Comment on I just cited myself.
barsoap@lemm.ee 3 months ago0.999… = 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”.
Tlaloc_Temporal@lemmy.ca 3 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.
barsoap@lemm.ee 2 months ago
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.
Tlaloc_Temporal@lemmy.ca 2 months ago
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.
barsoap@lemm.ee 2 months ago
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?