Imaginary numbers, why are they so useful

Okay, so maybe imaginary and complex numbers make sense to introduce and lead to a reasonable theory, but how could they possibly be useful? After all, in the real world we have real numbers of things (e.g. 3 frogs) and real amounts (16.2 dollars), not imaginary numbers of things. As it turns out, complex numbers are fantastically, staggeringly useful. This is especially true once we allow ourselves to start plugging complex numbers into functions (like , , , etc.) , and see what output they produce. For example, imaginary numbers give us a useful and surprising link between the exponential function and the sine and cosine functions, in Euler’s beautiful formula: