# Complex physical quantities

I have a question regarding complex physical quantities. Why do we consider only the real part of a complex physical quantity? Why not the modulus? Since, for $$z=a+bi$$, we have $$|z| = \sqrt{a^2+b^2}$$, and so the imaginary part contributes to the modulus, it is not clear to me when to use the modulus or the real part.

For example, the electric (or magnetic) field. And/or the Poyting vector. Below I take texts from Jackson, Classical Electrodynamics:

"Because the diffusion equation is second order in the spatial derivatives and first order in the time, it is convenient to use complex notation, with the understanding that the physical fields are found by taking the real parts of the solutions." (Jackson, 3rd. Edition, page 220)

"Then (6.131) can be written as $$\frac{1}{2} \int_V \mathbf{J^*\!\cdot E}\,d^3x+2i\omega \int_V (w_e-w_m)\, d^3x +\oint_S \mathbf{S\cdot n}\,da \tag{6.134}$$

... It is a complex equation whose real part gives the conservation of energy for the time-averaged quantities and whose imaginary part relates to the reactive or stored energy and its alternating flow." (Jackson, 3rd. Edition, page 265)

"...With the convention that the physical electric and magnetic fields are obtained by taking the real parts of complex quantities, we write the plane wave fields as $$\mathbf{E}(x,t)=\mathcal{E} e^{ik\mathbf{n\cdot x}-i\omega t} \ \ \ \ (7.8) \\ \mathbf{B}(x,t)=\mathcal{B} e^{ik\mathbf{n\cdot x}-i\omega t}$$" (Jackson, 3rd. Edition, page 296)

• In some applications like electronics we use the complex number to indicate the voltage and the phase, so it's really just a way of encoding two quantities into one complex number. Whenever you see the real part of a complex number being used in physics this is probably what is going on. In other applications like quantum mechanics we really are dealing with variables that are complex. To take this any further you will have to tell us in what areas of physics you are using the complex numbers. Jun 17, 2021 at 15:02
• In physics is common to use complex number in their "entirety", for example in quantum mechanics. Your question needs a context to be answerable. Jun 17, 2021 at 15:03
• Jun 17, 2021 at 18:37
• We generally consider the things we're interested in, and what we're interested in varies with time and circumstances. This applies even outside of physics. Jun 17, 2021 at 18:41
• @WillO: I am not trying to solve a specific problem or exam: I just want to discern and learn where/when to use real part or modulus. I gave the example of Jackson's text because It is where I found it more explicit. But when I was studying the propagation of light in a medium with complex permitivitty, it also came to my mind, mainly when I was writing the fortran program. Jun 17, 2021 at 19:02