# Notation of physical dimension of complex values

I can't find a good answer on the proper way to write the physical units for complex numbers.

\begin{align} z &= 707 \text{ mV} + 0.707\mathrm{i} \tag{1} \\ z &= (707 + 707\mathrm{i}) \text{ mV} \tag{2} \\ z &= 1 \text{ V} \, \angle 45^\circ \tag{3} \\ z &= (1 \, \angle 45^\circ ) \text{ V} \tag{4} \end{align}

I've seen all of these in the wild, and they all seem acceptable to me except for (4). What is the most correct way in cartesian and polar?

I'm ignoring exponential because there is only one way to write that.

• Voltage is not really complex. Complex voltage is just a mathematical convenience for keeping track of relative phase. – G. Smith Sep 22 at 21:24

• @jacob1729 Why you are so sure ? $$(a+ik) + (b+ik) = a + b + i 2k$$, so the result of addition of two complex numbers with same imaginary part can be interpreted as kind of complex number with two real parts, so 3D complex vector – Agnius Vasiliauskas Sep 23 at 8:43