26-04-2002 01:14									#91
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quote: Originally posted by Caesar the Great i guess we're just taken from one of those realities to another and that is consciousness... although wouldn't an exact copy of you be stuck somewhere (or more appropriately somewhen) at a point in time for eternity....

Its a philisophical question to ask whether an exact permutational duplicate of me, in every regard including neural construction and content, with an identical universe to my perception except for one atomic difference (like a speck of space dust having one less proton) I'd say that that was a clone... yet I would be different if I ever perceived the differing space dust particle. Therefore there is a probability that I am the same.

As to when... this exact state of the universe could have happened immediately after the inception of the universe, just like a pop-up book. So that me could exist any time.

Since I cannot perceive another useful quantum field state... it will remain a philisophical question.

	26-04-2002 01:19									#92
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quote: Originally posted by Anunikoba Sooo.. if a tree falls in a forest, and no one is around to hear it, then does it make a sound? Or, if a Quanta exists in a different reality, and we can't percieve that reality, does it exist at all?

well the mind blowing thing is that we can perceive quantum effects... they are just difficult to measure. It looks like quantum states leave artifacts in our reality.

If you place two uncharged utterly smooth plates 2 uf apart in a vacuum, they will move together. Why? because one half of a virtual particle appeared between them. They are using the property to work at sub nano scale, building new computers.

	26-04-2002 01:19									#93
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quote: Originally posted by Ecthelion Does the "what do you look like" thread contain a picture of Caesar? No!

I have a few good ones

	26-04-2002 01:21									#94
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quote: Originally posted by orange I have a few good ones

that depends on your definition "good" and my definition of "over my dead body"

	26-04-2002 01:24									#95
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Interestingly enough too.. QFT explains history... Your perception is the same going back on the state path that you travelled.

I.E. in QF infinite states are many single states in reverse.

	26-04-2002 01:40									#96
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quote: Incidentally, while pi's cool, e just sucks. I hate it. Stupid e.

I agree. Stupid e

Why has this thread taken 4 pages, when it needs a one word answer?
Yes.

	26-04-2002 01:44									#97
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quote: Originally posted by connorkimbro I'm gonna invent a new type of computer that uses every digit (0-9) instead of just 1's and 0's . . . and then try and open Pi in Notepad, to see what happens.

They're already making great progress into multi-state computers that would ditch binary.

	26-04-2002 01:50									#98
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quote: Originally posted by MrBaggins well the mind blowing thing is that we can perceive quantum effects... they are just difficult to measure. It looks like quantum states leave artifacts in our reality. If you place two uncharged utterly smooth plates 2 uf apart in a vacuum, they will move together. Why? because one half of a virtual particle appeared between them. They are using the property to work at sub nano scale, building new computers.

Thanx for your lessons in basic algebra and quantam mechanics- they are highly appreaciated. Hell! Even I read A Brief History of Time.

But my point is that when you speak of 'forwards-recursive infinite states', how relavant are all those other states of the quanta, if they are non-interacting with our percieved universe? (if, in fact, they exist at all)

quote: Originally posted by connorkimbro I'm gonna invent a new type of computer that uses every digit (0-9) instead of just 1's and 0's . . . and then try and open Pi in Notepad, to see what happens.

It will probably tell you the file is too large for notepad, and ask if you'd like to open it with Wordpad.



And its about damn time they abandoned binary!

	26-04-2002 01:50									#99
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Wombat: pi seems to be a subject that spawns discussion very quickly... And Squid hasn't yet even seen this thread...

	26-04-2002 01:58									#100
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quote: Originally posted by Anunikoba Thanx for your lessons in basic algebra and quantam mechanics- they are highly appreaciated. Hell! Even I read A Brief History of Time. But my point is that when you speak of 'forwards-recursive infinite states', how relavant are all those other states of the quanta, if they are non-interacting with our percieved universe? (if, in fact, they exist at all)

You can build a NMR component computer to perform quantum calculations, solve cryptographic problems and factors, and use other quantum behaviors such as teleportation (information teleportation) All QC concepts so far have either succeeded or are in the process being successfully developed. There is not yet one serious quantum computing concept that has failed. Solving decoherence is the issue; quata states have limited lifespans, so you have to create quantum error correction.

Just as modern planes prove the bernoulli principal, so do QC's prove quantum theory effects and lend serious weight to quantum theory.

	26-04-2002 01:59									#101
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Just back a bit on pi, the question if pi does contain itself. Answer: Exactly once, beginning with the first digit.
(I understand "contains itelf" as "there is some n such that beginning with the n-th digit you have a full representation of pi").
Proof:
The "once" is obvious, because starting with the first "3" you have a full representation of pi
Suppose there an n such that the n-th digit is 3, the n+1st is 1, n+2nd is 4 etc. (conveniently, let the first 3 be the digit number 0.) Then we know that the n+ith digit (i < n) equals the i-th digit. The 2n-th digit again is a 3. And so on. Which means in the end, pi is a periodic and thus a rational number, which it has been proved not to be. Therefore, there is no other digit than n=0 where you can find a copy of pi within pi.

quote: A number that can be expressed as a fraction

MrBaggins, this post seems not to be complete, I guess you have forgotten to fit in some pictures. Anyway, I should explain what algebraic numbers age: They are those numbers that can be expressed as the zeros of polynomials with integer coefficients.

A good hint: Don't mess with infinity if you haven't understood why there are as many integers as rational numbers and why there are more real numbers than rational numbers.

Two problems for beginners: Imagine you have a hotel with an infinite number of beds such that each bed is labelled by an integer number. Each bed is occupied.
1) One bus full of n tired guests arrives. How do you fit them into the hotel?
2) An infinite number (also labelled with distinct integer numbers) each with n tired guests arrive. How do you fit them into the hotel?

	26-04-2002 02:05									#102
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It's impossible to speculate on what Pi contains, since infinity is undefined.

	26-04-2002 02:06									#103
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quote: Originally posted by Adalbertus

corrected it missing two expressions at the bottom.

	26-04-2002 02:19									#104
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Actually, we don't really know if Pi really is an irrational number, do we?

Since we've never seen the end to it, we can't be sure it doesn't eventually start repeating itself and become, albeit extremely complex, rational.

	26-04-2002 02:19									#105
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quote: Originally posted by MrBaggins You can build a NMR component computer to perform quantum calculations, solve cryptographic problems and factors, and use other quantum behaviors such as teleportation (information teleportation) All QC concepts so far have either succeeded or are in the process being successfully developed. There is not yet one serious quantum computing concept that has failed. Solving decoherence is the issue; quata states have limited lifespans, so you have to create quantum error correction. Just as modern planes prove the bernoulli principal, so do QC's prove quantum theory effects and lend serious weight to quantum theory.

Once again, I am not debunking quantum theory (as incomplete as it is) as yes, it does work. It is this statement:

"Everything happens, and keeps on recursively happening.

We only perceive one of those realities."

that fails the proof test.

	26-04-2002 02:22									#106
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quote: Originally posted by Anunikoba Once again, I am not debunking quantum theory (as incomplete as it is) as yes, it does work. It is this statement: "Everything happens, and keeps on recursively happening. We only perceive one of those realities." that fails the proof test.

Actually, no... the fundemental principal of Quantum Field Theory and its divergence with General Relativity, is that there are infinities. QC leverages Quantum Field Theory behavior as much as Quantum Mechanics.

	26-04-2002 02:38									#107
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Maybe I should correct all the mistakes...

Nah, to much work .

	26-04-2002 02:40									#108
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quote: Actually, we don't really know if Pi really is an irrational number, do we?

I've never bothered so much about piology, but there exists a proof that pi is irrational (you can prove such things without checking all decimals), and there exists also a proof that every finite sequence of digits is contained in pi. I don't know where to find the proof, and I'm not so much interested in that topic as to search for it. Sorry.

Btw. If pi were a rational number, the "quadrature of the circle" would be possible, i. e. to geometrically construct a square with the same area as a given circle, using pen, ruler and compasses only. It has been proved to be impossible.

	26-04-2002 03:34									#109
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quote: Originally posted by Cesar the Great say you have a number line from negative infinity to infinity and you made a point for each rational number and each irrational number, the set of points for the rational numbers is A and the set of points for the irrational numbers is B. are there more elements in set A or set B?

You have more elements in set B.

Preliminaries:
First, the mathematician's notion of "has the same number of elements" is perhaps a bit counterintuitive. For sets in general, one talks about the cardinality, #A, of a set A. For finite sets, this is just the familiar number of elements. For infinite sets, you start with mappings. Take two sets A and B, then you can map each element of A to an element of B., in some way. (the familiar functions are examples of mappings, or numbering a set of six eggs is a mapping of the set {1,2,3,4,5,6} to the set of your eggs.)

Then you define mappings, who for each element of A have a unique image in B, as injective. More mathematically: A mapping f:A->B is injective if and only if for each a and b in A holds that if f(a) = f(B) then a=b.
The labelling of the eggs is injective. The sine function is not, as sin (0)= sin(pi), but 0 != pi.

Now back to the cardinalities: One defines for two sets A and B that #A <= #B if there exists an injective mapping f:A->B. This is exactly the way you would try to find out if you have more steaks or more plates. Further one defines #A =#B if #A <= #B and #B <= #A.

If you think a bit over it you will make the stunning discovery that there are as many even numbers as integers (see below for something even stranger). This seems to be odd at first, but there is no better way to define the "number of elements" for infinite sets.

Now we are prepared for infinity. First: There are as many rational numbers as positive integers.
Proof: Denote the set of positive integers by N and of positive rational numbers by Q. We already know that N is a subset of Q, therefore the injective mapping f:N->Q is trivially built. (note, there is no reason to exhaust Q with this mapping).
Then we know that each rational q can be expressed as a fraction n/m. Now set up the following scheme:

1/1 1/2 1/3 1/4 1/5 ....
2/1 2/2 2/3 2/4 2/5 ....
3/1 3/2 3/3 3/4 3/5 .....
4/1 4/2 4/3 4/4 4/5 ....

and so on. This scheme contains all positive rational numbers. Now start in the top left. Label the number 1/1 as 1. Then go down. Label 2/1 as 2. Then go right up, label 1/2 as 3. Go right, find 1/3 as 4. Go left down. 2/2=1/1, so skip this number. Again left down, 3/1 is number 5. And so on.:
_ _
|/ / / /
/ / /
/ / /
|_/

I'm not a good ASCII painter, but I think you see the path. Anyway, you label each of the positive rational numbers with a unique positive integer, so you have also #Q <= #N. The sets thus have the same cardinality, i. e. "number" of elements.

So, now about the positive real numbers, R. Let's assume we have already proven that each decimal number has a unique decimal expansion (except for 1=0.999999... and analogue). Then we assume we have successfully numbered all decimal numbers. Then we write them in a long list

3.1415926.....
2.7182818....
1.2000000....
etc.

Then we take the first number and change the first digit, take the second number, change the second digit, etc.
1.1415926...
2.4182818...
1.2100000...
etc.

and collect the changed digits to a new number, in this case 1.41... This number is clearly a real number and clearly not in our list. Which means there is no injective mapping f:R->N, or in other words #R > #N.

Now, the irrational I numbers are a subset of R only, such that R is the union of Q and I. Now, if I were countable (which means, #I = #N), then we could easily construct an injective mapping from R->N by mapping Q to the even and I to the odd numbers for example. Therefore, #I>#Q. (In fact, one can show, but I don't know, how atm, it's time for bed now, that #I=#R).

An interesting side note: It has been shown that it is impossible to know if there is a set A which fulfills #N < #A < #R.

Good night.

	26-04-2002 04:23									#110
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quote: Originally posted by orange I'm not saying it 'won't' but I am saying that it's more likely for conner's phone number to appear than it is for 12345678987654321 (repeated a thousand times over) to occur because of the SIZE of the number and repatative nature of the integer.

Look at the definition of an irrational number. Once you understand that, you will understand that likely is not the correct word. Both number DEFINITELY appear in Pi.

The proper word is frequency. Conner's phone number appears more frequently than your number.

Please note that both numbers appear an infinite number of times inside Pi.

	26-04-2002 04:44									#111
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"Does Pi contain my phone number?"

the answer is YES

It does. I won't tell you where it's at though. In fact, it contains both my cell phone number AND my regular phone line.

The following web page will search the first 100,000,000 digits of pi and attempt to find a string of numbers that you enter. So, there.

lol

http://www.angio.net/pi/piquery

	26-04-2002 11:25									#112
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quote: Look at the definition of an irrational number.

The statement "any finite sequence of digits can be found in the digits of pi" needs an independent proof. It cannot be deduced from the fact that pi is irrational.

The irrational number 1.01001000100001...
with always one 0 more between the 1's naturally cannot contain the number 42. Note, this is a decimal number. There exists also a binary number 1.01001000100001... which is bigger than the decimal number, it is about 1.28326... .


Slowly I begin to understand why a question which simply can be answered "yes" develops in a four page thread.

	26-04-2002 11:38									#113
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quote: Originally posted by connorkimbro Yeah, but it's not like 10 divided by 3 which gives you an infinite string of 3's. It's always different numbers, and never repeats a pattern.

If you can prove that Pi must contain every possible finite string of digits then I'll give you 10$

*Begins practicing acceptance speech for Fields medal*

	26-04-2002 11:41									#114
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quote: Originally posted by LulThyme If your phone number is finite in length then it is contained in Pi's decimals actually this is true for any irrationnal number's decimals and any phone number of finite length (not just yours ) I even think there are some sites where you can input a string of digits and it looks where in the digits of pi it appears for the first time. for 7 digits im not sure how efficient sucha site would be though ud have to look

I'm pretty sure this just ain't true.

Actually, I am sure.

The number:

0.10110111011110111110....

is irrational (and I'm pretty sure it's transcendental too), however it does not even contain every possible 1-digit string (doesn't contain digits 2-9).

	27-04-2002 07:19									#115
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quote: Originally posted by Juggernaut Actually, we don't really know if Pi really is an irrational number, do we? Since we've never seen the end to it, we can't be sure it doesn't eventually start repeating itself and become, albeit extremely complex, rational.

You can pretty easily prove that Pi is irrational from some of its series expansions (expression as an infinite sum). Proving that it's transcendental is slightly harder, but it's been done.

	27-04-2002 07:22									#116
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quote: Originally posted by pchang Look at the definition of an irrational number. Once you understand that, you will understand that likely is not the correct word. Both number DEFINITELY appear in Pi. The proper word is frequency. Conner's phone number appears more frequently than your number. Please note that both numbers appear an infinite number of times inside Pi.

Not necessarily true, AFAIK. Definitely not true simply because Pi is irrational.

	27-04-2002 08:44									#117
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Ugggggggggghhh. I smell lots of horribly false statements.

Interestingly enough, onr of our math-faculty lunches here had the topic "Is Pi normal?" Interesting that word hasn't come up yet, but normal means that for any string of length k in base b, there is 1/(b^k) chance of finding that string in all strings of length k in the decimal expansion. So in base 10, 1/10 of the digits are 2's, 1/10 are 3's, 1/100 of the 2-string patterns are "23," etc.

As KrazyHorse pointed out, proving pi is normal might get you a Fields Medal. But from what we've expanded so far and general evidence, pi at least "seems" random enough to be normal. Also, any random irrational number you choose is irrational (want to generate an irrational number? Roll a 10-sided die over and over again and write it down)- random selection of irrationals will give you probability 1 of picking a normal one, so the odds are on the side of pi being normal as well. The examples given of things like .010010001... are bizarre degenerate cases... which pi might be, but I doubt it myself.

Last edited by SnowFire on 28-04-2002 at 09:56

	27-04-2002 08:56									#118
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Probable, but not certain.

Just like I can show you that it's probable that any number you show me is irrational...

	27-04-2002 09:36									#119
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quote: Originally posted by SnowFire Also, any random irrational number you choose is irrational (want to generate an irrational number? Roll a 10-sided die over and over again and write it down)- random selection of irrationals will give you probability 1 of picking a normal one, so the odds are on the side of pi being rational as well.

You meant normal, yes?

	27-04-2002 09:50									#120
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quote: Originally posted by connorkimbro "Does Pi contain my phone number?" the answer is YES It does. I won't tell you where it's at though. In fact, it contains both my cell phone number AND my regular phone line. The following web page will search the first 100,000,000 digits of pi and attempt to find a string of numbers that you enter. So, there. lol http://www.angio.net/pi/piquery

quote: quote: The string 302837**** did not occur in the first 100000000 digits of pi after position 0.

not for me