I believe it was this one.
No, it was one posted by CD a few pages back - I’m just late to the party.
Ok I see. Well, read on. It has been fun.
I see that CT has been quiet for a couple of minutes. Not sure if he is doing some long division, making a flash video, composing another proof or conferring with 7777777 via the time cube.
BTW, Cognitive Tide you made a mistake in post 1701 in your cross multiplication.
Simple arithmetic is something you consistently get wrong.
They were quite well defined by Georg Cantor in the 19th century. Nobody on this board needs to reinvent that particular wheel.
I could also recommend to you any number of books on the subject. Understanding Infinity: The Mathematics of Infinite Processes would seem to be appropriate.
Um, you first.
Wrong.
“.000…0001” is meaningless. I know what you think it means, but it doesn’t.
Unless and until you can show where The Hamster King’s proof in post #1462 is in error (hint - you can’t) your posts have absolutely no credibility.
He left, hope he says “Hi” to the Queen of Sweden for us.
In some cases, not very long at all! For example, if I divide 1 by 3, this is what happens:
1/3 = 1 - 0.33 = 0.3; r = 0.1
So then we divide that rest:
0.1/3 = 0.1 - 0.033 = 0.03; r = 0.01
And already in the second step, we can see that no matter what we do, the same step reiterates - if you divide 1 by 3, you end up recursing into the next step, where you’re still dividing 1 by 3 - just one decimal point further to the left. It takes a grand total of 2 steps to recognize, “Wait, hang on, this goes on forever!” At no point does anything happen that could stop the algorithm. Simply by applying what we know about numbers, we can quantify this very efficiently -
10 DIV 1/3
20 SHIFT ONE DECIMAL PLACE
30 GOTO 10
Or something like that. We don’t have to empirically go through the entire thing, we can simply prove via simple logic based on the axioms that it never ends.
Well, in the sense that it’s fundamentally impossible (because no matter how you define x, x+1 will always both be larger and be an integer)…
But there’s no leap of faith involved. You have a pattern which is not randomly magicked into existence, but rather created as a result of an algorithm, and the algorithm shows you very clearly that the pattern will continue. This is like if you see a string of 1s and you know that they are being output by this program:
10 print 1
20 goto 10
No, there is no “leap of pattern recognition faith” in assuming that that code will continue to output 1s! Indeed, the leap of faith would be assuming that a program which by design only can output 1s would output anything else.
And it’s exactly the same with our “divide 1 by 3” program - in every step (decimal point), the list of actions performed and the inputs for those actions are the same - take input “3”, multiply by “0.3”, subtract from input “1”, receive output “1, 3” one decimal place further to the right, enter output. It doesn’t require faith to see that this continues forever, you simply have to understand what’s actually going on. It takes all of 2 steps to see the entire pattern and how it is built.
Also, what the hell does the NSA have to do with anything? No, seriously, what the hell? Care to drop the cryptic nonsense?
your premise seems to be wrong…
according to commonly accepted math,
.0111 = 1/90
which, multiplied by 81 gives
81/90
and (again, commonly accepted) equals
9/10 = .9
which, in no way gives us a valid starting point, because you’re stating that
.9<.1
Sorry, I omitted the squared part. still
81/(9090)=99/(9090)=9/(910)9(910)=1/10 * 1/10 = 1/100 = .01
and that still invalidates your starting point
I don’t believe it’s meant as their starting point. I believe they are stating the goal they wish to demonstrate in the remainder of the post, through transformation into equivalent statements ending in a manifestly true one.
I also don’t believe they do this correctly, of course. The transformation to “[.0111…][.0111…] < .000123456789 + .000000000091/81” is perfectly fine (though you and I know this is also actually an equality), but I don’t see why they then move to “.000123456789… < .000123456789… + .000…00091/81”. If I understood the motivation behind that move, I could respond to it better (again, you and I would likely see this as an equality, the infinitely many zeros making .000…00091 itself zero).
But since things took the “Only I understand recursion; the rest of you quiver in fear” turn, I’ve lost interest.
Indeed. I have always found it rather curious how some people latch onto recursion with some weird epiphany. Arts students especially.
Anyway, this whole thread is more of a spectator sport now. I just can’t be bothered. It is sort of fun for a while, like playing with a puppy. But then it gets tiresome.
(Don’t you have a thesis to write? :D)
The video is very, very annoying. The voice seems to be sythesized somehow and doesn’t quite match up with the images. Why did you link to it?
let p(n) be the statement 10[sup]-n[/sup]/3 = 10[sup]-n - 1[/sup] x 3 rem 10[sup]-n - 1[/sup] x 1
p(0) is true because 10[sup]0[/sup]/3 = 10[sup]-1[/sup] x 3 rem 10[sup]-1[/sup] x 1 is equivalent to: 1/3 = 0.3 rem 0.1
assume p(m) is true for some value of m, that is:
10[sup]-m[/sup]/3 = 10[sup]-m - 1[/sup] x 3 rem 10[sup]-m - 1[/sup] x 1
dividing both sides by 10 gives:
10[sup]-m - 1[/sup]/3 = 10[sup]-m - 1 - 1[/sup] x 3 rem 10[sup]-m - 1 - 1[/sup] x 1
equivalent to:
10[sup]-(m + 1)[/sup]/3 = 10[sup]-(m + 1) - 1[/sup] x 3 rem 10[sup]-(m + 1) - 1[/sup] x 1
which is the statement p(m + 1)
that is to say that p(m) => p(m + 1)
p(0) is true and p(m) => p(m + 1) therefore p(n) is true for all n
and so 1/3 = 0.3 + 0.03 + 0.003 + … = 0.333…
There’s 5 minutes wasted.
For a real world (that is not a measurement done in one’s mind as a thought experiment) finite precision measurement such as .2351 cm
The possible differences are:
1- .2351 cm = .7649 cm
999… - .2351 cm = .7648999… cm (an impossible answer because it suggests the measurement was made with infinite precision which it was not)
.9999 - .2351 cm = .7648 cm (using the same number of significant digits as the measurement yields a conflicting result)
Not to suggest that this talk of measurement has any relevance but isn’t 1.000… equally [no pun intended] infinite precision?
That is an excellent question in my opinion (I know everyone here thinks I am a moron so I won’t presume to think that my assessment of the question is all that relevant.)
One is the fundamental representation of interval for the reals. All operations therefore reference one in the sense that any operation would have to able to be expressed in terms of some expression involving 1.
This of course leads to the question of what is the least element for the reals?
Infinite precision is defined but the least element is finite precision (.0001 - a finite number of zeros). The concepts of magnitude and precision are embedded in the integer 1.
So the question is, is it implicit that 1, which is the fundamental representation of interval, is a concept of interval that is infinitely precise.
If 1 is thought to be infinitely precise then in my opinion it is unequivocal that .999… and one are two distinct members of the set.
Consider the integral from 0 to 1 of some function.
The integral is given by
∫f(x)dx
What is dx?
Is it a member of the reals?
f(x) defines a function consisting of real numbers.
How can you perform the integration if dx is not a member of the reals?
Technically, no. The actual answer is considerably more involved.
Depending on how you learned calculus, you learn the integral as a limit. There is a “dx” term in the integral series. And you take that limit as dx -> 0.
And depending on how detailed you got, you also learned that the integrability of functions can be a complex and difficult topic and that generally f(x) has to be “well behaved” for a certain value of “well behaved” for this limit to work.
Odd, considering your aversion to those self-same limits when applied elsewhere.
Except for countable objects (eg one apple, two trolls, three perfectly good proofs, etc), we cannot measure anything with infinite precision: a 3 foot plank is only approximately 3 foot long.
So why doesn’t the splendid world of integers come tumbling down by the same argument: “DOES NOT CORRESPOND TO REAL WORLD”.
Perhaps you’ll say we have all the classes of countable objects that serve as a real world model. Fair enough. And when we include the negative integers? Show me this -1 apple?
Or further still, what about the half-integers? 0; 0.5; 1; 1.5… etc? By your argument 0.5 as an exact measure cannot exist in the real world. Should we doubt that 0.5 + 0.5 = 1?