[MS]: These are some notes on my playing around with [Fraction Math] and the reference to the nice notes on Continued Fractions at http://www.inwap.com/pdp10/hbaker/hakmem/cf.html [http://www.inwap.com/pdp10/hbaker/hakmem/cf.html]. I have started playing with these things and resisted (for now) looking for other sources. Some of the "errors" I found in the reference may be due to the informal presentation of results there, which might be inaccurate or misunderstood by me: ''caveat emptor''. '''DEFINITIONS:''' a rational approximation p/q to a real number x is "best" iff, for every integer r and s, (s <= q) ==> (|x - p/q| <= |x - r/s|) It is "best on its side" if ((s <= q) & (sgn(x-p/q) = sgn(x-r/s))) ==> (|x - p/q| <= |x - r/s|) i.e, if no other fraction on the sameside of x with a lesser denominator comes closer. '''NOTATION:''' let us identify a (positive) real number x with its regular continued fraction representation x = {x[0] x[1] x[3] x[4] ...} Define the truncation of x after (n+1) terms a(x,n) = {x[0] x[1] x[3] x[4] ... x[n]} and let b(x,n,i) = {x[0] x[1] x[3] x[4] ... x[n-1] i} be a(x,n) with the last element replaced by 0 < i <= x[[n]]. Note that a(x,n) = b(x,n,x[n]) '''NOTE''': [[quotient_rep]] in [Fraction Math] computes the highest-order truncation requiring no integers larger than ''maxint'' in the rational representation p/q. ---- Item 101A (3) clearly says (AFAIU, not all claims reproduced here): The claims are: A - a(x,n) is "best" B - b(x,n,i) is "best" if i>1 C - b(x,n,1) is never "best" (note that b(x,n,i) is not defined when x[n] = 1) Let me provide counterexamples to both B and C, thus showing that they are not true. The example provided in the text actually contains counterexamples to B. Let x = pi = {3 7 15 1 292 ...} . b(x,0,2) = 2/1 = {2} is not best (3/1 is better) . b(x,1,2) = 7/2 = {3 2} and b(x,1,3) = 10/3 = {3 3} are not best (3/1 is better) . b(x,2,2) = 47/15 = {3 7 2} is not best (22/7 is better) For another counterexample to B, easier to follow by hand, consider x = 0.51 = {0 1 1 24 2} The number b(x,3,4) = 5/9 = {0 1 1 4} = 0.555... is not best, as 1/2 = {0 1 1} = {0 2} is better. For a counterexample to C, consider x = 7/10 = {0 1 2 3}; now b(x,2,1) = 1/2 = {0 1 1} = {0 2} is best. ---- So, in light of these counterexamples, one proposition I can hope could be true is: D - The set of "best on its side" approximations to 0 355/113 Now, 3.1416305 = {3 7 16 2 ...},and ''quotient_rep'' produced 355/113 = {3 7 16} = a(3.1416305,2) But the fraction 377/120 = {3 7 16 1} = b(3.1416305,3,1) is closer - a new counterexample to C. Remark that the next truncation involves integers that are too large: a(3.1416305,3) = {3 7 16 2} = 732/233