Maths/Stats challenge - 1 of 2 - polls

[Joe Horn]

(05-24-2022 07:31 AM)EdS2 Wrote:  I think my case, where there are three or more poll responses, is a bit more involved than the case of just two responses. In effect, we have multiple constraints all of which need to be satisfied.

Aha! I understand now.

For what it's worth, George Chrystal's "Algebra" (first published in 1889, available through Amazon from Dover Press, AMS Chelsea, and some other publishers) seems to cover this problem. Chapter X is called "Continued Fractions", with sub-section 10.15 called "Approximation by convergents". In Chapter XI ("Approximation of Irrationals by Rationals"), subsection 11.12 ("Simultaneous Approximation") begins:

Chrystal Wrote:So far we have been concerned with approximations to a single irrational number. ... [We now look at] the simultaneous approximation of k numbers (X1, X2 ... Xk) by fractions P1/Q, P2/Q, ... Pk/Q, with the same denominator Q (but not necessarily irreducible).

This is followed by Chrystal's typical dense mixture of English and math. I'll try to make sense out of it. Although this section is specifically about approximating irrational numbers, that's not essential to his process which uses continued fractions. If you have trouble finding this book, I can scan my copy (which is itself a printed scan) and post it here for you. In any case, it's nice to know that this problem was covered by an algebra "textbook" in 1889.