Eternal optimism. It’s also why we’re not a black clad army using fire to cleanse the world of immortal stupidity, so don’t take it away from us.
- is correct. And yes, there are other fractions whose decimal expansions I am unable to see.
No. Even if you go through ALL the finitely indexed digits, you don’t reach 1. I have repeated it so many times:
t = 0: I move my pen from point 0 to point 0.9 of the number line
t = 0.9: I move my pen from point 0.9 to point 0.99 of the number line
t = 0.99: I move my pen from point 0.99 to point 0.999 of the number line
…
I am NOT stopping at any finite step! We “reached infinity” in terms of how many '9’s there are, and still don’t reach 1.
If you stop, you’re not at infinity. There’s no “still don’t reach 1”.
Again, you’re trying to treat infinity as a really large finite quantity. This is a perfect example of why your question is not very well defined or useful.
You cannot reach infinity. Thinking about it in that way is dead wrong. You said earlier “There is an n-th digit for every n ∈ ℕ.” That means every n is a finite number. Although mathematicians dislike the simplistic phrase that infinity means endless, it is of help here in showing how this thinking cannot be applied to infinity. You cannot have “endless” and “reach” in the same concept.
Now take a break and answer my questions from before. If you reject that 0.999~ = 1, what math can you do? And if you can’t do any math, why aren’t you changing your thinking to a different technique that is useful? Your “math” is broken. Why not fix it?
If there are n iterations, and n ∈ ℕ, then indeed you’ve stopped after a finite number of iterations. If you’re trying to consider all n iterations, then this doesn’t make sense at all; for n = 500 and for n = 1,000 will give profoundly different answers. I think you’re restricting yourself by requiring n ∈ ℕ, there’s no reason why it should be in all cases. So in those cases where n !∈ ℕ, you’ll be coming up with bad results.
Do you know a bright 8-year-old who you can ask to explain long division to you?
Honestly and sincerely, biting my tongue so hard that it hurts, until you have read and understood the method for dividing one by three or until you are prepared to coherently and directly explain why the method is flawed you should absent yourself from this thread because, until then, you have zero, zilch and zip (what’s left after you take 0.999… away from 1) to say of the least interest.
Hey, netzweltler, want to see the ultimate result of your thinking? Look here.
Repeat to infinity! … would make a good sig.
Nevertheless, it is a mathematical truth that I have reached all the digits (every n-th digit for every n ∈ ℕ) by applying this process
t = 0: I move my pen from point 0 to point 0.9 of the number line
t = 0.9: I move my pen from point 0.9 to point 0.99 of the number line
t = 0.99: I move my pen from point 0.99 to point 0.999 of the number line
…
And this means reaching infinitely many digits, and it means completing 0.999…
I don’t care what math can be done or not. I don’t think it would change math no matter if I accept 0.999~ = 1 or not. I just want to clarify what is true about points on the number line, and I want to clarify if this point is well-defined or not.
I am considering n for all n ∈ ℕ. What’s wrong about that? That’s standard math.
What do you mean by n ∉ ℕ?
This sentence clearly encapsulates everything wrong about your approach.
And again, 0.99… is not a process. You don’t complete it. It’s the result of a process, and it’s by definition already complete.
Well, you’re spending an incredible amount of time on this so if you want to achieve clarity you should be clear that you are wrong and all the mathematicians in the world are right. It is clear that the point is well-defined. It is clear that you are using finite points to make a statement about infinity. It is clear you can’t change other peoples’ minds because they clearly understand your mistake. Clearly, reaching “infinitely many digits” has null content because it is impossible and every thinker from Zeno on understood that it was impossible and that the problem clearly couldn’t be approached that way. You’re a flat earther, in essence. Come on over to the round side.
Obviously I misunderstand was you mean by ℕ … I thought you meant natural numbers … however my argument still stands whichever number system ℕ represents …
In calculus, we have a thing called a differential, and my understanding is that every differential dx ∉ ℕ … and differential equations are fairly ubiquitous in standard math … perhaps you mean standard arithmetic? One of the many useful features of the calculus is it’s ability to handle infinity in a rigid way … Give it a try, I think you’ll be impressed …
Of course (0.999…) is well defined on the the number line … as well defined as 16/32’s … I think you mistake not finding these decimal numbers with them not existing … you’re just looking in the wrong place is all … can you at least try and stand over here and see what we’re seeing?
Is rigid calculus something to do with limited-slip differentials?
Don’t you think that math is a turn-on?
Ha ha, very funny, if you’re slipping your own bounds you won’t stay within your own area … there’s limits here ya know …
1/4 is not equal to .25000~ either, because (by the reasoning presented) you have to keep adding zeroes. No matter how many zeroes you add, you can always add one more. The task is never completed, and thus the two numbers aren’t equal.
That’s ridiculous. Apply the process suggested and you will see what really happens.
t = 0: I move my pen from point 0 to point 0.25 of the number line
t = 0.9: I move my pen from point 0.25 to point 0.250 of the number line
t = 0.99: I move my pen from point 0.250 to point 0.2500 of the number line
…
I have completed the task adding infinitely many zeros by t = 1. The position of the pen hasn’t changed and is equal to 1/4 from t = 0 to t = 1.
So, “by the reasoning presented” 1/4 is equal to 0.25000~.