@study @complex-numbers @py7002
31/01/23 14:00:33
- Try and figure out why we need complex numbers for describing AC signals.
- From a video I watched this morning. Complex numbers evolve as the set of numbers based on trying to determine the roots of 1.
- Because they don’t have a relation to the sets of numbers we already know, like rationals, irrational, integers etc. they are considered their own class, imaginary.
- Apparently Descartes called them imaginary in a derogatory way. That they shouldn’t really be taken seriously.
3Blue1Brown Video
- Create an imaginary axis, so that now, we have vectors, with a real part and an imaginary part.
- I think he’s kind of building up the operations with complex numbers, this shows that they may be useful (as opposed to offering them any grand intuitive meaning).
- i times z (an imaginary number) rotates it in the plane.Because of i’s defining feature
- Computational mechanism for rotations.
- 3 facts about multiplication in the i plane.
- z*i is rotation by 90 degrees.
- z*1 is z.
- z*(c+d+i) = cz + d(zi) distributive property.
- Multiplication works by morphing the plane or grid. So if we have a set of points on the complex plane. Multiplication is rotation.
- Unit circle in the imaginary plane.
- j is used instead of i in electrical engineering.
- cos + i sin something is sometimes written as cis of the angle.
- When multiplying complex trig functions, you’re rotating by the addition of the angles.
Got to midway through the next lecture.