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So reading chapter 9 in James W. Nilsson, Susan Riedel-Electric Circuits (10th Edition) book explains perfectly how can complex numbers be used to add and subtract sinusoidal waves as in the below figures:

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due to the property that adding the real parts of any two complex numbers is equivalent to adding the complex numbers as a whole and getting the real part of the result which is very intuitive and makes sense, however this is not true for multiplication or division(multiplying the real parts of two complex numbers is not equivalent to multiplying the complex numbers as a whole and getting the real part of the result), which makes me wonder how can Ohm's law for AC Circuits (V = I*Z) be possible using complex numbers to get the voltage by multiplying together the impedance and current where both are complex numbers.

user3407319
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    The reason this work is how phasors are defined to be self-consistent when treated as complex numbers -- are you familiar with that concept? Can you edit to better explain which logical steps you are having trouble understanding? – nanofarad Jan 10 '21 at 05:46
  • @nanofarad I mean phasors were made to add and subtract sinusoidal waves not to multiply them, right? So if we have two voltage equations or two current equations we can add them for example in KVL loop. But we can’t multiply phasors together, so how can we multiply impedance and current(both phasors) to get voltage? How is this equation valid using phasors? – user3407319 Jan 10 '21 at 09:21
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    $$(a+jb)(c+jd) = ac-bd + j(ad+bc)$$ Just do the math. – Andy aka Jan 10 '21 at 10:31
  • Just use the whole complex numbers in all the calculations. We only mess around taking the real part when we want to decompose the answer into the resistive part of the impedance, or the in phase component of a current. – Neil_UK Jan 11 '21 at 05:54

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