Whenever I think about complex numbers, I mostly think of them as an X/Y coordinate in 2d space, so I need to convert them to a vector (angle+magnitude), add the angle, and convert them back. I'm currently unable to grasp how the rotation equivalences map into the whole thing.
I have always been a fan of describing the complex numbers as 2d scaling and rotation matrices. If someone has enough familiarity with linear algebra, it is much more concrete and constructive. There is no 'we introduce this variable i with no motivation'.
You quickly get the understanding that there is a square root of the identity matrix. It then just takes a little work to show that the sum of two suchs matrices remains such a matrix.
It might make field extensions a bit more jarring for some students. But most students never need to get there.
I understand Gauss wanted to call the "imaginary" axis the "lateral" axis, which would have been a lot less confusing for math students. There is also the nice fact that understanding multiplication as rotation explains something else that students can find confusing: It makes sense that multiplying positive (real) numbers results in a positive real number, and it sort of makes sense that multiplying a positive real number times a negative real number gives a negative number, but much harder to understand why multiplying two negative numbers results in a positive number. But it's just a matter of 180° rotations.
If you want a more visual explanation of Euler’s formula, I wrote one a long while back:
https://docs.google.com/presentation/d/1oMNjkDp-LieSGnZEwNpceG8KTIvbnus9olu3KqnM5bg/
Whenever I think about complex numbers, I mostly think of them as an X/Y coordinate in 2d space, so I need to convert them to a vector (angle+magnitude), add the angle, and convert them back. I'm currently unable to grasp how the rotation equivalences map into the whole thing.
I have always been a fan of describing the complex numbers as 2d scaling and rotation matrices. If someone has enough familiarity with linear algebra, it is much more concrete and constructive. There is no 'we introduce this variable i with no motivation'.
You quickly get the understanding that there is a square root of the identity matrix. It then just takes a little work to show that the sum of two suchs matrices remains such a matrix.
It might make field extensions a bit more jarring for some students. But most students never need to get there.
I understand Gauss wanted to call the "imaginary" axis the "lateral" axis, which would have been a lot less confusing for math students. There is also the nice fact that understanding multiplication as rotation explains something else that students can find confusing: It makes sense that multiplying positive (real) numbers results in a positive real number, and it sort of makes sense that multiplying a positive real number times a negative real number gives a negative number, but much harder to understand why multiplying two negative numbers results in a positive number. But it's just a matter of 180° rotations.
This clarified the most fundamental question, that no one has ever answered to me till now. The shortest summary of this entire article!! though
Thank you, that’s gold.