One proof of irrationality of pi:

Look at the following integral


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

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

x(п-x) ≤ 0.5п(п-0.5п
0.5п(п-0.5п = 0.25пUP>2
0.25пUP>2 ≤ 3

x(п-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 п*3UP>n/n!

Now let's assume п is rational (and repeats): п=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):
(2pUP>4 - 360pUP>2qUP>2 + 3360qUP>4)/qUP>4

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

1/qUP>4 ≤ (2pUP>4 - 360pUP>2qUP>2 + 3360qUP>4)/qUP>4

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

1/qUP>n ≤ I(n)

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

1/qUP>n ≤ п*3UP>n/n!
1 ≤ п*(3q)UP>n/n!

But lim(x->∞ xUP>n/n! goes to 0. That would mean:

lim(x->∞ 1 ≤ п*(3q)UP>n/n!
gives
1 ≤ 0

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

[Edited on 9-4-2004 by amantine]

I dont get what you are trying to say.

Here is what I am trying to say:
PI is irrational. Because of this, it can only be represented decimally with an infinite number of integers. Due to the irrationality of PI, these integers have no over-arching pattern. Each n'th digit has an equal chance of being any integer between 0 and 9.

So basically you've got an infinite number of random integers. I am going to tell you that somewhere within the decimal expansion of PI you have:

[Where sets of three integers between 000 and 255 represent text characters, and 256 to 999 represent gibberish]

The entire text of the Bible.
The entire catalog of the Library of Congress.
The US Constitution.
The winning lottery numbers from the beginning of time to the end of time sequentially, for all lotteries, ever.
The stock prices for 5/1/1950, 5/1/2000, 5/1/2050, and 5/1/2100 all in a nice neat row, including a header that says "How to get rich if you know anything about number theory".

and a Document entitled "The future of Human existence" that accurately predicts the names of every person born since 50,000,000 BC through 7,000,000,000 AD, including geneological references and genetic structure, in alphabetical order.

The reason is, of course, that there are an infinite number of these integers in a random arrangement.

Proof: Take the Bible for instance. We know it starts with a capital letter, and though im not gonna bother to check, but lets say its a H. In order for PI to contain this letter, we would need the integer argument that corresponds to H. Starting anywhere, there is a 1 in 1000 chance we have a hit. So for every 1000 characters of PI you look at, you got a fair shot of hitting an "H". Now, lets say the next character of the Bible is "e". Well, we would first need to find an H, and then an e. This will happen whenever we have 6 integers precisely arranged, or once in every 1000000 integers of PI. Lets say the third character is "y". Well.. we will find the word "Hey" once in every 1000000000 integers of PI.

Since PI has an INFINITY of integers, no matter how complicated the sentence, book, or encyclopedia galactica we use, the probability of it being represented by successive integers is still LESS THAN INFINITY.

Q.e.d



[Edited on 10-4-2004 by Cascadego]

on a simplistic level of a message or code, yes. but what if, for example, that a binary represenation of pi calculated through a key such as a ratio in dna would yield a message in the form of a binary stream or a digital movie of sorts. some of the prior messages have some good ideas on things to try.

Originally posted by Cascadego
I dont get what you are trying to say.

Here is what I am trying to say:
PI is irrational. Because of this, it can only be represented decimally with an infinite number of integers. Due to the irrationality of PI, these integers have no over-arching pattern. Each n'th digit has an equal chance of being any integer between 0 and 9.

So basically you've got an infinite number of random integers. I am going to tell you that somewhere within the decimal expansion of PI you have:

[Where sets of three integers between 000 and 255 represent text characters, and 256 to 999 represent gibberish]

The entire text of the Bible.
The entire catalog of the Library of Congress.
The US Constitution.
The winning lottery numbers from the beginning of time to the end of time sequentially, for all lotteries, ever.
The stock prices for 5/1/1950, 5/1/2000, 5/1/2050, and 5/1/2100 all in a nice neat row, including a header that says "How to get rich if you know anything about number theory".

and a Document entitled "The future of Human existence" that accurately predicts the names of every person born since 50,000,000 BC through 7,000,000,000 AD, including geneological references and genetic structure, in alphabetical order.

The reason is, of course, that there are an infinite number of these integers in a random arrangement.

Proof: Take the Bible for instance. We know it starts with a capital letter, and though im not gonna bother to check, but lets say its a H. In order for PI to contain this letter, we would need the integer argument that corresponds to H. Starting anywhere, there is a 1 in 1000 chance we have a hit. So for every 1000 characters of PI you look at, you got a fair shot of hitting an "H". Now, lets say the next character of the Bible is "e". Well, we would first need to find an H, and then an e. This will happen whenever we have 6 integers precisely arranged, or once in every 1000000 integers of PI. Lets say the third character is "y". Well.. we will find the word "Hey" once in every 1000000000 integers of PI.

Since PI has an INFINITY of integers, no matter how complicated the sentence, book, or encyclopedia galactica we use, the probability of it being represented by successive integers is still LESS THAN INFINITY.

Q.e.d



[Edited on 10-4-2004 by Cascadego]

So basically....

in an infinite series of randomness absoleutly anything and everything that can be concieved of or every will be concieved of is possible. which is how i think we came to be.

we exist because we exist. because probability dictates us to exist.

I like pretzels.

-Durandal

gregg braden, author of, 'awakening to zero point' has a new book, ...'the god code', which is the result of research into 'reading' dna. his claim is that our dna is like a library(written in hebrew(of course)). the first message(the 'first' string on the 'first' strand) decoded by his team was "eternal god in the body.'
i'm gonna think this is so funny when i'm dead.