  Some properties of the optimal representation of numbers are investigated.
This representation, which is to the base-e, is examined for coding of
integers. An approximate representation without fractions that we call WF is
introduced and compared with base-2 and base-3 representations, which are next
to base-e in efficiency. Since trees are analogous to number representation, we
explore the relevance of the statistical optimality of the base-e system for
the understanding of complex system behavior and of social networks. We show
that this provides a new theoretical explanation for the nature of the power
law exhibited by many open complex systems. In specific, we show that the power
law distribution most often proposed for such systems has a form that is
similar to that derived from the optimal base-e representation.
