The Pi sequency revealed ?

Originally posted by foxtrot_uniform
im lost????????????????????????????????????????????????? isn't pi an infinite number? like no matter how high of a number you can think of you just add one and its bigger?

Well, something like that. PI has a continuous decimal without any known repeating pattern. Unlike, for example, the repeating pattern withing the decimal of:
3/10 = 3.33333.... (Pattern of repeat '3' forever)
or
1/7 = .142857142857... (Pattern of repeat '142857' forever)

There are many different patterns within PI but none that allow for a mathematical conclusion to it's value or cycle that will allow for it to be noted as a repeating decimal.

This is my way of explaining it anyway. You can look up a more 'official' definition on the net if you'd like.

why does 1/7 repeat and pi not repeat? there is a message in there somewhere.

Originally posted by Crysstaafur
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.

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), (www.psc.edu...), which is worse than 16 decimal places in base 10 (in scientific notation). So it would only go so far. Beyond that you would need to emulate arbitrary precision arithmetic, but 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.

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

why does 1/7 repeat and pi not repeat? there is a message in there somewhere.

Well it has to do with the relationship between 7 and 10. Base 10 is arbitrary, and in different bases 1/7 does not necessarily repeat (like in base 7 it is just 0.1)

Originally posted by spangbr
why does 1/7 repeat and pi not repeat? there is a message in there somewhere.

Well, 1/7 has a repeating decimal of .142857... There are lots of different decimals that repeat some kind of pattern like that.

For example:
21/11 = 1.90... Repeat of the 90

However, Pi cannot be represented 100% accurately by a fraction in that same way. There are only approximations for it like:
(22 / 7) It is accurate to 0.04025%.

(355 / 113) which is accurate to 0.00000849%

(104348 / 33215). This is accurate to 0.00000001056%.

Similar to Pi is Phi which also is a Non-Repeating decimal that continues forever. Phi has fractional approximations also like:
5/3, 8/5, 13/8, 21/13, 34/21

Back to Pi though. Using 22/7 we get:
3.1428571428571428571428571428571
Pi is:
3.1415926535897932384626433832795
So you can see that it is accurate up through '3.14' only then changes slightly. But it's close enough for most uses.

104348 / 33215 is:
3.1415926539214210447087159415927
Which you can see is accurate up through '3.141592653'

Does that answer your question???

Well it has to do with the relationship between 7 and 10. Base 10 is arbitrary, and in different bases 1/7 does not necessarily repeat (like in base 7 it is just 0.1)

This is why I think a focus is needed on the operations and the "extras" instead of the numbers. I figure if someone wanted to predict Pi, if its possible, they should look into the relationship between bases, operations and what they do and the nature of numbers. IMHO.

Originally posted by ktprktpr

Well it has to do with the relationship between 7 and 10. Base 10 is arbitrary, and in different bases 1/7 does not necessarily repeat (like in base 7 it is just 0.1)

This is why I think a focus is needed on the operations and the "extras" instead of the numbers. I figure if someone wanted to predict Pi, if its possible, they should look into the relationship between bases, operations and what they do and the nature of numbers. IMHO.

Good points. First of all, yes, you can alter the outcome by using different base values for your number system. Since as you've pointed out that using base 10 and dividing by 7 results in some complications. As far as changing the focus of our use of mathematics as a whole, dealing with basically the entire structure and philosophy behind numbers, you may want to look into something known as 'Dimensionless Mathematics/Numbers'. It is somewhat along those same line IMO and you might find it interesting.

Back to the bases thing. Some interesting number bases that have been used through history, for those interested, have been: 10, 12, 20, & 60. Which 60 and 12 are actually are still used today for some things as well like:
60 seconds to a minute
60 minutes to an hour
12 hours for each clock cycle (half of a 24 hour day.)
being that 5 x 12 = 60 you can see how they are in relation to each other. This continues into geometry where you have 360 degrees to a circle, which is 3 x120 or 6 x 60 etc.
So when you look at the clock and the hand is on 6, it is also 180 degrees, and 60 x 3 = 180.
When it's at 12, or 360 degrees it's also 120 x 3.

So you see the pattern use of certain numbers a bit there.

Originally posted by amantine

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]

Amantine, look :


4atan(1)-2atan(142857142857) =3.1415926535897932384626433832795 - 3.1415926535757932384626293832795
=0.000000000014000000000014

You see, you get a pattern when you withdraw 4atan(1)-2atan(142857142857).
You wont get any pattern if you try it with a random number, even if you try with 142855 or 142858 (142856 got also some weird properties, but we wont try to study 'em now).

Explaination :

142857 is known as the cycle number, and have many weird properties.

When you make 22/7=3.142857

Here are some other properties of the cycle number :


1 / 7 = 0.142857142857 ... = 142857 / 999999

1/7 = 1 * 0,142857 ... = 0,142857 ...
3/7 = 3 * 0,142857 ... = 0,428571 ...
2/7 = 2 * 0,142857 ... = 0,285714 ...
6/7 = 6 * 0,142857 ... = 0,857142 ...
4/7 = 4 * 0,142857 ... = 0,571428 ...
5/7 = 5 * 0,142857 ... = 0,714285 ...


22/7 = 3.142857...



143*999 = 142857



714285 - 142857 = 571428

You can try it with any n*142857 for n between 0 and 7...

142 + 857 = 999


857142 / 571428 = 1.5
Logic, it =6/4


142857 x 678 = 96857046 => 96 + 857046 = 857142
Try it with any number other than 678...

You can consider 142856 like the "little bro' " of 142857.

So what if you cut it in three parts ? :
14 28 56 ? 14 = 28/2 et 28 = 56/2 !

Amantine, I think you're knowledge is blinding you. You have learnt that pi is an irrational number so pi is an irrational number. But things can be differents, and sometimes, science block on a problem for centuries until just a small detail is revealed and unlock an entire our knowledge of the universe.

I have an approch of the universe really different from the actual knowledge teached in universities. And some recent discoveries showed me that I wasn't wrong about some thesys I know. For an example, I don't think that the earth have an eliptic rotation around the sun, but a spiralic one, and that's why you can't actually find pi as a rational number.
You understand what I mean ?

Originally posted by spangbr
why does 1/7 repeat and pi not repeat? there is a message in there somewhere.

I'll tell you :

Amantine explained that 2*atan(x)>pi when x>infinite.
It's an approximation explained by geometry, because you don't calculate the circumference of a circle, but the perimeter of a polygone of x sides. (In fact, they calculate the half (arc) of this polygon, and that's why you need to multiply it by 2 if you wan't to get pi).



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

Originally posted by Nans DESMICHELS
Amantine, look :
4atan(1)-2atan(142857142857) =3.1415926535897932384626433832795 - 3.1415926535757932384626293832795
=0.000000000014000000000014

Right. But you do admit that you are rounding, right? (That's not to say that the pattern is not there, but just for the record).

Amantine, I think you're knowledge is blinding you. You have learnt that pi is an irrational number so pi is an irrational number. But things can be differents, and sometimes, science block on a problem for centuries until just a small detail is revealed and unlock an entire our knowledge of the universe.

Do you agree with my previous post that the pattern you revealed has nothing to do with Pi?

HeirToBakossa >> I'll be agree with you if you agree with me.

Amantine explained that 2*atan(x)>pi when x>infinite.
It's an approximation explained by geometry, because you don't calculate the circumference of a circle, but the perimeter of a polygone of x sides.
(In fact, they calculate the half (arc) of this polygon, and that's why you need to multiply it by 2 if you wan't to get pi).

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

What I'll admit that there is a pattern like the one HeirToBokassa described exist. But it is hardly a pattern in the entire sequence of pi. After a short time the decimals lose their pattern.

And yes, I do have a problem with you claiming that pi is not irrational. Proofs don't just change from wrong to right in mathematics. If a proof is right and the proofs and axioms it uses are right, it is always right. The proofs exist.

The irrationality of pi is also implied by the Hermite-Lindemann theorem, but you probably won't accept that, because the Hermite-Lindemann theorem is based on the Lindemann-Weierstrass theorem which is based on Schanuel's Conjecture.

[edit on 29-7-2004 by amantine]

Originally posted by Nans DESMICHELS
I have an approch of the universe really different from the actual knowledge teached in universities. And some recent discoveries showed me that I wasn't wrong about some thesys I know. For an example, I don't think that the earth have an eliptic rotation around the sun, but a spiralic one, and that's why you can't actually find pi as a rational number.
You understand what I mean ?

Does your 'approch of the universe', including the idea behind your statement above have anything to do with Euclidean geometry/mathematics being a slightly incorrect method???

Originally posted by Nans DESMICHELS
Amantine explained that 2*atan(x)>pi when x>infinite.
It's an approximation explained by geometry, because you don't calculate the circumference of a circle, but the perimeter of a polygone of x sides. (In fact, they calculate the half (arc) of this polygon, and that's why you need to multiply it by 2 if you wan't to get pi).
[edit on 29-7-2004 by Nans DESMICHELS]

That's interesting. Are there any diagrams anywhere with which I can better understand that?

I think about it a different way:

if atan(a) = b
then tan(b) = a
and from definition of tangent
tan(b) = y / x (where x and y are the coordinates on the unit circle giving b as the angle)

now as you approach b = 90 degrees (or Pi/2 in radians), y approaches 1 and x approaches 0

so y/x approaches 1/0 (approaches infinity)
so tan( Pi/2 ) approaches infinity

So atan( a number approaching infinity) = a number approaching Pi/2

So, you tell me that the perfect circle cannot exist because you can't calculate it ?Humans are so silly sometime...

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

Sorry but can you explain me why we get 14 (2*7) in particular.


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

Originally posted by amantine
But it is hardly a pattern in the entire sequence of pi. After a short time the decimals lose their pattern.
[edit on 29-7-2004 by amantine]

Just to reiterate, the pattern has nothing to do with Pi. When you expand atan(x) (with x > 1) to its Taylor series, you get:

Pi/2 + [ some expression involving only constants and x, but not Pi]

When you subtract this from Pi/2 you get

[ some expression involving only constants and x, but not Pi]

So any pattern in that result has nothing to do with Pi, because there is no Pi in the expression

Also, check out the movie Pi en.wikipedia.org...(movie)

Not very mathematically rigorous (more like kookoo cloud land) but very entertaining for math enthusiasts.

Originally posted by amantine
If a proof is right and the proofs and axioms it uses are right, it is always right.

[edit on 29-7-2004 by amantine]

I'm sorry, amantine, but you have the proof under the eyes, and you just refuse to admit it.

I'll repeat myself but look :

When I withdraw 4*atan(1) with 2*atan(random number)
4*atan(1)-2*atan(123456789123)=0.0000000000162000001312799410639

I got a random pattern.

but :
When I withdraw 4*atan(1)-4*atan(142857142857 (the cycle number (1/7))
I got :

4*atan(1)-2*atan(142857142857)=0.000000000014000000000014

A pattern based on 14/2=7

That's not a coincidence.

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

Originally posted by Nans DESMICHELS
So, you tell me that the perfect circle cannot exist because you can't calculate it ?

Humans are so silly sometime...

Actually, as far as a 'Perfect Circle Existing', you don't even need math to realize that it's all but impossible. I'd say the fact that even using math we can't calculate it with perfect accuracy, just helps to prove the previous 'Philosophical/Logical' statement from a more 'Scientific' point of view.

IMO, a perfect circle is something that could only truly exist as a conceptual or imaginative idea, but not in reality.

Originally posted by Nans DESMICHELS

Originally posted by amantine
If a proof is right and the proofs and axioms it uses are right, it is always right.

[edit on 29-7-2004 by amantine]

I'm sorry, amantine, but you have the proof under the eyes, and you just refuse to admit it.

I'll repeat myself but look :

When I withdraw 4*atan(1) with 2*atan(random number)
4*atan(1)-2*atan(123456789123)=0.0000000000162000001312799410639

I got a random pattern.

but :
When I withdraw 4*atan(1)-4*atan(142857142857 (the cycle number (1/7))
I got :

4*atan(1)-2*atan(142857142857)=0.000000000014000000000014

A pattern based on 14/2=7

That's not a coincidence.

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

But tell me, it's not a demonstration of an axiom I've done upper ?

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

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

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

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

Originally posted by Nans DESMICHELS
Sorry but can you explain me why we get 14 (2*7) in particular.
[edit on 29-7-2004 by Nans DESMICHELS]

I have begun the explanation on page 2 of this thread, but have not completed it.

It has to do with
1)the taylor series of arctan
2) that 1/7 = 0.142857 142857 ...
3) that Pi/2 - atan(142857 142857 ...) = 0.000....0700....06999

I have shown how the first seven in the result comes about. I have not shown yet how the second seven (minus a small amount of error) comes about. I believe that it can be proven using Taylor's Remainder Theorem.

If anyone doubts that please let me know and I will (struggle with my inferior mathematical abilities to) finish the explanation.

Originally posted by Nans DESMICHELS
Amantine, can you explain me how I make this pattern ?

How can I get a rational number by withdrawing two irrational numbers ?
[edit on 29-7-2004 by Nans DESMICHELS]

Can you prove that the result is rational? It seems that you have rounded it, and that the real, unrounded, result is not a repeating or terminating decimal.

I strongly urge anyone investingating this to use a arbitrary precision calculator in their work. Remember, even 64-bit floating points only give you less than 16 decimals of precision in scientific notation.