This is how rotational matrix and complex numbers are related then. The rotational matrix can be represented as cos(θ)+isin(θ) in complex number notation, and complex numbers as [(a,-b),(b,a)] in matrix form (if you compare C to R², or well, you take (1,i) as basis in R²).
I'm enjoying your videos very much. You have a good manner of presentation, somewhat similar to Khan Academy, but I find KA a little too 'dumbed-down' and thus 'slow' to watch. (Don't get me wrong, I love Khan Academy!) Yours are a bit 'quicker', and not over-simplified/repetitive, but at the same time nicely intuitive and easy to understand. Anyway, I've only seen about 3 or 4 so far, but just wanted to let you know your vids are much appreciated! :-) PS: what software are you using to a) do the diagrams, and b) do the screen recording? Is it available for Windows? Do you use a drawing tablet/stylus, or mouse, or what? Thx! Cheers!
+Rob Harwood I use a Wacom Intuos pad for the pad and stylus (using the mouse would be embarrassingly sloppy), SmoothDraw 4 (free) for actually drawing, and Open Broadcast Software (free) for screen recording, and VideoPad (also free) editor for editing. It's all available for Windows and the actual recording of the video is easy, it's just the planning and research needed for the video that takes effort.
I saw this video and thought that it was going to be about the geometric interpretation of complex numbers as being exponents to another complex number. Still, this was an interesting insight. I actually used the idea of interpreting complex numbers as vectors when writing a calculator on desmos that would take an exponential number and raise to the power of another complex number. I've been studying the properties of this function (c = z^n) to see how the complex plane transforms based on the two inputs z and n. If you want you can check it out: www.desmos.com/calculator/ilgc3bewf2 (for x>0) www.desmos.com/calculator/sdgado2ier (for x
+BlueHawkPictures Yeah, it's interesting how going back to seemingly simple questions like how we should think of complex numbers can be profitable. You have the polar representation using exponentials, you have the vector interpretation, and you have a "scalar plus bivector" interpretation. That last one is a topic in the first geometric algebra video I made.
Being a simpleton, what advantage does this system have over x=cos theta, y=sin theta ? just don't get it. Any response in the way of clarification would be greatly appreciated, and thanks.
Usually, we have the coordinates (x, y) of a point before the rotation and we want to find the coordinates after the rotation. If you have a coordinate (0,y) of the point on the y axis, after the rotation it is not (0, cos theta). The x coordinate of the rotated point has become non-zero with respect to the original x axis. As shown in the video it has become -sin theta. Similarly, for the point (x,0) after the rotation, we obtain two non-zero values. The vectors (x,0) and (0,y) initially defined a certain point (x,y) on the plane being the projections of this point on the axes. After the rotation, we obtain two values per projection, so four values, which define the rotated point in the initial coordinate system.
I still would like to see how a complex exponential definition coincides with a real exponential definition, unless the latter definition was only good for the reals. Ex. 2^2.5 means that 2 is multiplied by itself 2.5 times. On the other hand, to think of e^i as e multiplied by itself i times is meaningless. Therefore, a need arises to redefine the exponentials in some different way--a more abstract and general definition.
@@Math_oma is a little bit convenient that the rotation used is a known identity isn't it? e^(pi/2*i)= i, but what would you do with an example that gives you e^(0.5073672*i)?
Thank you so much for the very detailed explanation. I was searching for such clarity.
Thankyou so much! Im in high school and preparing for the IMO. This helps so much!!
Cool, calm and complex does it.Thanks!
Amazing video! Thank you!
This is how rotational matrix and complex numbers are related then. The rotational matrix can be represented as cos(θ)+isin(θ) in complex number notation, and complex numbers as [(a,-b),(b,a)] in matrix form (if you compare C to R², or well, you take (1,i) as basis in R²).
Excellent presentation of the topics. Many many thanks. DrRahul Rohtak India
Wonderfull explanation
Very helpfull! Thank you very much!
Very nice explanation.
I'm enjoying your videos very much. You have a good manner of presentation, somewhat similar to Khan Academy, but I find KA a little too 'dumbed-down' and thus 'slow' to watch. (Don't get me wrong, I love Khan Academy!) Yours are a bit 'quicker', and not over-simplified/repetitive, but at the same time nicely intuitive and easy to understand. Anyway, I've only seen about 3 or 4 so far, but just wanted to let you know your vids are much appreciated! :-) PS: what software are you using to a) do the diagrams, and b) do the screen recording? Is it available for Windows? Do you use a drawing tablet/stylus, or mouse, or what? Thx! Cheers!
+Rob Harwood
I use a Wacom Intuos pad for the pad and stylus (using the mouse would be embarrassingly sloppy), SmoothDraw 4 (free) for actually drawing, and Open Broadcast Software (free) for screen recording, and VideoPad (also free) editor for editing. It's all available for Windows and the actual recording of the video is easy, it's just the planning and research needed for the video that takes effort.
Rob Harwood If Wacom is too pricy for you, check out Huion tablet, which costs less.
I saw this video and thought that it was going to be about the geometric interpretation of complex numbers as being exponents to another complex number. Still, this was an interesting insight. I actually used the idea of interpreting complex numbers as vectors when writing a calculator on desmos that would take an exponential number and raise to the power of another complex number. I've been studying the properties of this function (c = z^n) to see how the complex plane transforms based on the two inputs z and n. If you want you can check it out:
www.desmos.com/calculator/ilgc3bewf2 (for x>0)
www.desmos.com/calculator/sdgado2ier (for x
+BlueHawkPictures
Yeah, it's interesting how going back to seemingly simple questions like how we should think of complex numbers can be profitable. You have the polar representation using exponentials, you have the vector interpretation, and you have a "scalar plus bivector" interpretation. That last one is a topic in the first geometric algebra video I made.
Being a simpleton, what advantage does this system have over x=cos theta, y=sin theta ? just don't get it. Any response in the way of clarification would be greatly appreciated, and thanks.
Usually, we have the coordinates (x, y) of a point before the rotation and we want to find the coordinates after the rotation. If you have a coordinate (0,y) of the point on the y axis, after the rotation it is not (0, cos theta). The x coordinate of the rotated point has become non-zero with respect to the original x axis. As shown in the video it has become -sin theta. Similarly, for the point (x,0) after the rotation, we obtain two non-zero values. The vectors (x,0) and (0,y) initially defined a certain point (x,y) on the plane being the projections of this point on the axes. After the rotation, we obtain two values per projection, so four values, which define the rotated point in the initial coordinate system.
thank you this is very helpful for me really....
I still would like to see how a complex exponential definition coincides with a real exponential definition, unless the latter definition was only good for the reals. Ex. 2^2.5 means that 2 is multiplied by itself 2.5 times. On the other hand, to think of e^i as e multiplied by itself i times is meaningless. Therefore, a need arises to redefine the exponentials in some different way--a more abstract and general definition.
perfectly derivation. deriving using the angle addition theorem is meh.
you just couldn't do a more elaborated example wouldn't ya?, Why not rotating by 32.4 degrees instead of 90?
You should be able to do that based on the knowledge in this video.
@@Math_oma is a little bit convenient that the rotation used is a known identity isn't it? e^(pi/2*i)= i, but what would you do with an example that gives you e^(0.5073672*i)?
I really like your videos but that kind of examples are simply not real and too convenient
@@juanmanuelpedrosa53 Euler's formula to convert it to cos(0.5083672)+sin(0.5-83672)i, then multiply.
bruh