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The Pi sequency revealed ?

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posted on Jul, 29 2004 @ 03:41 AM
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So I must wait until you have biggers computers ?


[edit on 29-7-2004 by Nans DESMICHELS]

Anyway, I don't mind about that !

Stay with you're irrational number.

[edit on 29-7-2004 by Nans DESMICHELS]

You know, the problem of the rationality of PI (or in fact, the failure, or the aproximation of pi) is not important in everyday life maths and sciences, but when you wan't to study astrophysics and astronomy, the fact that pi is an "irrational number", or that pi will stay forever a never ending approximation can change many things.

[edit on 29-7-2004 by Nans DESMICHELS]

But i'm not surprised about that : Human mind cannot understand what he cannot explain.

And, also, that's funny but you call "irrational" a number you use everyday. That's not really serious. Amantine, you feel scientifically superior and by the way you make a mistake wich is not very scientific.


[edit on 29-7-2004 by Nans DESMICHELS]




posted on Jul, 29 2004 @ 03:51 AM
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Originally posted by Nans DESMICHELS
So I must wait until you have biggers computers ?

[edit on 29-7-2004 by Nans DESMICHELS]


No, if you use an arbitrary precision calculator, then you can get great results with your present computer. The problem is that if you are using a program that does not support arbitrary precision, then you will not get accurate results past 15 significant figures.

Try it, you'll see that the remainders do not end exactly with 14.



posted on Jul, 29 2004 @ 04:00 AM
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Originally posted by HeirToBokassa

Originally posted by Nans DESMICHELS
So I must wait until you have biggers computers ?

[edit on 29-7-2004 by Nans DESMICHELS]


No, if you use an arbitrary precision calculator, then you can get great results with your present computer. The problem is that if you are using a program that does not support arbitrary precision, then you will not get accurate results past 15 significant figures.

Try it, you'll see that the remainders do not end exactly with 14.


Can you tell me what program I should use ?

But even with "arbitry" precision, I'm sure you will get exactly 14. Or it will be really "arbitrary" precision.

The eloquence of the term used (irrational number, arbitrary precision), and there are many more examples like that. It's what I've called the "psychological failure" in moderns and classical sciences.
This is why I've turned back to this approach of sciences.



posted on Jul, 29 2004 @ 04:09 AM
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Okay, I'll close this debate with a chapter of "The little prince" (Antoine de Saint-Exupery). Some people thinks that it's a story for the kids, but there is a message inside this book, and it's not a message for kids.

I hope this will help you to better understand what I call "the psychological error of science" :



The sixth planet was ten times larger than the last one. It was inhabited by an old gentleman who wrote voluminous books.



"Oh, look! Here is an explorer!" he exclaimed to himself when he saw the little prince coming.

The little prince sat down on the table and panted a little. He had already traveled so much and so far!

"Where do you come from?" the old gentleman said to him.

"What is that big book?" said the little prince. "What are you doing?"

"I am a geographer," said the old gentleman.

"What is a geographer?" asked the little prince.

"A geographer is a scholar who knows the location of all the seas, rivers, towns, mountains, and deserts."

"That is very interesting," said the little prince. "Here at last is a man who has a real profession!" And he cast a look around him at the planet of the geographer. It was the most magnificent and stately planet that he had ever seen.

"Your planet is very beautiful," he said. "Has it any oceans?"

"I couldn't tell you," said the geographer.

"Ah!" The little prince was disappointed. "Has it any mountains?"

"I couldn't tell you," said the geographer.

"And towns, and rivers, and deserts?"

"I couldn't tell you that, either."

"But you are a geographer!"

"Exactly," the geographer said. "But I am not an explorer. I haven't a single explorer on my planet. It is not the geographer who goes out to count the towns, the rivers, the mountains, the seas, the oceans, and the deserts. The geographer is much too important to go loafing about. He does not leave his desk. But he receives the explorers in his study. He asks them questions, and he notes down what they recall of their travels. And if the recollections of any one among them seem interesting to him, the geographer orders an inquiry into that explorer's moral character."

"Why is that?"

"Because an explorer who told lies would bring disaster on the books of the geographer. So would an explorer who drank too much."

"Why is that?" asked the little prince.

"Because intoxicated men see double. Then the geographer would note down two mountains in a place where there was only one."

"I know some one," said the little prince, "who would make a bad explorer."

"That is possible. Then, when the moral character of the explorer is shown to be good, an inquiry is ordered into his discovery."

"One goes to see it?"

"No. That would be too complicated. But one requires the explorer to furnish proofs. For example, if the discovery in question is that of a large mountain, one requires that large stones be brought back from it."

The geographer was suddenly stirred to excitement.



You can find the entire text there >> www.angelfire.com...

[edit on 29-7-2004 by Nans DESMICHELS]



posted on Jul, 29 2004 @ 04:14 AM
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Originally posted by HeirToBokassa
I can see a programming language being useful to corroborate the apparent pattern. But even 64-bit arithmetic will only give 51 decimal places in base 2 (in scientific notation)


You cannot have decimal places in binary. You can have binary digits in binary.
Decimal places only exist in base 10.

UBASIC can handle 2600 digits of base 10 non-floating math. This form of BASIC is intended for Huge value Number Theory(All reals and complex numbers)

If you need floating point, many other languages can suffice. I mentioned BASIC in general as it is very easy to learn, speed is not an issue, accuracy however, is. There are expansion libraries for it written in asm, that can handle huge floating point values. Example: Quickbasic(Double Precision = 64-bit floating point) and perhaps a Qb friendly library off of Simtel or futurebasic, may have the cajones to handle it by way of Sci notation.


Beyond that you would need to emulate arbitrary precision arithmetic


I have already discussed a way to do just that in my prior post.



I'm guessing that UBASIC and XBASIC are slower than C (which is widely perceived as the second fastest language in the world, second only to assembly language). GNU calc is written in C, can support scripting of its own, and also has shared libraries to write C programs with calc functionality.


UBASIC is an interpreter for MS-Dos. It will be a lot slower than compiled C code, but somewhat comparable to Interpreted C. X-Basic will be much faster than UBASIC(at a loss of accuracy for different reasons) because it is a compiler. It will still be a tad slower than Visual C++, as it is a P(re:token based)-code compiler and not a native (re:asm) compiler. Accuracy is the issue, not speed.

This also assumes the user in question can program in C. This goes against the idea of simplicity, as handling the algorithm to achieve said investigation would be difficult enough in it's own right.



But I think it would be cooler to just prove with the taylor series the convergence to the pattern.


That I can concur with! As well as other patterns of interest, as it can be made to be flexible with any reiterary mask. *cough* arrays/pointers *cough*



Should speed become an issue, then I would recommend BCX, "Basic to C Translator". Name says it all. It also includes a C compiler for win32, LCC, so you can make win32 exe files. Thereby avoiding a total code rewrite.

The only reason I even brought up about using BASIC, to handle this task, is in it's simplicity and that it would also be possible to increase accuracy through arbitrary means, such as, stashing the values into a string array, so the accuracy would only be limited by string array memory, instead of relying on the limit of decimal places in any other data format.(singe,double, or triple precision floating values) To each their own...


[edit on 29-7-2004 by Crysstaafur]



posted on Jul, 29 2004 @ 04:16 AM
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"But you--you come from far away! You are an explorer! You shall describe your planet to me!"

And, having opened his big register, the geographer sharpened his pencil. The recitals of explorers are put down first in pencil. One waits until the explorer has furnished proofs, before putting them down in ink.

"Well?" said the geographer expectantly.

"Oh, where I live," said the little prince, "it is not very interesting. It is all so small. I have three volcanoes. Two volcanoes are active and the other is extinct. But one never knows."

"One never knows," said the geographer.

"I have also a flower."

"We do not record flowers," said the geographer.

"Why is that? The flower is the most beautiful thing on my planet!"

"We do not record them," said the geographer, "because they are ephemeral."

"What does that mean--'ephemeral'?"

"Geographies," said the geographer, "are the books which, of all books, are most concerned with matters of consequence. They never become old-fashioned. It is very rarely that a mountain changes its position. It is very rarely that an ocean empties itself of its waters. We write of eternal things."

"But extinct volcanoes may come to life again," the little prince interrupted. "What does that mean-- 'ephemeral'?"

"Whether volcanoes are extinct or alive, it comes to the same thing for us," said the geographer. "The thing that matters to us is the mountain. It does not change."

"But what does that mean--'ephemeral'?" repeated the little prince, who never in his life had let go of a question, once he had asked it.

"It means, 'which is in danger of speedy disappearance.'"

"Is my flower in danger of speedy disappearance?"

"Certainly it is."

"My flower is ephemeral," the little prince said to himself, "and she has only four thorns to defend herself against the world. And I have left her on my planet, all alone!"

That was his first moment of regret. But he took courage once more.

"What place would you advise me to visit now?" he asked.

"The planet Earth," replied the geographer. "It has a good reputation."

And the little prince went away, thinking of his flower.



[edit on 29-7-2004 by Nans DESMICHELS]



posted on Jul, 29 2004 @ 04:17 AM
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Originally posted by Nans DESMICHELS
Can you tell me what program I should use ?

But even with "arbitry" precision, I'm sure you will get exactly 14. Or it will be really "arbitrary" precision.


1) Here is an explanation of what arbitrary precision means: mathworld.wolfram.com...

Arbitrary-precision arithmetic consists of a set of algorithms, functions, and data structures designed specifically to deal with numbers that can be of arbitrary size.

Note that it most certainly does not mean arbitrary results. It means that the precision is arbitrary -- it has no pre-programmed bound. The only limitations are the computer's memory and processing power (not even the computer's word size).

I'm sure you agree that Pi does not fit into 15 decimal places. Therefore, because of the detail required for this research, arbitrary precision (or very high fixed precision ... I believe amantine uses 512 bit precision which should give more than 100 significant figures) is necessary.

If you are suggesting that my results will be "arbitrary" meaning incorrect because I am using an arbitrary precision calculator, you are mistaken. Please see page 2 where I computed pi to 100 decimal places (and I can do 1000 or 10000 too, it all depends on what you set epsilon to -- the user is in complete control)

2) I recommend GNU calc
www.gnu.org...

You need to be running linux. If you're running windows, then it should work if you first set up Cygwin (www.cygwin.com...) but it will take some patience.

GNU MP Bignum Library (www.swox.com... ) is another choice (again requires linux or windows with cygwin). With this one you must write your own C programs.

I do not know of any free and dependable arbitrary precision calculator for windows. I think Matlab, Maple, and/or Mathematica may support it. I recommend GNU calc with Cygwin if you run windows, but you have to be patient in setting it up and learning how to use it.

3) See page 2 of the thread where I listed my results. The number did not end with a perfect 14.



posted on Jul, 29 2004 @ 04:53 AM
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Originally posted by Nans DESMICHELS
And, also, that's funny but you call "irrational" a number you use everyday. That's not really serious.[edit on 29-7-2004 by Nans DESMICHELS]

What are you talking about??? The term "irrational number" was not invented by Amantine. See mathworld.wolfram.com... for an explanation.

What do you expect, that we should invent a new expression?


Amantine, you feel scientifically superior and by the way you make a mistake wich is not very scientific.

I find your tone very childish.



posted on Jul, 29 2004 @ 07:03 AM
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Originally posted by Nans DESMICHELS
But tell me, it's not a demonstration of an axiom I've done upper?


I don't know what you mean, could you rephrase that?


Amantine, can you explain me how I make this pattern?


The pattern that only exists in the first 20 or so decimals? Sure, for x > 0:

atan(x) = pi/2 - atan(1/x)
pi/2 - atan(1/x) = pi/2 - ( sum (from k=0 to infinity) (-1)^k/( (2k+1)*x^(2k+1) ) = pi/2 - ( 1/x - 1/(3x^3) + 1/(5x^5) - 1/(7x^7) + ... )

As HeirToBokkasa said, pi/2 - atan(x) doesn't depend on pi:

pi/2 - ( pi/2 - ( 1/x - 1/(3x^3) + 1/(5x^5) - 1/(7x^7) + ... ) ) = 1/x - 1/(3x^3) + 1/(5x^5) - 1/(7x^7) + ...

If we approximate the infinite sum with the first four terms we get:

1/142857 - 1/(3*142857^3) + 1/(5*142857^5) - 1/(7*142857^7) = 7.0000069998926663306693490823372 64396625042271255536887394412749*10^-6

That were your pattern comes from. From the fact that 1/142857 =7.00007*10^6. The other terms add little to the sum in the beginning of the decimals, but in the end they add enough to destroy the pattern.


How can I get a rational number by withdrawing two irrational numbers?


It's actually possible to do this, but not with pi. You can do it with logarithms and with roots. If you want, I show these examples.

I would appreciate it if you didn't insult me.



posted on Jul, 29 2004 @ 04:06 PM
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Originally posted by Crysstaafur
You cannot have decimal places in binary. You can have binary digits in binary.
Decimal places only exist in base 10.

Sorry, in case anyone was confused by this, I meant digits after the "dot". These are commonly referred to as decimal places even in bases other than 10 (for example see
scitec.uwichill.edu.bb...
and
www.cs.fiu.edu...
but yes, technically they should not be called "decimal places"... perhaps "binary places"... do you know what I should call them?


UBASIC can handle 2600 digits of base 10 non-floating math. This form of BASIC is intended for Huge value Number Theory(All reals and complex numbers)

That's quite impressive. I still recommend arbitrary precision, as they handle both integer and floating-point math with no predefined limit on precision. GNU calc is good because you can use it without programming, and GNU MP BigNum is the fastest around for arbitrary precision math.



I have already discussed a way to do just that in my prior post.

Your string method is not practical for real computation because it is extremely inefficient both for storage and for processing.

EDIT: Another flaw in your string method is the reliance on base 10 mode which is not good. I recommend looking at how GNU MP BigNum takes care of business.

[edit on 29-7-2004 by HeirToBokassa]



posted on Jul, 30 2004 @ 02:11 AM
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Originally posted by amantine
I would appreciate it if you didn't insult me.


I didn't insulted you, i just said that you feel scientifically superior, and it's an egoist approach, and not really scientific.


You show me demonstrations that you have learnt from you're teachers,
but you never take the time to, at least, check my results.


You have learn many things mechanically, and you think you're knowledge is big, but you don't know nothing. You're like a geographer who have never moved from his deck and think he claim to knowing the world.

But, be scientific, experiment, verify once and twice, and you will see, you will fall on the same results than me...


I was the first surprised by these results, but think about it : 142857 is not a common number, it's the cycle number, and pi is the circum ratio. These numbers are connected, that's logical.


[edit on 30-7-2004 by Nans DESMICHELS]



posted on Jul, 30 2004 @ 03:05 AM
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Originally posted by HeirToBokassa
but yes, technically they should not be called "decimal places"... perhaps "binary places"... do you know what I should call them?


That would seem to be suitable, since a 'decimal point' in binary is a
binary point.

My curiousity is piqued about BigNum, thx for mentioning it.



posted on Jul, 30 2004 @ 04:46 AM
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I know that you think that PI is an irrational number containing an infinity of expressions (and in fact, all the possible expressions), and no pattern.

What I think is that the "pi tie" (PI decimals, It's a gameword use to designed pi decimals, but it rhyme in french, not in english, you can translate it to : "the longer is the tie, the better is the pi."
) is an expend of the cycle number 142857...and this expand can contain ALL possible expressions.



Read this article about the cycle number :

www.articlesforeducators.com...

At the end of the article, the author explain that :


I've found several numbers like this. One of them is so large it takes an entire sheet of paper to print it! In fact, it's so large that it contains every four digit telephone number!


[edit on 30-7-2004 by Nans DESMICHELS]

[edit on 30-7-2004 by Nans DESMICHELS]



posted on Jul, 30 2004 @ 07:13 AM
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This may a little off topic, but I think it's pretty weird that the first 144 numbers of the PI sequence add up to 666.
Just like the words "do not fear" are 365 times in the bible.
It's just weird.



posted on Jul, 30 2004 @ 11:39 PM
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Originally posted by Jakko
but I think it's pretty weird that the first 144 numbers of the PI sequence add up to 666.

Why is that weird? Is it the fact that the first 144 numbers, rather than say the first 143 numbers, add up to 666? Or is it just that the first m numbers add up to 666?

Because if it's the second outcome you're talking about, that's not really weird. If you considering accumulating the sums of the digits of pi, like:

3 --> 3
3.1 --> 4
3.14 --> 8
3.141 --> 9
...

etc., then you will see that at some point you have to get a number from the set [658, 659, ..., 665, 666] because there's no way to skip from a number less than 658 to a number greater than 666. (This assumes the pattern doesn't terminate, and Pi doesn't -- at least not before 144 places)

Also, if you first get a number from that set less than 666, you still have the chance of landing on 666 with a subsequent accumulation.

I wrote a program to figure out the probability that a random nonterminating sequence of non-zero integers would accumulate, at some point, to any particular sum. (The "non-zero" part is acceptable because any non-terminating sequence of numbers including zeroes can be mapped to a non-terminating sequence of numbers not including zeroes by removing the zeroes, and the list of accumulations achieved would not change because zero doesn't contribute to the accumulations.)



The result seems to be that the probability converges to 18% for numbers past 56 (such as 666)

1 0.111111
2 0.123457
3 0.137174
4 0.152416
5 0.169351
6 0.188168
7 0.209075
8 0.232306
9 0.258117
10 0.175686
11 0.182861
12 0.189462
13 0.195271
14 0.200033
15 0.203442
16 0.205139
17 0.204702
18 0.201635
19 0.195359

...
664 0.2
665 0.2
666 0.2
...

EDIT: whoops! my code was wrong -- what you see above is the corrected version, and the sum converges to 20% rather than 18%


[edit on 30-7-2004 by HeirToBokassa]

[edit on 31-7-2004 by HeirToBokassa]



posted on Jul, 31 2004 @ 03:27 AM
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HTB (or Amantine) : Why don't you try to verify by yourself my results ?

[edit on 31-7-2004 by Nans DESMICHELS]



posted on Jul, 31 2004 @ 03:38 AM
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Originally posted by Nans DESMICHELS
HTB (or Amantine) : Why don't you try to verify by yourself my results ?
[edit on 31-7-2004 by Nans DESMICHELS]


1) The pattern has nothing to do with Pi.

2) Regarding the pattern itself, I showed on page 2 of this thread that the pattern never ends cleanly with an even "14", but rather an irregular sequnce starting with "13999..."

Maybe I don't understand what you mean correctly -- which results are you talking about?



posted on Jul, 31 2004 @ 04:41 AM
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Originally posted by HeirToBokassa

Originally posted by Nans DESMICHELS
HTB (or Amantine) : Why don't you try to verify by yourself my results ?
[edit on 31-7-2004 by Nans DESMICHELS]


1) The pattern has nothing to do with Pi.

2) Regarding the pattern itself, I showed on page 2 of this thread that the pattern never ends cleanly with an even "14", but rather an irregular sequnce starting with "13999..."

Maybe I don't understand what you mean correctly -- which results are you talking about?


Nope, verify by yourself, the pattern end clearly with 14, and it has no correlation with the kind of calculator you use.


He is telling me that I should have better result with a double precision calculator than with a single precision one. He will never suppose think that I've tryied yet, with Simple, Double and infinite precisions.

[edit on 31-7-2004 by Nans DESMICHELS]

[edit on 31-7-2004 by Nans DESMICHELS]



posted on Jul, 31 2004 @ 08:30 AM
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Originally posted by HeirToBokassa
Why is that weird? Is it the fact that the first 144 numbers, rather than say the first 143 numbers, add up to 666? Or is it just that the first m numbers add up to 666?


It's weird that it's 144 and not for example 156, adding up to 666.



posted on Jul, 31 2004 @ 09:28 AM
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How about the first calculation of Pi in Literature?


1 Kings 7:23 He [Solomon] made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim [diameter = 10] and five cubits high. It took a line of thirty cubits to measure around it. [circumference = 30]


The entire article can be found at

www.yfiles.com...




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