In a good physical theory dimensionless quantities, such as the ratio m_p/m_e of the mass of the proton to the mass of the electron, do not depend on the system of units being used. This paper demonstrates that one widely used method for defining measures of entanglement violates this principle. Specifically, in this approach dimensionless ratios E(ρ)/E(σ) of entanglement measures may depend on what state is chosen as the basic unit of entanglement. This observation leads us to suggest three novel approaches to the quantification of entanglement. These approaches lead to unit-free definitions for the entanglement of formation and the distillable entanglement, and suggest natural measures of entanglement for multipartite systems. We also show that the behaviour of one of these novel measures, the entanglement of computation, is related to some open problems in computational complexity.