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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.
I get that ever model is wrong, but some are useful.
There is nothing wrong about decimal notation. It is correct. There's also nothing wrong about Roman numerals... they're just awkward AF.
Basic decimal notation doesn’t work well with some things, and insinuates incorrect answers.
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.
0.999… has no smallest digit, thus the carry operation fails to roll it over to 1.
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)
Someone using only basic arithmetic on decimal notation will conclude that 0.999… is not 1.
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?
simply accept that it has to be the case because 0.333... * 3. [...] That is a correct mathematical understanding
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.
Why is basic arithmetic so sacred that it must not be besmirched?
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".
catches people because using arithmetic properly leads to an incorrect understanding of repeating decimals.
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.