This is the flawed system, there is no method by which 0.999… can become 1 in here.
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.
My issue lies entirely with people who use algebraic or better logic to fight an elementary arithmetic issue.
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.
Instead of telling those people they’re wrong, focus on the flaws of the tools they’re using.
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.
Maths teachers are constantly wrong about everything
Very rarely wrong actually.
the abomination that is PEMDAS
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.
https://www.youtube.com/watch?v=lLCDca6dYpA
…oh wait I remember that
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.
until you gave up
I didn’t give up, you did.
I suggest we don’t do it again but instead, you review the thread
I suggest you check some Maths textbooks, instead of listening to a Physics major.
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.
0.999… = 1 requires more advanced algebra in a pointed argument,
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.
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
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.
