Does the real part contain all the physical information?

As physicists, when working with complex numbers, we are told that the meaningful physical quantity is just the real part of the complex number. This detailed and pedagogical paper demonstrates that one must be very careful in applying adage. For example, both the real and imaginary parts of the complex polarisation charge on plasmonic nanostructures are relevant, and taking only the real part results in a misleading interpretation of physical phenomena. Polarisation charge formed on nanostructure surfaces upon optical excitation provides a useful tool to understand the underlying physics of plasmonic systems. Plasmonic simulations in the frequency domain typically calculate the polarisation charge as a complex quantity. In this paper, we provide a pedagogical treatment of the complex nature of the polarisation charge and its relevance in plasmonics, and discuss how naively extracting the real part of the complex quantities to obtain physical information can lead to pitfalls. We analyse the charge distributions on various plasmonic systems and explain how to understand and visualise them clearly using techniques such as phase-correction and polarisation ellipse representation, to extract the underlying physical information