A multi-element directional coupler is considered as a cascaded set of 2-element directional couplers, each of which, at the design frequency, is perfect in match and directivity. A characteristic impedance matrix and a voltage transfer matrix may be defined for each of the basic 2-element couplers, and these are analogous to the characteristic impedance and the propagation coefficient in iterative 4-terminal network theory.The voltage transfer matrix is shown to be a linear function of the cosine and sine of a certain angle which is related in a simple way to the coupling factor of the elemental coupler. This function has algebraic properties analogous to those of complex numbers. In particular, the use of de Moivre's theorem is shown greatly to simplify the design of multi-element couplers. An example is given of the design of a 5-slot 3 dB coupler (a binomial slot hybrid).