
Doesn't pass scrutiny, because then either 0.333... /= 1/3 or 3 /= 3 (or both). It simply cannot be the case when looking at the whole system, as opposed to only the single question 0.999... ?= 1 and trying to glean something from that. Context matters: Any answer to that question has to be consistent with all the rest you know about the natural numbers. And only 0.999... = 1 fulfils that.
Why are you making this so complicated?
This is my point, using a simple system (basic arithmetic) properly will give bad answers in specifically this situation. A correct mathematical understanding of arithmetic will lead you to say that something funky is going on with 0.999... , and without a more comprehensive understanding of mathematical systems, the only valid conclusions are that 0.999... doesn't equal 1, or that basic arithmetic is limited.
So then why does everyone loose their heads when this happens? Thousands of people forcing algebra and limits on anyone they so much as suspect could have a reasonable but flawed conclusion, yet this thread is the first time I've seen anyone even try to mention the limitations of arithmetic, and they get stomped on.
Why is basic arithmetic so sacred that it must not be besmirched? Why is it so hard for people to admit that some tools have limits? Why is everyone bringing in so many more advanced systems when my entire argument this whole time is that a simple system has limits?
That's my whole argument. Firstly, that 0.999... catches people because using arithmetic properly leads to an incorrect understanding of repeating decimals. And secondly, that starting with the limits of arithmetic will increase understand with less frustration than throwing more complicated solutions around.
My argument have never been with the math, only with our perceptions of it and how we go about teaching it.
It isn't. It's convenient. Toss it if you don't want to use it. What's not an option though is to use it incorrectly, and that would be insisting that 0.999... /= 1, because that doesn't make any sense.
A notational system doesn't get to say "well I like to do numbers this way, let's break all the axioms or arithmetic". If you say that 0.333... = 1/3, then it necessarily follows that 0.999... = 1. Forget about "but how do I calculate that" think about "does multiplying the same number by the same number yield the same result".
Repeating decimals aren't apart from decimal arithmetic. They're a necessary part of it. If you didn't learn 0.999... = 1, you did not learn decimal arithmetic. And with "necessary" I mean necessary: Any positional system that supports expressing rational numbers will have repeating digits. It's the trade-off you make, by fixing the divisor (10 in our case), to make numbers easily comparable by size, because no number can divide any number cleanly because there's an infinite number of primes. Quick, which is the bigger number: 38/127 or 39/131.
Any notational system has its awkward spots. You will not get around awkward spots. Decimal notation has quite few of them, certainly fewer than Roman numerals where being able to do long division earned you a Ph.D. If you can come up with something better be my guest, I already linked you to a starting point.
When there's an incorrect answer it's because the user has made a mistake.
They were wrong, and I told them where they went wrong (did something to one side of the equation and not the other).
The system I'm talking about is elementary decimal notation and basic arithmetic. Carry the 1 and all that. Equations and algebra are more advanced and not taught yet.
There is no method by which basic arithmetic and decimal notation can turn 0.999... into 1. All of the carry methods require starting at the smallest digit, and repeating decimals have no smallest digit.
If someone uses these systems as they were taught, they will get told they're wrong for doing so. If we focus on that person being wrong, then they're more likely to give up on math entirely, because they're wrong for doing as they were taught. If we focus on the limitstions of that system, then they have the explanation for the error, and an understanding of why the more complicated system is preferable.
All models are wrong, but some are useful.
What do you mean not taught yet? There's nothing in the meme to indicate this is a primary school problem. In fact it explicitly has a picture of an adult, so high school Maths is absolutely on the table.
In high school we teach that they are the same thing. i.e. limits of accuracy, 1 isn't the same thing as 1.000..., but rather 1+/- some limit of accuracy (usually 1/2). Of course in programming it matters if you're talking about an integer 1 or a floating point 1.
The only people I've seen get things wrong is people not using the systems correctly (such as the alleged "proof" in this thread, which broke several rules of Maths and as such didn't prove anything), and it's a teacher's job to point out how to use them correctly.
I mean those more advanced methods are taught after basic arithmetic. There are plenty of adults that operate primarily with 5th grade math, and a scary number of them do finances...
This isn't about limits of accuracy, we're working with abstract values and ideal systems. Any inaccuracies must be introduced by those systems.
If you think the system isn't at fault here, please show me how basic arithmetic can make 0.999... into 1. Show me how the carry method deals with Infinity correctly. If every error is just using the system incorrectly, then a correct use of the system must be applicable to everything, right? You shouldn't need a new system like algebra to be correct, right?
According to who? Where does it say what it's about? It doesn't.
You still haven't shown why you're limiting yourself to basic arithmetic. There isn't anything at all in the meme to indicate it's about basic arithmetic only. It's just some Maths statements with no context given.
Different systems for different applications. Sometimes multiple systems for one problem (e.g. proofs).
Limits of accuracy isn't algebra.
According to me, talking about the origin of the 0.999... issue of the original comment, the "conversion of fractions to decimals", or using basic arithmetic to manipulate values into repeating decimals. This has been my position the entire time. If this was about the limits of accuracy, then it would be impossible to solve the 0.999... = 1 issue. Yet it is possible, our accuracy isn't limited in this fashion.
Because that's where the entire 0.999... = 1 originates. You'll never even see 0.999... without using basic addition on each digit individually, especially if you use fractions the entire time. Thus 0.999... is an artifact of basic arithmetic, a flaw of that system.
Then you agree that not every system is applicable everywhere! Even if you use that system perfectly, you'll still end up with the wrong answer! Thus the issue isn't someone using the system incorrectly, it's a limitation of the system that they used. The correct response to this isn't throwing heaps of other systems at the person, it's communicating the limit of that system.
If someone is trying to hammer a screw, chastising them for their swinging technique then using your personal impact wrench in front of them isn't going to help. They're just going to hit you with the hammer, and continue using the tools they have. Explaining that a hammer can't do the twisting motion needed for screws, then handing them a screwdriver will get you both much farther.
It never was, and neither is the problem we've been discussing. You can talk about glue, staples, clamps, rivets, and bolts as much as you like, people with hammers are still going to hit screws.
Right. So not according to the meme, which doesn't tell us where the 0.999... comes from. Nor the 1 - could be an integer, floating point, or an estimation. Thanks for playing.
The system works perfectly, it just looks wonky in base 10. In base 3 0.333... looks like 0.1, exactly 0.1
Oh the fundamental math works fine, it's the imperfect representation that is infinite decimals that is flawed. Every base has at least one.
That does very accurately sum up my understanding of the matter, thanks. I haven't been adding on to any of the other conversation in order to avoid putting my foot in my mouth further, but you've pretty much hit the nail on the head here. And the higher mathematics required to solve this halting problem are beyond me.