Given any two fractions n1/d1 and n2/d2, the mediant is defined as the fraction / n1 n2 \ n1 + n2 M( -- , -- ) = ------- \ d1 d2 / d1 + d2 Graphically, we can regard each fraction n/d as a vector with the components (d,n), and then the mediant is simply the vector sum of two given fractions, as shown below The locus of fractions with a given numerical value q is obviously the straight line through the origin with slope q, so it's clear that the value of the mediant is strictly between the values of the two original fractions. Also, the mediant has the same value as the average of the two vectors, since the ratio of (n1+n2)/2 and (d1+d2)/2 has the same value, (n1+n2)/(d1+d2), as the mediant. It's also clear that for any two positive real numbers w1 and w2 the weighted mediant defined as / n1 n2 \ w1 n1 + w2 n2 M_{w1,w2}( -- , -- ) = ------------- \ d1 d2 / w1 d1 + w2 d2 has a value that is strictly between the values of the arguments. Incidentally, notice that it's important to distinguish between the concepts of "fraction" and "ratio". A fraction is an ordered pair of real numbers, whereas a ratio is a single real number. We can regard each ratio as an equivalence class of fractions. For example, the ratio 3 includes the fractions 3/1, 4.5/1.5, 12/4, and so on. However, the various members of an equivalence class yield differents results under the mediant operation, as shown by the fact that M(3/1,1/1) = 4/2 whereas M(6/2,1/1) = 7/3. We can obviously generalize the mediant concept to any number of fractions n1/d1, n2/d2,..., nk/dk, and define the mediant as / n1 n2 nk \ n1 + n2 + ... + nk M( -- , -- , ..., -- ) = ------------------ \ d1 d2 dk / d1 + d2 + ... + dk Again we see that the mediant represents the vector sum of the individual arguments, interpreted as vectors, and again the sum lies along a line through the origin that also passes through the geometrical midpoint (center of gravity) of the points (dj,nj), j=1,2,..,k. It follows that the value of the mediant is strictly between the extreme values of the arguments. (This inequality was first pointed out by Cauchy.) Furthermore, the same is true even if we apply arbitrary positive "weights" w1, w2,...wk to give the weighted mediant / n1 n2 nk \ w1 n1 + w2 n2 + ... + wk nk M_{wj}( -- , -- , ..., -- ) = --------------------------- \ d1 d2 dk / w1 d1 + w2 d2 + ... + wk dk Since the general mediant always yields a result whose value is strictly within the range of the arguments, we can use mediants to define a contractive mapping, and iterate this mapping in a way similar to the celebrated arithmetic-geometric mean, or James Gregory's geometric-harmonic mean. (See Iterated Means.) To give a simple illustration (suggested by D. G. Morin), suppose we wish to compute rational approximations to the square root of 2. If we begin with two numbers whose product is 2, and produce successive pairs of numbers that are progressively closer to each other AND whose product remains equal to 2, we will approach values equal to sqrt(2). To accomplish this, we can apply the mapping (f1, f2) -> (M, 2/M) where M = M(f1,f2) Thus, beginning with 1/1 and 2/1, we produce the sequence of pairs 1/1 2/1 3/2 4/3 7/5 10/7 17/12 24/17 41/29 58/41 etc. The same approach can be applied to roots of any order. For example, to compute rational approximations of the cube root of N, we can begin with the three fractions a c Nb --- --- ---- b a c whose product is N, and then map them to the triple of weighted mediants a+c+Nb Na+c+Nb Na+Nc+Nb -------- , --------- , ---------- b+a+c Nb+a+c Nb+Na+c whose product is also N, and which has the same form as the original triple, i.e., a' c' Nb' --- , --- , ---- b' a' c' where c' = c + Na + Nb a' = c + a + Nb b' = c + a + b In matrix notation this can be written as _ _ _ _ _ _ | 1 N N || c | | c' | | 1 1 N || a | = | a' | |_ 1 1 1 _||_ b _| |_ b' _| Letting L denote the coefficient matrix on the left, we see that L^n approaches a counter-symmetrical matrix whose components differ by a factor of N^(1/3) in both the vertical and horizontal directions. For example, with N=2 we have _ _ | 4159 5240 6602 | L^7 = | 3301 4159 5240 | |_ 2620 3301 4159 _| If we let s = 1/N and q = N^(0/3) + N^(1/3) + N^(2/3) then the nth power of (L/q) approaches _ _ | s^(3/3) s^(2/3) s^(1/3) | lim (L/q)^n = | s^(4/3) s^(3/3) s^(2/3) | n->oo |_ s^(5/3) s^(4/3) s^(3/3) _| The analagous results apply to higher order system, such as generating the 4th roots of N by computing the powers of _ _ | 1 N N N | | 1 1 N N | | 1 1 1 N | |_ 1 1 1 1 _|

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