I read the sentence as : It’s not like … that it is “getting closer to” 1 “as you go”. AHunter3’s negating a mistaken notion, not advocating it.
But, netzweltler, while you’re here, please take the time to answer my questions. This kind of behavior absolutely fascinates me and people like you won’t explain it.
Why are you spending so much time arguing this with us? What are you getting out of it? You say you’re not using it for math. So what are you doing? You can’t convince the entire world of mathematicians, professional and amateur, that you’re right and they’re wrong. 1 is *defined *as the answer. You can’t argue that a definition is wrong. You can substitute a different definition and get a different set of mathematics, but a definition is by definition always right.
You’ve posted 79 times so far. You’ve made the same argument 79 times. You’re not convincing anyone who disagreed the first 78 times. So why are you doing this? Please help me understand. That would be the most wonderful thing you could do to lessen our ignorance.
Eight, sir; seven, sir;
Six, sir; five, sir;
Four, sir; three, sir;
Two, sir; one!
Tenser, said the Tensor.
Tenser, said the Tensor.
Tension, apprehension,
And dissension have begun.
—The Demolished Man
I tried to get through this thread, but it was too painful due to all of the various sidetracks. In case this wasn’t one of the usual arguments, the one that worked best for me back in the day was “For any two different real numbers, there are an infinite amount of real numbers between them. Name any one real number between 0.999~ and 1.0”
Of course, I’ve seen absolutely bizarre attempts to counter that. My favorite was 0.999~1. Um, okay.
Depends on what “not exist” means in mathematics. True is that we haven’t reached 1 because none of the steps on the list shows how this happens. Reaching 1 would be the only condition to define a residual distance of 0.
What I always insisted on is that 0.999… is not a point on the number line. So, it is not a point in the interval [0, 1) either (this time correct wording). [0, 1) is simply the union of the intervals [0, 0.9], [0.9, 0.99], [0.99, 0.999], …
How does the limit 1 come into play? All we can see is that it doesn’t. And again:
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
…
There is no action on the list that includes the limit.
As long as it is not clarified that (0.999…) is a point on the number line? Why should we find a point between point 1 and a non-point?
I would like to see a counter-proof to my claim, that we don’t reach 1 even after completing infinitely many steps by t = 1.
Greater than any element of { 0.9, 0.99, 0.999, … }, yes. But take into account, that there is no segment in 0.999… which you cannot find already in one of the elements of the set { 0.9, 0.99, 0.999, … }.
Well, I am convinced that my argument is correct. And I am interested in the counter-arguments coming up during this thread. As soon as I can see that we are stuck in a loop I might stop participating this thread.
I am not convinced that we are stuck in a loop yet.
Okay, let me bring back another statement of yours then:
Here you’re saying we are passing infinitely many digits. 0.999… is infinitely many digits. Are you saying we pass a point that doesn’t exist on the number line?
You can’t have it both ways. If 0.999… is not on the number line there is a number with a finite number of 9’s that is the largest number in the interval [0, 1>.
Now it’s possible to make a number system that has such numbers, but then we’re no longer dealing with the reals, and you’ve made no attempt at introducing the consistent and meaningful definition of infinitesimals required for such an expansion beyond the reals.
Or put another way: If 0.999… is a number it is on the number line. If it’s not a number, your entire argument becomes the boring old “there are no infinities!”.
Correct me if I’m wrong, but originally you claimed you cannot find this point on the number line using you methodology … you can certainly insist all you want to … but that doesn’t make it true …
Sure it does … as t gets closer and closer to 1, the number gets closer and closer to 1 … do you agree? … now compare that to the definition of a limit … once again you’re treating the infinitesimal as an extremely small amount, but an infinitesimal is infinitely small … it has no “amount”, or to say an amount of 0 …
Again and again and again … just because this number isn’t clearly on the number line in your mind doesn’t mean it’s not on the number line … by this same reasoning you would be claiming π is not on the number line because you can’t explicitly find yourself … hopefully you can see how ridiculous such a claim is … of course π is a number, how can you justify saying it’s not?
The limit as t approaches 1 of your summation is equal to 1 … this is when you move your pen an infinitely short distance, or a distance of zero … I’ll have to leave the proof of an infinitely short distance being equal to zero to your second year calculus instructor …
We are definitely stuck in a loop here … we say such-and-such is defined this way … and you say it’s not … it is … it’s not … it is … it’s not … it is … it’s not … etc etc etc …
for ($i = 1; $i <= 10; $i = 1) {
echo "it is ... it's not ... <br>";
}
But we’re not at any of the steps. We’ve done them all. The steps are over. We’ve completed every single step. So no matter how arbitrarily small you set the distance between 1 and 0.99… I can point to a step that is smaller. The only reasonable conclusion, then, is that by completing infinitely many steps, the distance is zero. Because, again, if it weren’t zero, we wouldn’t be done yet, because at every step it comes closer to zero.
You’re right. Because the limit is outside that list. The list goes on forever; we’re looking at what happens after that point.
In mathematics, we say that the limit of a series is defined as thus:
lim from n to infinity of the sequence a(n) = L when for every real number ε > 0, there exists a natural number N such that for all n > N, we have | a(n) − L | < ε.
This only applies to 1. Resolve the above equation for your sequence, {0.9, 0.99, …}, and you’ll find that once again, we reach the same conclusion: ε becomes arbitrarily small. No matter how close to zero you take it, you can always find an N such that all ns larger than N lead to a(n) - 1 being smaller than ε. This doesn’t work with any other number.
But you’re still looking at finitely many steps. Again, what is the distance between 1 and the point we reach after infinitely many steps? Is it 0.00001? Nope, because we passed that step ages ago. Is it 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001? No, again, we passed that step. It does not matter what distance you claim it to be, you cannot find a distance we have not already gone smaller than. Ergo, the distance must be zero.
Let me try to change the subject by posing a challenging problem loosely related to this .9999… question.
The set of infinite sequences of the digits 0-9 is closely related to the set of real numbers in the half-open interval [0,1), e.g. the sequence 3,3,3,3,3,… corresponds to 1/3 and so on.
Construct a bijection between the two sets.
At first glance, it seems like you’re done at once. 3,3,3,3,3,… ←→ the real number 1/3, and so on.
But this doesn’t quite work. 2,0,0,0,0,… → 1/5 and 1,9,9,9,9,9,… → 1/5. This duplication means we do not have a perfect bijection.
Can anyone fix this problem and find a perfect bijection? I do not have an answer yet. (I can do it with some difficulty for base 2 or base 6, but not yet base 10. :mad: )
I came up with a solution. (Indeed, a solution I’d found, but since forgotten :smack:, when I posed the same problem to myself years ago.)
[SPOILER]
We write Diff(x) if there are two decimal representations of the real number x, i.e. if x = A × 10[sup]-B[/sup] for integers A,B.
We write Seq(x) to denote the string of decimal digits in a decimal rendition of x, e.g. Seq(1/7) = 1,4,2,8,5,7,1,4,2,8,5,7,…
When Diff(x), Seq(x) denotes the representational sequence ending in 000…; Seq9(x) will denote the alternate sequence ending in 999…
We now describe a f(x) which provides a bijection between the reals in the half-open interval [0,1) and the set of infinite sequences of decimal digits.
If NOT Diff(x) then f(x) = Seq(x).
If Diff(x) and x < 1/2. then f(x) = Seq(2x)
If Diff(x) and x > 1/2. then f(x) = Seq9(2x-1)
If x = 1/2 then f(x) = 9,9,9,9,9,9,…[/SPOILER]
netzweltler, I must confess that I’ve lost track of what your argument is. So I’m going to try to restate it, and you can tell me if I have it right. OK?
As I understand it, your argument is that, while any finite sequence of digits has meaning, and corresponds to a real number, no infinite string of digits can represent a number, and that there are therefore some numbers which simply cannot be represented by a string of digits. 0.9 is an infinite string of digits, and therefore is an example of a non-number, as is 0.3, which means that there is only one string of digits which represents 1, and no string of digits which represents 1/3.
You agree that 1/3 is a number.
You don’t agree that 0.33333… is a number.
You agree that 0.1(base3) is a number.
But 0.33333…(base 10) and 0.1(base3) are the same number! And both are the same number as 1/3. And 10/30. And π/3π. And 1.00000…/3.0000…
All those are different ways of representing the same number. Doesn’t it seem strange to you that in base 10 mathematics we can represent some numbers that according to you couldn’t be represented in base 3, and that in base 3 we can represent some numbers that we can’t represent in base 10?