I can manipulate complex numbers and functions. (chapter 13)
Big hint:
$$Ae^{i\theta}=Ae^{i(\theta + 2n\pi)}$$
editRead chapter 13 sections 1 and 2
editblog post of mine about complex numbers in quantum mechanics
editgreat quote about negative integers
editFirst we talked about how humanity has invented numbers. The two common guesses at the order were:
and
with unanimous consensus on the first and last ones. I asked if they could imagine pi rocks or 3+7i rocks, noting that we don't really use complex numbers for counting.
During the quiz (curve fitting again) I jotted down the things I thought we should hit today:
I asked them to write the two that they were least confident about. Euler got a lot of votes until I reminded/told them that \(e^{i\theta}=\cos\theta+i\sin\theta\). Mostly it was about roots and rotations.
I asked them to draw a random z on an Argand diagram and then to draw where i times z would be. We then talked about why multiplying by i is the same as a positive 90 degree rotation. I asked what multiplying by a random z would do and we talked about rotation, stretch, and translation (noting that you don't really need the translation part).
We were running out of time so we quickly hit powers (\(z^n=r^n(\cos n\theta+\sin n\theta)\)) and roots. They asked for a resource on roots.