The Pi sequency revealed ?, page 2

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§á^2 + 24
I(3) = -24§á^2 + 240
I(4) = 2§á^4 - 360§á^2 + 3360
I(5) = 60§á^4 - 6720§á^2 + 60480
I(6) = -2§á^6 + 1680§á^4 - 151200§á^2 + 1330560

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

x(§á-x) ¡Ü 0.5§á(§á-0.5§á )
0.5§á(§á-0.5§á ) = 0.25§á^2
0.25§á^2 ¡Ü 3

x(§á-x) ¡Ü 3

Now the other part:
0 ¡Ü sin(x) ¡Ü 1

sin(x) ¡Ü 1

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

Now let's assume §á is rational (and repeats): §á=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 ¡Ü (2p^4 - 360p^2q^2 + 3360q^4)/q^4

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

1/q^n ¡Ü I(n)

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

1/q^n ¡Ü §á*3^n/n!
1 ¡Ü §á*(3q)^n/n!

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

lim(x->¡Þ ) 1 ¡Ü §á*(3q)^n/n!
gives
1 ¡Ü 0

This is ofcourse not true. That problem is our assumption that §á 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...

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.

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]

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.

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