>>11114471Saying that complex numbers were designed to multiply as rotations and that Euler's formula is trivial is an anachronism that explains nothing of what OP is asking.
>>11114061Your intuition of the complex numbers is right and it's how complex numbers should be taught and understood.
Imaginary numbers were first used to solve cubic equations by "abusing" taking roots of negative numbers, as if they were "real numbers." Some people at the time taught this was some bullshit hack, so they called them imaginary numbers.
The connection to the plane, trigonometry and rotations came almost 200 years after this, with Euler's formula as the crystallization of the efforts of that time.
The beauty of it is the realization that a hack used to solve equations was actually deeply connected to numbers in the plane, rotations and trigonometric functions.
Euler's formula, like all great ideas, took a life of its own, becoming the de facto way to think about all of this, and the fact that you are now here asking how such a trivial fact is considered beautiful is proof that it worked well.
