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Theoretical point of you. Please try to take a square root of negative frequencies and it will take you to strange place. Some 300 years ago it was necessary to introduce a variable called $j$ Nevertheless, Laplace transform, transforms time-domain signal to $s$-domain where $$s=\sigma+j\omega$$ where as Fourier transform, to frequency domain $j\omega$


One answer is very simple: Information. An AC signal simply can't be quantified with a single number. Sum two 100hz 1V signals and you might get anything between 0 and 2 depending on phase. Complex numbers solve this problem by 'carrying around' two pieces of information all the time. Poles and zeros are similar - their frequency doesn't tell you ...

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