netzweltler, please pay attention to this. If this were Ancient Greece, your “intuitive” approach to the apparent contradictions and paradoxes of infinities might be right up there on the leading edge of mathematics & philosophy. But many great minds have brought us a long way since then, among other things developing rigorous methods to deal with infinities in various contexts.
By analogy, you’re arguing on about the same level as “if we evolved from monkeys, why are there still monkeys”.
The problem of notation. 1/3 really is 0.3333…, but the issue you may have is that 0.3333… seems to go on forever. However, 1/3 in base 3 is written as 0.1. No problems with notation.
So, rather than 0.333…, 0.666…, 0.999… when writing 1/3, 2/3, 3/3 in base 10, let’s instead write it in base 3:
0.1, 0.2, 1.
No problem with notation at all, and the “problem” with infinite sequences goes away. And, since we all agree (I hope) that 1/3 in base 3 is 0.1, then we all see that there’s no issue with 0.99999… = 1, since we all realize it’s just a notation problem, not a mathematical problem.
What’s 0.333…+0.333…+0.333… equal to? Well, in base three, 0.1 + 0.1 + 0.1 = 1. The same number in different bases are still the same number, so 0.333…+0.333…+0.333… = 0.999… = 1.
Of course you reach 1/3 when we consider all the lines … you ask to pick just one line, but that’s not how infinity works … your “for each” is invalid, it should be “for each and every” … whichever line we find will only represent a very very large finite number.
Well, since you’ve declared to hold to your flawed logic and will be repeating infinitely, I have no choice but to answer:
Beetlejuice
Beetlejuice
Beetlejuice
Can you tell us what other number than 1/3 can be expressed by 0.333~?
If it isn’t 1/3, then what do you think it is? Can you construct this point?
The problem is that, if you could, you would also be showing that 0.000~ is > 0, and that’s nonsense.
FTR, I found your argument cogent and utterly persuasive.
Well, we were both born, SDMB-wise, in March of 2002, so, as twins, it only makes sense that we’d find each other persuasive.
Now you’re just being an idiot.
(Don’t make me post a smiley.)
My join date is also shown as March 2002. But I really joined several months earlier. All my early posts and my join date were lost in the winter of missed content. So I was born again in March 2002. Maybe you guys were as well. Sometimes facts aren’t facts.
My point entirely! Before the great virtual death of 2002 I had been very witty and universally liked. It’s had having to drag your zombied corpse back from oblivion.
Hey, I resemble that remark!
This thread has taken a weird turn.
If you start moving your pencil on the calendar, you’ll reach my join date at t=6 years.
Ahem! 5.999… years!
That’s actually kinda creepy …
If you avoid using infinite sequences then the problem is gone. I agree. This doesn’t mean that the “problem” with infinite sequences goes away.
How do you distinguish “for each” and “for each and every”? By the way, whichever digit we find in 0.999… will only represent a finitely indexed digit - even if there are infinitely many of them.
As soon as we can define a point 0.333~ on the number line we are able to define what number can be expressed by 0.333~.
It is true that we complete 0.333… if we consider all the lines. This doesn’t mean that we reached 1/3, because none of the infinitely many lines shows how we get there.
netzweltler, If I understand your argument correctly (one cannot place 0.999… on the number line because it isn’t equal to any of {0.9, 0.99, 0.999, etc}) then similarly one cannot place any other repeating decimal nor any irrational number.
But as shown by RitterSport if we consider bases other than 10, a given repeating decimal can be transformed into a non-repeating decimal and easily placed on the number line. Numbers cannot pop into and out of existence according to how many fingers we have.
π, e, √2 and at least a couple more irrational numbers are perfectly well defined despite your supposed limitation.
PS: Did you duck the question: “What is the decimal expansion of 1/3?”
I don’t think so. See my answer to Trinopus’ question.
All I have seen what RitterSport has shown is that 1/3 can be written 0.1 in base 3. So what? Both notations are well-defined points on their number lines.
Post #296? – with all due respect, you ducked the question there too. I think I see why…
The so what is :
1/3 = 0.1 (base 3) – a point easily placed on the number line
and
1/3 = 0.333… (base 10) – a point you claim is impossible to place on the number line
So, now it is quite pivotal that you state in plain and direct language what you think the decimal expansion of 1/3 is, because clearly you reject the second of these two assertions or you wish to claim that numbers flip-flop in-and-out of existence according to which base we choose to name them in.