In this note it is shown, by a counterexample, that the familiar theorem on the continued fraction expansions of equivalent numbers does not hold when these notions are extended to complex numbers. Two real numbers x, x' are called equivalent, x-x', if (I) x' = (ax + b)l(cx +d) forsomea, b, c, deZ, ad-bc =1. Consider also the continued fraction (CF) expansion of a real number (2) x = (ao, al, , an-1, xJ), xn,1 = an-1 + 1/xn, where xn is the nth complete quotient of x. It is a standard theorem in CF's that x-x' if and only if, in the CF expansions of x and x', there exist m, n such that am+k=an+k for all k>O-more briefly, xm=xn. Hurwitz, in a paper [2] on the CF (where the partial quotients an may be negative) proved that essentially the same result carries over. That is, x-x' if and only if there exist m, n such that xm= ?x , where these are complete quotients of the nearest integer CF's. Hurwitz also defined [1] a complex generalization of the nearest integer CF (it might be called the Gaussian CF) by which a complex number x is expanded in a simple CF as in (2) with partial quotients an in Z[i]. There is an analogous notion of equivalent numbers as in (1), where a, b, c, d eZ[i], ad-bc=+I, +i. Although this complex CF has many analogies to real CF's, the expected theorem on equivalent numbers fails, as shown by a COUNTEREXAMPLE. Let Q2 = j(i+ (43 +28i)1/2), A = (5-i+Q2)/(4-i), B=(3+2i+Q2)/4. Then A--B, in fact A=(2B-i)/(B-i). However the CF expansions of A and B, which are periodic, are distinct: A = (2 + i, 3i, -1 + 2i, -I + 2i,3, -2-i), B = (2 + i, -2 + i, -2 + i, I 2i, I 2i, I + 2i). Thus A m = + Bn, or even + iBn, never holds. Received by the editors May 8, 1973. A MS (MOS) subject classifications (1970). Primary IOF20; Secondary 12A05.