Ah … calculus uses limits, not number lines … please try again, and please use math instead of hokey irrelevant grammar school descriptions.
There’s something back there about all infinite strings being undefined as numbers, except of course things like pi, cause you can just stop at 3.14159 and that’s close enough for most purposes …
…and they have said that they don’t know what the decimal expansion of 1/3 is (I took that to mean they felt it wasn’t 0.333… – notwithstanding that when one moves one’s pen from 0 to 0.9 one passes right through it – because, for pen reasons, it isn’t a number).
I have seen lots of claims that 1 = 0.999… but you can be the first one who solves this:
If the claim is that we can reach a point 0.999… on the number line if we move 0.9 + 0.09 + 0.009 + … digit by digit, then the task is to define a procedure (a list) that shows how this point can be reached.
It isn’t. Infinities can’t be dealt with this way. Are you ready to acknowledge we are indeed going in circles yet?
We know you’re confused but we’re not sure how.
Are you claiming that 0.9999 is not a number?
How about 0.3333 ? Does it avoid the problem you perceive with 0.9999 or does it suffer from the same problem?
How about π = 3.141592653… which also has an infinite sequence in its decimal expansion? Is it OK, or does it also fail?
Don’t just keep repeating your rubric; answer these questions so we can move forward.
(0.3, 0.33, 0.333, …) does suffer from the same problem as (0.9, 0.99, 0.999, …). The limit that cannot be reached on the number line is 1/3 then instead of 1.
The infinite sequence of 3.141592653… does suffer from the same problem. The limit that cannot be reached on the number line is π then.
So … you can’t “move your pencil” to 1/3. You can’t “move your pencil” to π.
Perhaps the problem isn’t the numbers, but rather with your pencil and how you are trying to move it.
Of course it is a problem how I am trying to move it. I can reach 1/3 if I am moving it in a single step. Please define the procedure how to move the pen to 1/3 in infinitely many steps!
And you can reach 0.9999 in a single step.
(Unless you can produce some cogent response, this will be my last post.)
Please define a situation where this would be an interesting question.
At best, our hero is simply repeating (ad infinitum) Zeno’s so-called paradox – despite cogent arguments that this approach is flawed, and has been known to be flawed for some centuries (I’d suppose that Zeno himself knew his argument was flawed).
These discussions aren’t always the most amusing, but this one has become one of the most boring.
You just did, with an infinite number of steps … if you’re not getting there, then your pen is what’s fucked up …
This is like a riddle to you, you state something impossible and demand we make be possible … ridiculous … your logic is unsound, your premise is juvenile and if you think you’ll chase everyone away … [wolfish grin] … you’d be wrong …
You’d have to pass through 1/6 though. And before that, you’d have to pass through 1/12. And before that…
How do you get from 0.3 to 0.33 anyway? You first have to move the pen from 0.3 to 0.315, then to 0.3225, then to 0.32525, then to …
You may think that 0.33 is already a defined point on the number line, but so is 0.333…
Your inability to treat infinities as anything other than a big finite doesn’t change this.
Well … we never stop appending 9’s to the decimal … but if we did stop, see see, then it’s not 1 … like duh …
If the claim is that we can reach a point 0.333… on the number line if we move 0.3 + 0.03 + 0.003 + … digit by digit.
If this is a procedure of infinitely many steps, at what step number do you pass 1/6, or 1/12?
You cannot get from a point of the number line to any other point of the number line in infinitely many steps.
You can easily get there in a finite number of steps. So, to go from 0.3 to 0.33 in a single step is no problem. Show the procedure how to get there in an infinite number of steps!
And if we don’t stop we see we are nowhere on the number line.