Comment on I just cited myself.
Tlaloc_Temporal@lemmy.ca 5 months agoDecimals 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.
those systems are giving an incorrect answer
When there’s an incorrect answer it’s because the user has made a mistake.
Instead of telling those people they’re wrong
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.
not taught yet
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.
There is no method by which basic arithmetic and decimal notation can turn 0.999… into 1.
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.
If someone uses these systems as they were taught, they will get told they’re wrong for doing so
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.
What do you mean not taught yet?
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…
limits of accuracy
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?
barsoap@lemm.ee 5 months ago
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.
SmartmanApps@programming.dev 5 months ago
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.
barsoap@lemm.ee 5 months ago
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.
SmartmanApps@programming.dev 5 months ago
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.
Tlaloc_Temporal@lemmy.ca 5 months ago
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
barsoap@lemm.ee 5 months ago
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”.
Tlaloc_Temporal@lemmy.ca 5 months ago
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.