I’m going to share these with as many people as I can. It is criminal with how high the quality of your explanations and animations are and how much work you clearly put in that you are so sorely under appreciated in terms of raw “channel numbers”.
A fantastic series to say the least. You manage to bring seemingly unrelated ideas together and crystallize them to a structure so clear it almost seems obvious in retrospect. Bravo, and thank you so much!
Fantastic explanation. Most of the mathematics teachers (not all) evasing such explanation by giving another formula without graphical representation. But you have done a great job.
I'm somehow stuck at the step where you've got rid of the 1/i expression in the parentheses. How does -(1/i)*sin(theta) turn into i*sin(theta)? How does the sign get flipped?
Good question, and sorry that I don't make this more explicit in the video. The trick is actually very simple: When you have a 1/i, you can just multiply the numerator and denominator both by i. The numerator is then i*i, which is just -1 by definition. So you get i/(-1), which is the same as -i. That's why the sign gets flipped.
Excellent video with really good visuals. Even if I knew the subject, it was really well explained. However, try to be a little more dynamic with your voice ! Take example on 3b1b or Mathologer, you can understand the subject they present even more thanks to their intonation. You don’t wanna sound like a boring lecture on an interesting topic in university. Also, it would be better to have a longer part on the roots of unity. After 19 minutes on the exponential, it’s weird to have only 1 minute on the roots of unity. We don’t really understand where it comes from and we don’t have time to understand the link with the video. Even with these little problems, I really like the channel and you all the videos so far. I cannot wait to see the videos on more complex subjects lol.
Then you won't learn by eigenchris, in his tensor calculus and relativity series, that's they're style, man. Leave them be. Also, that part about the roots, from what I understood will be delayed until the video(s) on lie groups and symmetries, be patient.
Grant Sanderson is indeed blessed with the perfect voice for educational videos. Unfortunately, I have to make do with what I have. I'm afraid you'll just have to get used to my monotonic droning ;-) The roots of unity will come back a few more times, e.g. in the video on subgroups. Coming soon.
Thanks for mentioning this. Needham shows a "right angle spiral" that converges onto the unit circle, but he proves this using formulas, not visually. Still, it's a nice visual representation of the theorem. I seem to remember that 3B1B also uses this spiral in one of his videos.
Very easy to understand and i feel enlightened! Have employed eulers formula for rotation for a long time but never understood it until now. Hope you can extend to 3d rotation and quarternions which can be viewed as a higher dimension complex number
A "straight angle" in English refers to 180 degrees or pi radians. But around 4:30 you seem to use "straight angle" to mean "right angle", which is 90 degrees or pi/2 radians. (As an aside "arithmetic" has stress on "met" when it's an adjective, but on "rith" when it's a noun (so it almost fits into the English pattern of initial-stress-derived nouns), so that 0:38 takes an extra second for a native speaker of English to parse.)
Thanks for the correction. English is not my native language, as you probably guessed. I sometimes translate too literally from Dutch. I'll have to be more careful with that.
@6:36 - I don't think Euler's formula is elegant. Using i^x is better. Where x is a right angle. Also sin() and cos() should respect right angle values. Then there is no need for a pi symbol in there, which has no meaning except when required for rotational distance calculations. I think equations would be cleaner with right angle powers. And remove the 2pi which has little value except for chaing the base symbol, which isn't even needed if cos() and sin() were understood to use right angle entry. Math is like traditionally dumb. They love pi because they do. They put it in where it's not needed and celebrate memorizing digits like it means something.
I personally would rather work in fractions of a full circle, so i = a^(1/4). Unfortunately... the number a that could satisfy that equation is... _1,_ which isn't the best choice for the base of an exponential meant to generate angles. I'll also mention that for the ℂomplex numbers' cousins (one of which was briefly mentioned in the previous video before being discarded to due the presence of zero-divisors), neither have any value of φ where their "imaginary" unit, which I'll call "j" for now, is e^jφ.
THANK YOU ! WOW ! You're a GREAT Teacher !!
cant wait for your next video
I’m going to share these with as many people as I can. It is criminal with how high the quality of your explanations and animations are and how much work you clearly put in that you are so sorely under appreciated in terms of raw “channel numbers”.
Thanks, I really appreciate your effort to share our videos. And your feedback is very kind and energizing.
The sequence of videos continues
to be inventive as well as informative.
I am happy to be following it.
A fantastic series to say the least. You manage to bring seemingly unrelated ideas together and crystallize them to a structure so clear it almost seems obvious in retrospect. Bravo, and thank you so much!
Thank you, I really appreciate the feedback.
Fantastic explanation. Most of the mathematics teachers (not all) evasing such explanation by giving another formula without graphical representation. But you have done a great job.
Thank you!
I have to say, the many rotations explanation is probably the best I've heard!
Thank you! Beautiful explanation! ❤
Good stuff right here, thank you for this project. Looking forward to more.
duidelijk uitgelegd, aangename stem. Goed tempo om het uit te leggen!
Blij dat je het goed vond!
Can't wait the video about Lie groups!)
Euler's formula also has an important role in geometric algebra. Eager for that part.
I'm somehow stuck at the step where you've got rid of the 1/i expression in the parentheses. How does -(1/i)*sin(theta) turn into i*sin(theta)? How does the sign get flipped?
Good question, and sorry that I don't make this more explicit in the video. The trick is actually very simple: When you have a 1/i, you can just multiply the numerator and denominator both by i. The numerator is then i*i, which is just -1 by definition.
So you get i/(-1), which is the same as -i. That's why the sign gets flipped.
Great video as always
Thanks! Glad you liked it.
Excellent video with really good visuals. Even if I knew the subject, it was really well explained. However, try to be a little more dynamic with your voice ! Take example on 3b1b or Mathologer, you can understand the subject they present even more thanks to their intonation. You don’t wanna sound like a boring lecture on an interesting topic in university. Also, it would be better to have a longer part on the roots of unity. After 19 minutes on the exponential, it’s weird to have only 1 minute on the roots of unity. We don’t really understand where it comes from and we don’t have time to understand the link with the video. Even with these little problems, I really like the channel and you all the videos so far. I cannot wait to see the videos on more complex subjects lol.
Then you won't learn by eigenchris, in his tensor calculus and relativity series, that's they're style, man. Leave them be. Also, that part about the roots, from what I understood will be delayed until the video(s) on lie groups and symmetries, be patient.
Grant Sanderson is indeed blessed with the perfect voice for educational videos. Unfortunately, I have to make do with what I have. I'm afraid you'll just have to get used to my monotonic droning ;-)
The roots of unity will come back a few more times, e.g. in the video on subgroups. Coming soon.
@@linuxp00 Eigenchris is one of my favorite channels and a big inspiration for All Angles.
Needham gives a partially geometric interpretation of the Taylor series argument in Visual Complex Analysis, p 12-14.
Thanks for mentioning this. Needham shows a "right angle spiral" that converges onto the unit circle, but he proves this using formulas, not visually. Still, it's a nice visual representation of the theorem. I seem to remember that 3B1B also uses this spiral in one of his videos.
Very easy to understand and i feel enlightened! Have employed eulers formula for rotation for a long time but never understood it until now. Hope you can extend to 3d rotation and quarternions which can be viewed as a higher dimension complex number
Very clearly explained.
Thank you for the positive feedback.
Great content here! Thanks for the intuition
A "straight angle" in English refers to 180 degrees or pi radians. But around 4:30 you seem to use "straight angle" to mean "right angle", which is 90 degrees or pi/2 radians.
(As an aside "arithmetic" has stress on "met" when it's an adjective, but on "rith" when it's a noun (so it almost fits into the English pattern of initial-stress-derived nouns), so that 0:38 takes an extra second for a native speaker of English to parse.)
Thanks for the correction. English is not my native language, as you probably guessed. I sometimes translate too literally from Dutch. I'll have to be more careful with that.
again, thank you for such a beautiful video❤
Thank you right back for your kind comments.
Suurepärane video! Tänud. An excellent video, many thanks!
thanks
👍👍👍
👏👏👏👍👍👍
what shall i study to deeply understand the sin and cos
Trigonometry. But it would also help to study the physics of waves.
@@AllAnglesMath thanks
This man said straight angle.
I have since been told it should be "right angle".
I don't get a joke
That's the joke, release a video on April, 1st, to make you think there is one
@6:36 - I don't think Euler's formula is elegant. Using i^x is better. Where x is a right angle. Also sin() and cos() should respect right angle values. Then there is no need for a pi symbol in there, which has no meaning except when required for rotational distance calculations. I think equations would be cleaner with right angle powers. And remove the 2pi which has little value except for chaing the base symbol, which isn't even needed if cos() and sin() were understood to use right angle entry. Math is like traditionally dumb. They love pi because they do. They put it in where it's not needed and celebrate memorizing digits like it means something.
Oh, Man... I got it! April Fools to you, too!
I personally would rather work in fractions of a full circle, so i = a^(1/4). Unfortunately... the number a that could satisfy that equation is... _1,_ which isn't the best choice for the base of an exponential meant to generate angles.
I'll also mention that for the ℂomplex numbers' cousins (one of which was briefly mentioned in the previous video before being discarded to due the presence of zero-divisors), neither have any value of φ where their "imaginary" unit, which I'll call "j" for now, is e^jφ.
Tiny rotation gang represent!