TITLE: How can a complex exponential represent a real world quantity?
QUESTION [9 upvotes]: Equations containing complex exponentials are mysterious. The complex exponential merely embodies a complex number but in a more compact form where doing maths is easier. Right? If this complex exponential represents a sinusoid then why can't we just write it as a $A\sin(\omega t+\varphi)$ rather that a weird exponential with $j$ in the power?
OK, lets take example of Fourier analysis and other fields of electronics where this is used. I am confused what complex exponentials actually mean and why they show up in mathematical equations in engineering.
Somebody may answer and say that Oh, the complex exponential can be visualized as a helix. Well, sure, but what does it mean when it is used in an equation containing physical quantities as it is used in engineering for example?

REPLY [2 votes]: Adding to Michael Hardy's answer, multiplying by $i$ only means you're trying to rotate something by $90^{\circ}$. It doesn't mean that the quantity doesn't exist, it just means you're achieving a rotational motion mathematically. A really simple (and visual) explanation can be found here.
Now substituting values for $\theta$ in $e^{i\theta}=\cos{\theta}+i\sin{\theta}$ results in a unit circle plotted by points that are rotated in the counter-clockwise direction! $e^{-i\theta}$ just means you're achieving clockwise rotation.