The Pi sequency revealed ?

what are you talking about.

No need to respond.....I'll get me coat..!!

lol.. more commonly known as 22/7 ??
EDIT : i dont get it..
EDIT : Why would pi be rational if a circle is meant to be an infinitely perfect shape/curve?

[edit on 18-6-2004 by quiksilver]

Originally posted by amantine

Originally posted by Nans DESMICHELS
4atan(1)-2atan(142857142857)

0.000000000014000000000014

No, it approximates that. The real answer is: 1.400000000001399999999978533333333 26613333333386981333333646746666665 28794666665250090666669509998222284 00325902221823327094691342655094489 78243061285172058104162119194660237 75810799130537397842460345463572786 42768243070097724234732764952218828 60815089318196497016898223287070808 00499675285137518904359280141809953 69450977776856994023529684191017152 64480512615163104521082110292790539 77859809823095001437644765545856567 51930523580865859801162359386628497 07024483383867440604191331571072722 57703436365728995411837*10^-11

Nope !

2atanfunny ! It make SATAN !

(14285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285 71 428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571 42857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857)

0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000014

[edit on 18-6-2004 by Nans DESMICHELS]

[edit on 18-6-2004 by Nans DESMICHELS]

Originally posted by amantineI quote my post in another thread about the irrationality proof of pi we learn at school. A similar proof can be found here.

One proof of irrationality of pi:

Look at the following integral


That iintegral has as values
I(2) = -2��^2 + 24
I(3) = -24��^2 + 240
I(4) = 2��^4 - 360��^2 + 3360
I(5) = 60��^4 - 6720��^2 + 60480
I(6) = -2��^6 + 1680��^4 - 151200��^2 + 1330560

All sums of powers of ��. We will determine a maximum for the integral:

x(��-x) �� 0.5��(��-0.5�� )
0.5��(��-0.5�� ) = 0.25��^2
0.25��^2 �� 3

x(��-x) �� 3

Now the other part:
0 �� sin(x) �� 1

sin(x) �� 1

Filling this in and calculating the integral gives you an upperlimit for I(n) of ��*3^n/n!

Now let's assume �� is rational (and repeats): ��=p/q, where p and q are two whole numbers and the expression has been simplified as much as possible.

That means for example with I(4):
(2p^4 - 360p^2q^2 + 3360q^4)/q^4

Because this is an positive numbers that has been simplified as much as possible we can say:

1/q^4 �� (2p^4 - 360p^2q^2 + 3360q^4)/q^4

This is true for every I(n), so you can say:

1/q^n �� I(n)

We determined a upper limit for I(n), so 1/q^n is always smaller than that:

1/q^n �� ��*3^n/n!
1 �� ��*(3q)^n/n!

But lim(x->�� ) x^n/n! goes to 0. That would mean:

lim(x->�� ) 1 �� ��*(3q)^n/n!
gives
1 �� 0

This is ofcourse not true. That problem is our assumption that �� is rational. Q.E.D.

[edit on 18-6-2004 by amantine]

I don't see any proof there that PI is an irrational number...

I'm sorry Amantine, I'm sure you are qualified in sciences, but you know that science got to be always put in questions.

You know that only six centuries ago, scientifics used to tell that earth was flat like a plate......actually, people are calling "irrational" a number they use everyday...
who's irrational ?!

[edit on 18-6-2004 by Nans DESMICHELS]

And also you can't say there will never be a pattern, just that you cannot predict that there will be one, even after calculating the first 200 billion digits. So you can't say that PI is an irrational number. You can say that PI is considered as an irrational in the actual maths and calculations state of art...

Follow me ?

Aren't all trigonometric functions based on PI? Hence, that's why you had lookup tables in old trigonometry text books to look up the number because most calculations resulted in irrational numbers?

Please help me if I'm wrong.
Thanks,
TC

It's a valid proof using a method called 'indirect proof'. You assume that the opposite of what you're trying to prove is true and then show that a contradiction follows from that. I assume that pi is rational and show that then a contradiction occurs.

In mathematics, things don't get disproven. You build on certain axioms regarding proofs in general and some specific of the part of mathematics you're dealing with. A proofs is always right when you assume the axioms.

Pi also has to be irrational, because it is proven transcendental as well (although this is based on Schanuel's Conjecture). If that conjecture is true, that pi has to be transcendental and the conjecture is probably true.

It was a good shot NANS, but this isn't it. It's definitely neat, but in science and math you often make mistakes, and there are often false paths - this is one of them.

It shows that you have great creativity and drive to try to solve the problem, but it wasn't really the answer. Good job, let's move on to the next conjecture.

By the way.. I think I came up with something neat for the separation of primes, but it is way too hard to explain in text, so, until I've made a full and comprehensive document on it.. it'll remain in my head. One day though!

Originally posted by mOjOm


How about this one:
If you add the first '144' Pi decimals it adds up to 666 too.

Numbers are fun, but I still don't get what you're trying to do here exactly.

The odd part about that number is that 144 is one of the bibles magic numbers. 12*12

1,3,6,7,11,12,40 are all magic numbers. That is freaky huh? Maybe there is something more too all the numbers?

As for pi (if you are a radical aethist stop reading here) perhaps it in and of itself is the sign of a higher power. Something so important and usable but never truely solvable. For its proff is always over the next hill.

What I try to show is that when I withdraw 2*atan(142857) (the cycle number) with 4*atan(1), I don't obtain a random pattern, like if I'd tryed it with an random number (like 123456) but a pattern based on 14 (2*7).

It happens to know some things about numbers, but let's consider the funny part of them.....

Well if you start with a prime number, let's say 13 and then multiply it with another prime number, let's say 5.

What is the answer?? Of course, 65!!!!!

And if you add another prime number, let's say 3 it makes 69!!!!!!!!!!!!!!

Ouaou!!!! Well done you have the magic number !!!!! 69!!!

If you multiply now 69 with 11, another prime, you get 759!!! and if you take away another prime number, let's say 93, then:

759-93 = 666!!!!!!!!!!!!!!!

666 THE NUMBER OF THE BEAST!!!!! Oh my God, i am so scared!!!!

We all know how evil the women are!!!

I think you all agree, or at least you have to agree with this statement!!!

Let me then prove to you that women are evil, using mathematical equations and symbols.

We know that women are time consuming and money waisting, actually you can say that women are both time and money waisting, acceptable.

Therefore women must be equal to the direct product of money and time

women = money * time (* means multiplied)


But we all know that time is money, therefore we can replace in the above equation

women = money * money = money^2 ( money^2, means squared)

But again we all know that the root of evil is the MONEY, therefore,

Women = money^2= ( square root of evil)^2

And therefore WOMEN = EVIL

Originally posted by Nans DESMICHELS
What I try to show is that when I withdraw 2*atan(142857) (the cycle number) with 4*atan(1), I don't obtain a random pattern, like if I'd tryed it with an random number (like 123456) but a pattern based on 14 (2*7).

But you get no pattern!

2*atan(142857)-4*atan(1) = -1.400001399978533266133 86981646745287932500935 10060003255030689421953 92640559560018346998172 96760987030106919761449 25416671142553211696211 02789916674946864235622 96803221683803679003027 42987675626799381259190 82110329013780901933689 08062311724918095442846 02634536873407845182857 86358507317199224784755 80649146979159478478303 78804016442667667647713 89643581674673143787091 38988280685101261294611 55329834697617088081177 89875914968137546780706 40264440647976192329759 38446888611607283675115 43765404459420145074732 8921679*10^-5

It is the precision of your calculator that's the problem. The problem comes from the fact, as I have stated before, that lim(x->infinity) 2*atan(x) = pi. And since 4*atan(2) = pi, the larger number you choose for x the more 2*atan(x) - 4*atan(2) approximates 0. Just admit that you made a small mistake, it happens to everyone.

TN1, I sure hope you're being sarcastic (what else can it be?).

[edit on 28-7-2004 by amantine]

First of all I want to point out that it seems very clear, reading Nans DESMICHELS's previous post, that she believed that there was some significance that the arctan of a number formed by repeating the series 142857 came close to PI.

arctan(142857)*2=3.14157865............................. = PI

and

I'm trying to understand why when I do :

2*atan(142857142857142857)
I'm close to PI

and when I do 2*atan(142857142857142857142857)
I'm still closer to PI.

There is no significance, of course, as others pointed out. However, the twice repeated 14s separated by zeroes (ignoring some amount of error after the second 14)(negative error, so it actually is 13999...) is interesting.

I'm using the GNU calc program, which supports arbitrary precision.
First I set the epsilon value for transcendental functions quite low (10^-100).

; epsilon(10^-100)
; display(110)

Then I calculate pi using a built-in trig function. If there are any claims that GNU calc is not precise as advertised, this should settle it:

; acos(-1)
3.1415926535897932384626433832795028841971693993751058209749445923O7816406286208998628034825342117068

Compare this with the first few digits of pi (I got them from www.cecm.sfu.ca...)

3.1415926535897932384626433832795028841971693993751058209749445923O78164062862089986280348253421170679...

Okay, so calc is working. (I replaced one of the 0s with an O because of some ATS julian date script bug which tries to replace some numbers with *)

Now let's check out this pattern:

; 4*atan(1) - 2*atan(142857142857142857)
0.0000000000000000140000000000000000139999999999999997853333333333333326613333333333333386981333333334

Close, though not exact. But note that it is much closer to the pattern than the following (varying last digit):

; 4*atan(1) - 2*atan(142857142857142858)
0.0000000000000000139999999999999999160000000000000002753333333333333344253333333333333088081333333334
; 4*atan(1) - 2*atan(142857142857142856)
0.0000000000000000140000000000000001120000000000000006673333333333333350133333333333333095921333333334

Now we go further:

; epsilon(10^-1000)
; display(1010)

; 4*atan(1) - 2*atan(142857142857142857142857142857142857142857)
0.0000000000000000000000000000000000000000140000000000000000000000000000000000000000139...

; 4*atan(1) - 2*atan(142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857)
0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000140000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000139...

Note that I do not imply that the 9s at the end are repeating (they aren't).

There are many interesting properties of the sequence 142857 found by googling it, but I could not find Nans's discovery anywhere on the internet. Of course, the internet is not a good place to look for such a thing, but I have no reason not to take Nans's word that she discovered this. Congratulations Nans!

Now I notice that the number of zeros between the decimal and the first 14 is the same as the number of zeros between the first 14 and the second 14 (or virtual 14 as it's actually 13999...)

# of reps of the sequence; # of zeroes
3; 16
7; 40
15; 88

So it seems to be:

n; 6n-2

I'm guessing this property could be derived from the taylor series of arctan, but I'm too rusty with my math and too lazy. Nans, why not give that a shot?

Sadly, I was right.

See www.efunda.com...

Notice that for arctan(x), if x > 1, the first constant term is pi/2. This is followed by a series whose only variable is x. Thus, the remainder of 0.00...01400...139.. has nothing to do with Pi, but is only yet another peculiar attribute of the number 142857.

EDIT: Remember that 1/7 = 0.142857142857142857142857....

Also remember that, because of how the question was phrased, we could be more interested in the remainder/2 which would be 0.000...070...0699...

So let w = 0.142857142857142857142857.... to m decimal places
and x = w * 10^m

Then 1/x = 1/w * 10^-m = 7 * 10^-m

Thus the first term of the taylor expansion creates the first seven in the remainder/2.

What's left is to prove that the sum of all the other taylor series terms equals 7*10^-2m. I'll pass

Side note: as m approaches infinity this remainder will of course go to zero.

[edit on 28-7-2004 by HeirToBokassa]

im lost????????????????????????????????????????????????? isnt pi an infinite number? like no matter how high of a number you can think of you just add one and its bigger?

Originally posted by foxtrot_uniform
im lost????????????????????????????????????????????????? isnt pi an infinite number? like no matter how high of a number you can think of you just add one and its bigger?

Please see my last post. The phenomenom has nothing to do with pi, but rather with the peculiar number 142857

I am not suggesting that Pi is predictable or repeating

I'm just starting with basic math (Intermediate Algebra, thank you very much ), but I was just thinking about the sequence 142857. The other day I read that parabolas, circles and asymptotes are really just cross sections of right cones (conic sections). So obviously quadratic formulas map to a higher order surface, of which we're seeing a slice.

So it occurred to me that the cyclic sequence could be take as a cross section of a higher topology. Now I may be confusing arbitrary polynomial results with a constant cyclic number, but I can extend this to all cycle numbers. So all cyclic numbers can be said to be a cross section of some part of higher order topology.

Now the term topology is being horribly abused here. It would be more accurate to say that cyclic sequences might map to some kind of pattern. Patterns are created by observers (there are no objective patterns, right?) and are derived from operations.

So, I find the value in cyclic numbers as something that would tell us more about the operations we use in this world to manipulate numbers. Cyclic numbers are not going to reveal some secret pattern in themselves. They are but a ripple in an operational surface that we should look closer at.

I believe that abstract algebra deals with this in some respect, in that different forms of addition (rings, etc) are defined. Here we take a closer look at operations.

Oh, and an interesting page on cyclic sequences is here: www.atarimagazines.com...

I will concur with both parties that the discovery of whether or not pi is rational or irrational can only be achieved accurately through reiteration. This is what the limit of x approaches infinity achieves using the above metioned integral. Likewise I can see what Nans is attempting, however she is only using a 2nd iteration. If would recommended investigating a manner where you could loop your formula continuously, and refeeding the equation each time, and have each result display. UBASIC should be useful for your purposes and it would be good to use in order to compare/contrast against the proof that Amantine used. It is a programming language, which would be great for this sort of thing, since you would be able to loop and refeed qutie easiliy, and even perhaps test for certain circumstances in each iterative loop, only to be displayed in real time.

Here is the link to UBASIC for MS-Dos
archives.math.utk.edu...

If you need something even more powerful, you may have to look around for something that has higher than 32-bit precision math. I think X-BASIC *may* have some 64-bit functions, but I could be wrong too.

Anyhoo, you may have to make a string tool, that would take in the number before it's dec places have filled their limit of accuracy, convert it into a string and stash in an array, then have the number variable cleared, and somehow shift the decimal places in the formula* to get ready for the next string of decimal places, rinse and repeat. *(divide it by a number that is multiplyed by 10)

I hope this is helpful.