Infinite is infinite. What am I getting past? The problem is that mathematics doesn't understand infinite. It doesn't understand that 0.[n]1 exists, which is to say 0.[0..]1 exists, which is to say 1 - 0.9.. = 0.[0..]1
You cannot get to 0.999 without the 8, therefore the 8 is required, therefore the 8 must be in the final solution. It's a requirement. It's the same as 1/3. The remainder is a requirement, thus it must be in the final solution.
By using your limit system of manipulation, I showed that 0 < 0. I think the grumors started going hot and heavy around this time and you never replied to it.
I did miss this, but: 0 < 0 is true due to: 0.0.. < 0.00.. < 0.000.. Zero is special, like it is already.
1.9... < 1.99... 1 + .9... < 1 + .99... subtract .9... from each side 1 + .9... - .9... < 1 + .99... - .9... 1 + (lim(.9...) - lim (.99...)) < 1 + (lim (.99...) - lim (.9...)) 1 + 0 < 1 + 0 1 < 1
1 < 1 is true here because 1.0.. < 1.00.. < 1.000.. We will need better precision, though, because math folks will struggle understanding concepts, no matter how often shown.
How about this, we say if taking the limit on an infinitely repeating decimal, the operator becomes <= No different really than flipping the operator due to negatives.
Because in pure mathematics, there are no significant digits. 1 is exactly 1. pi is exactly pi. e is exactly e. i is exactly i. So when I divide 1 by 3, I am able to do it as far as I want, infinite number of times, and never have to worry about significant digits. By notation standards, .9... is simply a decimal followed by infinite 9s. It is a shorthand, nothing more, nothing less. I can write it as .9... or .99... or .999999..., it is all the same by definition of notations. 1 is simply 1.0000..., but that is to unwieldy to write and no one does it, and simply writes 1. Now, in engineering, or any of the sciences, yes, you cannot say 1/3 of a 1 foot board is exactly .33333.... of a foot long. Because the ruler you used to measure that board gave you 1 foot, but if it only reads inches, you are at best guessing between 11.5 and 12.5 inches and averaging it out to 12 inches or 1 foot. But in mathematics, because it is all in the abstract, we are not limited by this. 1.0... doesn't mean there is some chance that it is really 1.01. Nor does 1.00... definitely KNOW it isn't 1.01 but it might be 1.001. It is notation, because we cannot write out an infinite number of zeroes, and nor do we wish to do so. So what you are wanting to do is introduce something that is not there, notably significant digits. Which is fine, but it really isn't going to get us anywhere mathematically. What you want to do is change the standard notation to something else so that .9... is an ambiguous value, and .99... is less ambiguous, but still ambiguous in its exact value. But we cannot have a common discussion if we cannot agree on the language.
I've already said it isn't going to get anywhere mathematically. It's going to make no difference, except that it will be correct. There is no problem where 0.9.. works better than 1. There is no problem where 0.[0..]1 works better than 0. That doesn't make them equal, it just makes them practical. Mathematics is supposed to be abstract, which is what is truly gained. Abstract, and not practical. Significant digits brings abstraction. Losing them means practicality. Don't confuse the two.
How does significant digits bring abstraction? I see it as bringing the opposite. I cannot have a perfect circle in reality, it is impossible, because of significant digits and precision. In the abstract, I can have a perfect circle because I say I do.
By acknowledging that you cannot have a perfect circle, but these 25 digits of precision are good enough, so I say I do. Which is different than saying “I cannot have a perfect circle, here it is.” You have the abstract and the caveat. The only thing missing today is the caveat. We get the caveat sometimes. Providing it always doesn’t limit abstraction.
I must be getting slow because I have read and re-read this a dozen times and it makes zero sense to me. "I cannot have a perfect circle, here it is" seems like a contradiction and part of some nonsensical limerick.
Exactly, that's what math does. That's why its nonsense. "I cannot reach infinity: here it is in this case, it's 1." Math often says: "I cannot have a perfect circle. Here it is, here is the equation."
Are you a finitism proponent? Do you not agree with math that works in the abstract and cannot be done outside of the mind? Because as far as I am concerned there is no contradiction in saying "I cannot construct a perfect circle in reality with a compass and a pencil, but in my mind, as a pure mathematical construct, I can." If you limit yourself to physically constructable geometry or counting, you throw out a large chunk of mathematics that much of our modern science is built upon.
Something else I meant to mention about this statement: lim (.9..) = 1 is that lim (x->y) c = c where x approaches y and c is a constant. This would imply that .9.. = 1, which you say is not true, so I am not sure that methodology works, either.
You can have abstraction and correctness, and if you require significant digits as part of mathematics, there will be a well defined notation that is abstract and correct in less than a decade. Probably closer to 3 years.
It doesn’t imply that 0.9.. = 1, it implies the LIMIT is equal to 1 The limit is a function, and it’s saying the function with these parameters is equal to 1. The individual parameters may not equal 1 on their own, but once through the function, the function as a whole does.
