Interestingly, the denominators you get for the golden ratio are the same as the denominators for the convergents to sqrt(5) = 2 phi - 1. That is, the same sequence of denominators arises from the best rational approximations of 2 phi - 1 and the worst of phi. Actually, I don't know how interesting that is. What are the denominators for the worst rational approximations of 2 phi - 1, and do they arise as the best of anything else? On Fri, Jan 27, 2017 at 7:11 PM, Mike Speciner <ms@alum.mit.edu> wrote:
Yes, approximating pi with cf terms 3,7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2, I get 1 2 3 4 11 18 25 32 39 46 53 166 279 392 505 618 731 844 957 ...
On 27-Jan-17 18:13, James Propp wrote:
Interestingly, it seems that nobody's studied the WORST rational approximations to pi!
I'd better explain what I mean by this before anyone shouts "Hey, a *million* is a really bad rational approximation to pi!"
Let nint(x) denote the nearest integer to x. For each denominator q, look at the discrepancy |q Pi - p| where p = nint(Pi q); call this disc(q). The best rational approximations to pi are those for which disc(q) is smaller than disc(1),...,disc(q-1), so it's natural to define the worst rational approximations to pi as those for which disc(q) is greater than disc(1),...,disc(q-1).
If I haven't goofed, the denominators of the worst approximations to pi are *1,2,3,4,11,18,25,32,...* (a sequence that isn't in OEIS).
Before I compute more terms, can anyone corroborate my numbers? It seems improbable to me that nobody's thought to look at this before.
(If we play this same game with the golden ratio, we get *1,4,17,72,305,1292,...* which is in the OEIS (A001076 <http://oeis.org/A001076>).)
Jim _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun