Your semantic quibble doesn’t change the mathematical reality that 0.999… = 1. That’s it.
Really?! Bolt can do it but the maths doesn’t hold?! I would worry if my clever arguments were contradicted by simple real-world examples – specifically I would worry that my clever arguments were brown, smelly, steaming and arranged in a heap.
Still, is it not obvious that at time t = x he is at x decametres?
Also, if we plot the points {(0, 0), (0.9, 0.9), (0.99, 0.99) …} on a graph, they certainly seem to be heading towards (1, 1).
I said before I believe that you are claiming* that because 1 doesn’t appear in the “list” on the LHS it can never appear on the list on the RHS. You are supposing your conclusion and begging the question.
- it is hard to tell exactly what you are arguing (or why) – your one argument seems to be one is not in the list, you are right it isn’t – but then it was you who decided it would be a good idea to write down 0.999… The contradiction that you believe to have discovered is simply that your efforts to list all the digits of this number are futile.
Yes, the easy way to dismiss the claim that the sequence never reaches “1” because you can’t point to the specific point in the sequence where that happens is to remind the claimant that you can’t get to the point where it happens. Thus, all iterations of the sequence being finite in the number of "9"s involved, the response is simply to say, “you haven’t gone far enough, yet. Call me when you get to the infinityeth term.”
Then you agree that 1 - (0.999…) = (0.000…) = 0 ?
Of course it’s obvious. 0.999… is simply not describing how this happens.
Absolutely not. 0.999… means infinitely many digits following.
The list
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
…
means infinitely many lines following. There are no efforts. It’s all done already. 0.999… is complete and the list is complete - at t = 1.
The question doesn’t make sense as long as we cannot find a point 0.999… on the number line. You need two points to define a distance.
I don’t suppose you’ll accept the grammar school arithmetic that if a - b = 0, then a = b.
If a and b are well-defined points on the number line, like a = 1 and b = 1, of course I accept grammar school arithmetic.
I suppose you hold that 0.333… and 1/3 are different too (but you won’t say different by what(?)).
Incidentally, what do you think the decimal expansion of 1/3 is?
What if b = 3/3, or e/e, or (0.999…), or any other symbol we define as having a value of one?
Now I have this running through my head: THAT'S INFINITY - By Carol Weiss - YouTube
And again we are dealing with the problem:
t = 0: I move my pen from point 0 to point 0.3 of the number line
t = 0.9: I move my pen from point 0.3 to point 0.33 of the number line
t = 0.99: I move my pen from point 0.33 to point 0.333 of the number line
…
For each of the infinitely many lines is valid, that I get closer to 1/3 but don’t reach 1/3. Tell me if you can find a line for which this does not hold true.
I cannot see why 3/3 or e/e is not well-defined.
Even (0.999…)/(0.999…) is a well-defined point on the number line.
Which is the same as t=0.999… where the value is t=0.999… = 1. You’re not making a cogent point to anyone except yourself.
Would it help if you thought about the problem in base 3 instead of base 10? In base 10, 1/3 = 0.33333…, but in base 3, 1/3 (base 10) = 1/10 = 0.1 (base 3). Similarly, 2/3 (base 10) = 2/10 = 0.2 (base 3), and 3/3 (base 10) = 10/10 = 1 (base 3). If you had a number line that went 1, 2, 10, 11, 12, 20, etc., you could certainly find 0.1 on that line.
For the ten millionth time. Each of those points are finite. Just like the n you used earlier. You are trying to make a point about infinity by using a series of finite points. And you are failing. That should tell you that your tools are inadequate to the task. You should therefore immediately stop what you are doing and look for tools that are adequate to the task. The limit, e.g.
Using the right tools is critical to all of math. You cannot use Euclid’s geometry to determine areas on the surface of the earth: it is the wrong tool for the job, no matter how good and useful it is for other tasks.
You have been told repeatedly that you are going about the task in the wrong way. Arguing repeatedly in return that your method gives you a different answer is utterly futile. Of course it gives you the wrong answer. You are using the wrong tool. The discussion ends right there.
Well, maybe a thread titled “An Infinite Question” requires an infinitely repeated answer…
The infinitely repeated answer is:
For each of the infinitely many lines is valid, that I get closer to 1/3 but don’t reach 1/3. Tell me if you can find a line for which this does not hold true.
Ok. But what kind of problem is this a solution for?