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Does complex numbers have physical dimensions? Is it sensible to talk about the dimensional analysis of $Z$ where $Z$ is the impedance of a mechanical oscillatory system? Or is it the $|Z|$ which has dimensions. [ $|Z|$ = modulus of $Z$ ]

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Some quantities are complexified for mathematical convenience and only the real part retains a physical meaning. When you have a general phasor, like an oscillating potential or current, you can think that the amplitude is rotating on the complex plane, so that both the real and imaginary part have the same physical dimension, and the actual phenomenon is just the projection to the real axis.

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  • $\begingroup$ but does that mean that the complex number has a dimension in itself? $\endgroup$ – Apiastos Feb 18 '15 at 3:48
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    $\begingroup$ What do you mean by that exactly. You can assign physical dimensions to complex numbers like you do with reals, So you can think that phasors are like rotating their real part into an invisible imaginary axis, which is a real multiple of I and as such has the same dimension as the real part. $\endgroup$ – Phoenix87 Feb 18 '15 at 8:52
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In oscillators, we have a fourier transform of the green's function that have complex poles $\omega_0 + i\Gamma$ . The real part of this pole is a frequency $\omega$ (have units too) and the imaginary part is the inverse of the mean life $\tau$ of the oscillation $\omega_0$. This pole is a complex numeber that has physical dimensions. (more here)

Complex numbers is completely natural. Sometimes we are interested in the length of this numbers, and sometimes in the projection of this numbers in some axis.

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Strictly speaking, complex numbers do not have dimensions, because pure real numbers don't either. E.g. neither $2.45$ nor $2+3i$ represents a concrete physical quantity, so they are both dimensionless.

But complex-valued physical quantities can certainly have dimensions. For example, electrical impedance is complex-valued and has the dimensions of resistance (Ohms in SI units). That's because a complex-valued physical quantity technically equals a dimensionless complex number times a dimensionful base unit (like "1 Ohm").

The real and imaginary part of any complex-valued physical quantity must have the same dimensions, or it wouldn't make any sense to add them together (since the "conversion factor" $i$ is dimensionless). Another way to see this is that the norm $|Z| := \sqrt{(\text{Re }Z)^2 + (\text{Im }Z)^2}$ would not be defined if $\text{Re }Z$ and $\text{Im }Z$ had different dimensions.

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