I stated only that it produces each next digit, you entered the "from the previous one." As n grows, it leaves unchanged a finite string of consecutive digits following 0. The index of the last digit of this string is a monotone increasing function of n. Therefore, you can use this formula to determine any digit with absolute certainty.
Now, the fact that it takes 10 million terms to get the 14th digit is irrelevant for this conversation. We are talking about the ability to determine the digits of PI, not the practical computation of these digits on contemporary computers. Whether it takes 2 terms or 2 trillion is of no consequence. The formula provides constructive order for the digits of PI.
While it is an advanced undergraduate proof to show that PI is irrational, this does not mean that the digits of PI are without "order."
First, recall that irrational simply means that PI cannot be expressed in the form PI = P/Q for P and Q!=0 integers. Alternatively, this means that the digits of PI will never form a repeating sequence, such as 237/137 which is
But this does not mean that PI has no order! For example, a special value of the famous Riemann Zeta Function gives PI by the formula
PI*PI/6 = 1 + 1/4 + 1/9 + 1/16 + 1/25 +...
So if you want to calculate the digits of PI, just keep adding 1/n^2 on the right hand side. Isn't this an ordering of the digits? Does this have enough meaning to defunct movies such as PI - who rotate around a mythical quest to find some religious numeracy within the digits?
I think so. However, it comes down to this question, how do we define order? I think that order is best defined as the lack of disorder, the lack of randomness. Just as trust is best defined as the lack of distrust. In this definition, we have found order in PI, we have an analytic formula (above) to produce each next digit! Thats as far from random as you get.
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I stated only that it produces each next digit, you entered the "from the previous one." As n grows, it leaves unchanged a finite string of consecutive digits following 0. The index of the last digit of this string is a monotone increasing function of n. Therefore, you can use this formula to determine any digit with absolute certainty.
Now, the fact that it takes 10 million terms to get the 14th digit is irrelevant for this conversation. We are talking about the ability to determine the digits of PI, not the practical computation of these digits on contemporary computers. Whether it takes 2 terms or 2 trillion is of no consequence. The formula provides constructive order for the digits of PI.
While it is an advanced undergraduate proof to show that PI is irrational, this does not mean that the digits of PI are without "order."
...
First, recall that irrational simply means that PI cannot be expressed in the form PI = P/Q for P and Q!=0 integers. Alternatively, this means that the digits of PI will never form a repeating sequence, such as 237/137 which is
1.729927007299270072992700... (keep adding 72992700)
But this does not mean that PI has no order! For example, a special value of the famous Riemann Zeta Function gives PI by the formula
PI*PI/6 = 1 + 1/4 + 1/9 + 1/16 + 1/25 +
So if you want to calculate the digits of PI, just keep adding 1/n^2 on the right hand side. Isn't this an ordering of the digits? Does this have enough meaning to defunct movies such as PI - who rotate around a mythical quest to find some religious numeracy within the digits?
I think so. However, it comes down to this question, how do we define order? I think that order is best defined as the lack of disorder, the lack of randomness. Just as trust is best defined as the lack of distrust. In this definition, we have found order in PI, we have an analytic formula (above) to produce each next digit! Thats as far from random as you get.
Here is the zip code for Robert Harrison & Assoc so you can be sure your gifts get there:
2550 S TELEGRAPH RD STE 275
BLOOMFIELD HILLS MI 48302-0908
This tidbit from the usps.