It might have been a reason, mathematicians wanting everything well defined etc. But here's a better way to think about it: On real number line, addition defines shifts and multiplication defines scaling. If you are in two dimensions, what is the equivalent? We define a 2 dimensional number such that multiplication defines scaling + rotation. The complex in complex number should not be read as complicated but like duplex or two things intertwined together.
The next question is why bother? What's the point? Turns out that important real life signals, like AC voltage and current, are sinusoidal. And real life electrical machines shift the phase of these signals. By using complex numbers to represent these signals, you can continue to use simple maths of DC circuits to analyze AC circuits. So you'd can still use V = IR, but R of a AC machine like motor will be impedance (generally called Z), represented by a complex number.
I found first few pages of MD Alder's complex analysis for Engineers indispensable in demystifying this complex stuff. Here's a quote from first paragraph "If Complex Numbers had been invented thirty years ago instead of over three hundred, they wouldn't have been called `Complex Numbers' at all. They'd have been called `Planar Numbers', or `Two-dimensional Numbers' or something similar, and there would have been none of this nonsense about `imaginary' numbers"
The next question is why bother? What's the point? Turns out that important real life signals, like AC voltage and current, are sinusoidal. And real life electrical machines shift the phase of these signals. By using complex numbers to represent these signals, you can continue to use simple maths of DC circuits to analyze AC circuits. So you'd can still use V = IR, but R of a AC machine like motor will be impedance (generally called Z), represented by a complex number.
I found first few pages of MD Alder's complex analysis for Engineers indispensable in demystifying this complex stuff. Here's a quote from first paragraph "If Complex Numbers had been invented thirty years ago instead of over three hundred, they wouldn't have been called `Complex Numbers' at all. They'd have been called `Planar Numbers', or `Two-dimensional Numbers' or something similar, and there would have been none of this nonsense about `imaginary' numbers"