Exactly! I hate the Taylor Series explanation of Euler's formula. "Oh, just take the most analytic definitions of e^x, sin(x), cos(x), mix-n-match, and it works!" All symbols, no intuition.
If we see each concept individually (continuous growth + rotation) we can deduce that we get something like "continuous rotation" or a circular orbit.
And if we use a complex number (a + bi, not purely imaginary i) we get a spiral pattern. Euler's Formula becomes "obvious" dare I say ("obvious after the greatest mathematician figured it out for us".)
If we see each concept individually (continuous growth + rotation) we can deduce that we get something like "continuous rotation" or a circular orbit.
And if we use a complex number (a + bi, not purely imaginary i) we get a spiral pattern. Euler's Formula becomes "obvious" dare I say ("obvious after the greatest mathematician figured it out for us".)