Henry Baker wanted to approximate reals x by "pythagorean rationals" x =approx= a/b with squareroot(a^2+b^2)=integer. Obviously, yes, any such pythagorean (a,b) does have a continued fraction (namely just use the CF for a/b); and as Asimov remarked the set of rational points on the unit circle is dense, hence any x can be approximated arbitrarily closely by a pythagorean rational. The pythagorean rationals form a group in the sense that any two rational points on the unit circle can be multiplied (viewed as complex numbers). That is, if v+iw has v,w rational with v^2+w^1, then view that (v,w) as the rational (v*LCM(vdenominator,wdenominator)) / (w*LCM(vdenominator,wdenominator)). Euclid found all pythagorean triples (a,b,c) namely a = (m-n)*(m+n), b=2*m*n, c=m^2+n^2 which enables easily finding arbitrarily many arbitrarily close pythagorean rational approximations to x by this method: 1. use CFs to find large a,b so x =approx= a/b 2. solve for m,n satisfying above two simultaneous quadratic equations (as real numbers) 3. round off m,n to nearest integers 4. find resulting pythagorean a,b.