Free Republic Browse · Search News/Activism Topics · Post Article

To: Alamo-Girl; Doctor Stochastic
A string of numbers extracted from the extension of pi would falsely appear random where they are in fact highly determined.

Consider the number
0.12345678910111213141516171819202122232425....

It is highly determined, in fact one can come up with a formula for the n-th digit.

Given any number, it is obvious that it appears infinitely many times in this decimal.

It is conjectured that pi has the same property; it is known that almost all real numbers do.

Dr S: Anything to add to AG's assertion? or mine?

726 posted on 12/13/2005 3:32:58 PM PST by Virginia-American

To: Virginia-American

It's possible to generate a system that nearly "appears random." That's what Pseudo-Random-Number-Generators do. "Random" is more appropriately applied to a process, not the result of such process.

No written string of numbers is "random"; it merely is. The best one can do is say that a string of numbers obeys certain laws that randomly generated strings do.

Champernowne's number (.12345678910..., as you gave) does have the property that "any string occurs with the proper frequency." Thus (if mapped to a alphabet), it would contain the complete works of Shakespeare, the complete works of Shakespeare with one error, etc. (and the complete text of "Contact.") However, it can be proved (somewhere, I don't have access to review journals) that Champernowne's number does not obey the Law of the Iterated Logarithm. (I don't know how to generate a number that does except by ad hoc post hoc adjustments to the output.)

804 posted on 12/13/2005 6:33:48 PM PST by Doctor Stochastic (Vegetabilisch = chaotisch ist der Charakter der Modernen. - Friedrich Schlegel)

To: Virginia-American; Doctor Stochastic