3Blue1Brown on YouTube did a nice video explaining the equivalence of various notations for vectors. Math, Physics, and Engineering all prefer different notation for the same thing.

edit: Can't find a link to the video right now.

Look for 'Essense of Linear Algebra' (a series) on the 3Blue1Brown channel on YouTube.

As a fellow ME, I can tell you that the complex number is used to convey phase (angle) information. It's useful when there are two or more signals being looked at. The angle between the two signals (say voltage and current) is important when looking at power developed in the system (as well as other reasons).

If you were to look at more than one sine wave of the same frequency, it is important to also look at the angle between the two.

For example, take a three phase system, where each of three wires measures 120 Volts RMS to neutral/ground.
But the voltage measured between them is 208 Volts RMS.
The math only works out if you take into account the phase angles involved.

You don't have to use complex numbers. The basic way is to use differential equations. These are hell to solve, though. By transforming the equations into the complex symbolic space, they turn into normal algebraic equations that are much easier to work with.

Years back a friend of mine was in university to become a mechanical engineer; he introduced me to polar co-ordinates, which I found mud simple to use. I must readily admit however that I never tried to use these in situations where discrete "sine waves of the same frequency" had to be compared / worked with...

CR

"As iron sharpens iron, so one person sharpens another." [Proverbs 27:17, NIV]

The complex number is a very useful instrument for a.c. calculations. At first it has to be a single harmonics-the fundamental- for the system frequency. Introducing an imaginary number of square root of -1 .The result is not an actual vector but it is what is called a "phasor" represented in a trigonometric circle-Cartesian plane. Some of mathematical operators for vectorial theory could be applied some other does not.[For instance no vectorial multiplication or gradient, curl and divergence as in vectors].
Usually in a Cartesian plane the abscissa is what is called "real" like resistance and active power and on ordinate the reactance or reactive power. You may expand the area and use other way.
For instance if you have 3 balanced voltages[R,Y,B] you may take R as basis and consider VR=V+j0 then VY=V.cos(-120o)+jV.sin(-120o)=VY=-0.5.V-j0.866.V and VB=-0.5.V+j0.866.V.
By using complex number all the electrical calculations are simplified [ like load, short-circuit or voltage drop calculation].
They are more sophisticated complex number functions as Hankel function for instance, but they are not usually in the electrical engineering calculations.
A Microsoft Excel spreadsheet presents built-in complex functions as IMSUM,IMSUB,IMPRODUCT IMDIV and other in order to facilitate the complex number mathematics.

This is no different than complex analysis of power spectral densities of random vibrations in mechanical analysis. The Fourier Transform of random vibration results in complex magnitudes as befitting the phasing of the sinusoidal components within the vibration. Any reconstruction of time-domain vibration from PSDs require the complex values.

TTFN (ta ta for now)
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If you know about Euler's equation, then you know the answer. Google Charles Steinmetz for more. He developed the ac circuit theory we use today (for steady-state solutions) and got us out of the time domain.