'Euclid tries to deal with noncommutativity' A real number can be approximated by a single rational number which is a ratio of integers. The algorithm to produce this rational number is the oldest non-trivial algorithm: Euclid's algorithm which is a greedy algorithm for removing the largest integer part; inverting the remainder, removing the largest integer part, etc. The efficiency of the algorithm is ever-so- slightly improved by using a 'rounding' division instead of the greediest division, but the greediest algorithm still produces the same rational approximation to the real number. So let's move to the complex numbers and their approximation by a ratio of Gaussian integers. Presumably the same arguments continue to work, but I've not seen any discussion of the exact number of steps, or whether the sequence of convergents is always the same. But now we want to consider the approximation of quaternions having real coefficients by an infinite product of 'integer' quaternions, where by 'product' we include both quaternion multiplication and quaternion division. Due to noncommutativity, we may have to extend our product on either the left OR the right. Note that the *product* itself is important; we can't simply produce 4 rational numbers which approximate the 4 real coefficients of the given real quaternion, and *add* them together. We want an actual sequence of integer quaternions representing a quaternion *product*. Here's a potential algorithm: 1. Start with the approximation a=1. 2. If we pick a quaternion c to extend our current approximation, we have *four* choices to evaluate: extending on the left -- e.g. a:=ca or a:=(1/c)a -- or extending on the right a:=ac or a:=a(1/c). We somehow choose the 'greediest' 'integer' quaternion c, and the greediest extension method of the four, which produces the largest reduction in the difference between the norm of the given real quaternion and the norm of the approximating product of integer quaternions. 3. Rinse and repeat. There is a small question about whether to use quaternions with actual integer coefficients or the Hurwitz quaternions with half-integer coefficients; presumably one or the other will work 'better' and produce more elegant results. ------ So now comes the question: how to make the 'right' choice at each step? Also, IS there a method of 'cheating', whereby we can build up our approximating quaternion by multiplying and dividing by rational integers and then putting the results together with a fixed number of full quaternion multiplications and divisions? I.e., given 4 rational numbers q0,q1,a2,q3 which approximate the coefficients of the real quaternion, is there a easy way to construct a quaternion *product* of 'integer' quaternions to produce the final value q0+q1*I+q2*J+q3*K ? --- Finally, assuming that one has such a quaternion product sequence, how does one show that it is the *shortest* such sequence? I.e., is there any way to do 'collapsing' moves?