Re: [math-fun] Solving polynomial equations with roots, etc.
Date: Tue, 01 Dec 2009 20:49:57 -0800 Subject: Re: [math-fun] Solving polynomial equations with roots, etc. From: "Stephen B. Gray" <stevebg@roadrunner.com> Henry Baker wrote:
Isn't this what the Borwein expansions are all about?
At 06:07 PM 12/1/2009, you wrote:
from SBG ...
Date: Mon, 30 Nov 2009 19:44:17 -0800 Subject: Re: [math-fun] Solving polynomial equations with roots, etc. From: "Stephen B. Gray" <stevebg@roadrunner.com>
On a different subject, for anyone to comment on.
Everyone knows that sqrt(2), pi, and all irrational numbers have no decimal strings that repeat an infinite number of times. That's not necessarily the same as the decimal expansion being totally without any pattern. That is, is there any way to predict the next digits of say sqrt(2) or pi without doing one of the usual computations? In other words, is it possible to "know" the entire decimal expansion of any "ordinary" irrational? Is anything known about patterns in "regular" irrational expansions?
I'm excluding numbers invented for the sole purpose of being irrational or transcendental and with an obvious pattern like .101001000100001.... or .123456789101112..... ), etc.) I know about the question of "normal" expansions but that has little to do with my question.
Any info will be appreciated.
Steve Gray
If by the Borwein expansions you mean the formula for computing hex digits without knowing the preceding ones, yes, I know about that. I also know about the contined fractions having simple patterns. Assuming that the "value" of the irrational is given only by the decimal expansion, is it possible to "know" the exact value by knowing the pattern? This is part of a theological issue which actually means very little to most people. (And it may be meaningless.). (I'm not a believer.)
On Tue, Dec 1, 2009 at 8:59 PM, <rcs@xmission.com> wrote:
Date: Tue, 01 Dec 2009 20:49:57 -0800 Subject: Re: [math-fun] Solving polynomial equations with roots, etc. From: "Stephen B. Gray" <stevebg@roadrunner.com>
Henry Baker wrote:
Isn't this what the Borwein expansions are all about?
At 06:07 PM 12/1/2009, you wrote:
from SBG ...
Date: Mon, 30 Nov 2009 19:44:17 -0800 Subject: Re: [math-fun] Solving polynomial equations with roots, etc. From: "Stephen B. Gray" <stevebg@roadrunner.com>
On a different subject, for anyone to comment on.
Everyone knows that sqrt(2), pi, and all irrational numbers have no decimal strings that repeat an infinite number of times. That's not necessarily the same as the decimal expansion being totally without any pattern. That is, is there any way to predict the next digits of say sqrt(2) or pi without doing one of the usual computations? In other words, is it possible to "know" the entire decimal expansion of any "ordinary" irrational? Is anything known about patterns in "regular" irrational expansions?
I'm excluding numbers invented for the sole purpose of being irrational or transcendental and with an obvious pattern like .101001000100001.... or .123456789101112..... ), etc.) I know about the question of "normal" expansions but that has little to do with my question.
Any info will be appreciated.
Steve Gray
If by the Borwein expansions you mean the formula for computing hex digits without knowing the preceding ones, yes, I know about that. I also know about the contined fractions having simple patterns. Assuming that the "value" of the irrational is given only by the decimal expansion, is it possible to "know" the exact value by knowing the pattern? This is part of a theological issue which actually means very little to most people. (And it may be meaningless.). (I'm not a believer.)
What do you mean by "pattern"? Do you mean an algorithm for producing each digit given only its index? If so, then any computable number works, whereas uncomputable numbers (e.g. the complexity of the rational approximations is unbounded) don't. -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
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