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.

  • 1
    $\begingroup$ Voltage is not really complex. Complex voltage is just a mathematical convenience for keeping track of relative phase. $\endgroup$ – G. Smith Sep 22 at 21:24

(2) is correct.

(1) is incorrect. It makes it look like you're adding two things, one of which has units of potential and the other being unitless. You can't add things that have different units.

I've never seen (3) or (4), so I think they're probably nonstandard.

  • 3
    $\begingroup$ What do you mean: "...won't work in three dimensions"? There aren't three dimensional complex numbers. $\endgroup$ – jacob1729 Sep 22 at 22:12
  • $\begingroup$ @jacob1729: Thanks, total brain fade there :-) $\endgroup$ – Ben Crowell Sep 23 at 2:35
  • $\begingroup$ + a similar notation could still work for quaternions $\endgroup$ – Akerai Sep 23 at 6:58
  • $\begingroup$ @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 $\endgroup$ – Agnius Vasiliauskas Sep 23 at 8:43
  • $\begingroup$ I've never seen (3) or (4), so i think they're probably standard in fields I'm not familiar with. $\endgroup$ – Emilio Pisanty Sep 23 at 9:08

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