We show that the sequence ( |Im(z n )|) is never monotone when z is a non-real complex number, and ( |Re(z n )|) is never monotone, unless z=re i?% , where %=kรm for some odd m, and r is sufficiently large or sufficiently small. Similar results hold for the length of the projection of z n onto any line through the origin.
1999 Academic Press
The purpose of this note is to prove a simple, amusing and apparently new application of some familiar number theory. The most interesting special case is the following: if z is a non-real complex number, then the sequence (|Im(z n )| ) is not monotone; in fact, it increases and decreases infinitely often. This oscillation contrasts with the familiar fact that (|x n | ) is monotone for real x.
Recall that if l ; denotes the line in C through the origin with angle ; to the real axis, then the distance from a point w to l ; is equal to |Im(e &i?; w)|. This distance is |Im(w)| when ;=0 and |Re(w)| when ;= 1 2 .
Theorem. Let z=a+bi=re i?% be a non-real complex number (b{0), and let u(n)= |Im(e &i?; z n )| = |r n sin ?(n%&;)|. Then the sequence (u(n)) increases and decreases infinitely often (indeed, each event occurs with positive probability), unless the following conditions are met: %=kรm, gcd(k, m)=1, m; ร Z and r is sufficiently large or sufficiently small. In these cases (u(n)) is monotone increasing or monotone decreasing.
Proof. Note first that u(n)=0 if and only if n%&; # Z; under our hypotheses, % ร Z, so (n\1) %&; ร Z, hence u(n\1)>0.