I always think it's better to simply discuss 2D vectors with the add and multiply rule (adding angles) first. Then we can clearly see why the horizonal axis is equivalent to the real number line. Then we define "i" to be the 90 degree vector. Starting with i = sqrt(-1) is backwards, from the point of view of understanding what is going on!!! For example at 5:30 you say "it is useful to think of i as a 90 degree rotation..." but it would be better to talk about rotations first, then define i as the 90 rotation. Then the whole thing is clear to everybody.
He explains that at 11:57 The location is the input, the color is the output, e.g. red represents 0 degrees output. In the case of iz, the red will be at the 270/-90 degrees input because those values multiplied by i will be rotated 90 degrees counterclockwise to 0 degrees. You probably already know it, just leaving this comment for others because I was confused too.
Very nice visualisations! It would be interesting if when you model the 3D geometry of the function that there was a projection onto the plane of some sort for reference. For example in Section "3D phase portraits" if you used a collimated light source from above the object, the shadow would be an accurate projection. Or alternatively map the image of the phase portrait onto the plane.
Yes, you're right, and it is the convention in modern text books. It has some advantages and disadvantages. My statement is meant to emphasise that i is not equal to the negative square root, not that it is the only definition. I deliberately went non-conventional here, to avoid getting bogged down in how roots work. The i^2=-1 definition can be counter-intuitive for students until the complex square root operation is correctly defined.
[x, -y]^n= (x+i*y)^n [y, x] Now go crazy. Don't limit yourself to two dimensions with that function. sqrt(y^2+z^2). Just think about that... A 2D-rotation matrix in R^n space. Easier to learn than quaternions and you are not limited to R^4.
I'm pretty new to making educational videos, so feedback is welcome.
keep going? :D
You will be the next 3Blue1Brown
So good! The 3d graphics are amazing
Keep it up! These videos are incredible, the potential reach of these videos is immense
jolly good show..
Very nice 3D graphic!
Wow...this is great!!
Great video, thank you. I am looking forward for complex integration.
Out for real this time! :D
Great stuff as always!
I love your videos so much. Please keep making them. I learn so much! You are such a great teacher.
This is genius! I've been struggling to make a mandelbrot set by hand, but this video surely helps alot!
I want to do that too!
This series is awesome!
Nice one 👍
I always think it's better to simply discuss 2D vectors with the add and multiply rule (adding angles) first. Then we can clearly see why the horizonal axis is equivalent to the real number line. Then we define "i" to be the 90 degree vector. Starting with i = sqrt(-1) is backwards, from the point of view of understanding what is going on!!! For example at 5:30 you say "it is useful to think of i as a 90 degree rotation..." but it would be better to talk about rotations first, then define i as the 90 rotation. Then the whole thing is clear to everybody.
That 3D graphs are *tasty*
13:42 wow...it represnts soo many graphs of cos(z) when seen with different angles that's really interesting to know
Great^3 +i1!
Loving the visuals! Wouldn't the rotation @13:32 be in the opposite direction? Just a sanity check since I'm still new to mappings
He explains that at 11:57 The location is the input, the color is the output, e.g. red represents 0 degrees output. In the case of iz, the red will be at the 270/-90 degrees input because those values multiplied by i will be rotated 90 degrees counterclockwise to 0 degrees. You probably already know it, just leaving this comment for others because I was confused too.
Very nice visualisations! It would be interesting if when you model the 3D geometry of the function that there was a projection onto the plane of some sort for reference. For example in Section "3D phase portraits" if you used a collimated light source from above the object, the shadow would be an accurate projection. Or alternatively map the image of the phase portrait onto the plane.
2:17 Im not so sure i "is always defined as the positive square root of -1". Actually, i is defined better by i^2 = -1
Yes, you're right, and it is the convention in modern text books. It has some advantages and disadvantages. My statement is meant to emphasise that i is not equal to the negative square root, not that it is the only definition. I deliberately went non-conventional here, to avoid getting bogged down in how roots work. The i^2=-1 definition can be counter-intuitive for students until the complex square root operation is correctly defined.
Thank you so much.
can I get a 3d object file for the 13:50 cosz figure so that I can resin 3d print it?
If you took a regular equation like x = vt and replaced t with s*exp( iu ) what would that mean?
finally!
[x, -y]^n= (x+i*y)^n
[y, x]
Now go crazy. Don't limit yourself to two dimensions with that function. sqrt(y^2+z^2). Just think about that... A 2D-rotation matrix in R^n space. Easier to learn than quaternions and you are not limited to R^4.
didn't realize there were complex numbers down under 🤔
Pleas , : upload rest videos
❤
The Imaginary Part is 3. Not 3i
when eugene khutoryansky got that rtx
reupload?
Yes, I made a big typo in the first, labelled a graphic completely wrong.
🤓.. yeah, you lost me at...... the variable named... i 😃