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I just cited myself.
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this post was submitted on 27 Jun 2024
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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.