this series, having watched it twice now, is beautiful. i was a first year student watching first time, and understood it, 2nd year, got a deeper understanding of what is happening, about to watch it going into my third year. you are an amazing teacher.
Indeed, I'd love to do a series on differential geometry (my home field) and on category theory (a great love of mine). And connections to physics are always cool. It's just a question of time. Stay tuned...
Thanks! On my channel my videos are organized into a few different playlists, which is a good way to look throught them. Note that within each playlist the order may not be correct---I haven't fixed that yet.
Things are definitely different if you use quaternions, etc. The equation x^2 + 1 = 0 actually has an infinite number of solutions in the quaternions (anything of the form ai +bj +ck with a^2 + b^2 + c^2 = 1, which forms a sphere). Bezout's theorem requires an algebraically closed field, which in particular requires commutativity, and that is violated by the quaternions.
Awesome thanks for the response. Have you ever considered doing videos in other math fields? In particular I think you did a great job explaining differential forms in that video series and would love to hear your explanations of say differential geometry, category theory, or whatever else and their applications to physics. Thanks again!
I'm gobsmacked at how much you've produced here. Loved the Reimann Hypothesis series and now discovering Hodge. Do you have any sort of syllabus or suggested order for viewing your lessons? Thanks so much for the effort!
Another great video. One question though: if you use higher dimensional complex numbers (e.g. quaternions, octonians, etc.) does that imply that there are more solutions to something like y = x^2 +1? Are +/- j, +/- k also solutions? And if so does this violate the degree = number of zeroes conjecture? Thanks, and keep up the good work!
Why can't we draw the mentioned graph in 3D instead of 4D? I mean, you could use xyz graph and in function, e.g. y=x²+1, use the x axis for real numbers, z axis for imaginary and y axis for the outcome value.
@@masicbemester if x is real, so is x^2 +1, making z always 0, so it would only require two dimensions. I think they meant the function x^2 +1, for any complex x, which would require four dimensions since not every complex number squared is real (for example 1+i).
When in hi school by the same principle I was trying to invent something that will solve the equation modulus of 1= -1. 15 years later still nowhere :)!
Usually an equation is expression1=expression2, both having some variable x. And 3 is a solution of the equation if when you substitute x for 3 you get that the left side equals the right side. In the equation 1=-1, where is the x? There isn't any. It's either true or it isn't true. A solution is something that makes it true, a non-solution is something that doesn't. If 1=-1 is true, then ANYTHING is a solution. If 1=-1 is false, then ANYTHING is a solution. But for the familiar notion of 1, and the familiar notion of -1, 1=-1 is FALSE. Just plain false. That's all. For the FAMILIAR NOTION of 1 and -1 that is.... Now... So give up on find a solution for the familiar notion of it. But now you have a path open to you! Just find a solution for other notions of 1 and -1. One such notion is called Z2. I don't know how to explain Z2 quickly but it's not complicated. In Z2 you define 2=0. And then you'll get 1=-1. And 3=1. Actually, in the universe of Z2 you only have two numbers, the number 0, and the number 1. For example -1 is just a different name for 1 here. And 2 is just a different name for 0 here. So this universe only has two elements, maybe you'll find that a bit borign not sure. If I misunderstood what you wanted by equation MODULUS then I'm sorry.
La conjecture de Hodge c'est l'ensemble des valeurs de i^ 2 dans l'ensemble même de i Zeta au cube définit la valeur du temps dans l'espace même recroquevillé.
At least for me the video is a few seconds out of sync, and it makes it confusing with all the "this line versus this line" commentary not matching the cursor.
This is cool. Take the x and turn it into a I, Y=i to the 3rd minus the i. Now has to the second. Now you y= the the I to the second. Turn y into a I. Now you have i equals i to the second. Take the I one the second , and divide it by 2. Now you have I=I.
this series, having watched it twice now, is beautiful. i was a first year student watching first time, and understood it, 2nd year, got a deeper understanding of what is happening, about to watch it going into my third year. you are an amazing teacher.
Indeed, I'd love to do a series on differential geometry (my home field) and on category theory (a great love of mine). And connections to physics are always cool. It's just a question of time. Stay tuned...
i hope you'd also connect it with complexity theory
Thanks! On my channel my videos are organized into a few different playlists, which is a good way to look throught them. Note that within each playlist the order may not be correct---I haven't fixed that yet.
Things are definitely different if you use quaternions, etc. The equation x^2 + 1 = 0 actually has an infinite number of solutions in the quaternions (anything of the form ai +bj +ck with a^2 + b^2 + c^2 = 1, which forms a sphere). Bezout's theorem requires an algebraically closed field, which in particular requires commutativity, and that is violated by the quaternions.
Awesome thanks for the response. Have you ever considered doing videos in other math fields? In particular I think you did a great job explaining differential forms in that video series and would love to hear your explanations of say differential geometry, category theory, or whatever else and their applications to physics. Thanks again!
I'm gobsmacked at how much you've produced here. Loved the Reimann Hypothesis series and now discovering Hodge. Do you have any sort of syllabus or suggested order for viewing your lessons?
Thanks so much for the effort!
Another great video. One question though: if you use higher dimensional complex numbers (e.g. quaternions, octonians, etc.) does that imply that there are more solutions to something like y = x^2 +1? Are +/- j, +/- k also solutions? And if so does this violate the degree = number of zeroes conjecture? Thanks, and keep up the good work!
Why can't we draw the mentioned graph in 3D instead of 4D? I mean, you could use xyz graph and in function, e.g. y=x²+1, use the x axis for real numbers, z axis for imaginary and y axis for the outcome value.
you need tornado variable. not yet invented.
Toxic Singed Main because the results can be complex numbers
@@arnouth5260 isn't that what they said?
y + zi = x²+1, where the input x is real and the complex output is y + zi
@@masicbemester if x is real, so is x^2 +1, making z always 0, so it would only require two dimensions.
I think they meant the function x^2 +1, for any complex x, which would require four dimensions since not every complex number squared is real (for example 1+i).
@@arnouth5260 oh right. How did I not take this into account 😅
I'll see if anyone else has the same issue. It seems to work OK for me.
Mr Incredible math meme got me here
same here
When in hi school by the same principle I was trying to invent something that will solve the equation modulus of 1= -1. 15 years later still nowhere :)!
Usually an equation is expression1=expression2, both having some variable x. And 3 is a solution of the equation if when you substitute x for 3 you get that the left side equals the right side.
In the equation 1=-1, where is the x? There isn't any. It's either true or it isn't true. A solution is something that makes it true, a non-solution is something that doesn't. If 1=-1 is true, then ANYTHING is a solution. If 1=-1 is false, then ANYTHING is a solution.
But for the familiar notion of 1, and the familiar notion of -1, 1=-1 is FALSE. Just plain false. That's all. For the FAMILIAR NOTION of 1 and -1 that is....
Now... So give up on find a solution for the familiar notion of it.
But now you have a path open to you! Just find a solution for other notions of 1 and -1.
One such notion is called Z2.
I don't know how to explain Z2 quickly but it's not complicated. In Z2 you define 2=0. And then you'll get 1=-1. And 3=1. Actually, in the universe of Z2 you only have two numbers, the number 0, and the number 1. For example -1 is just a different name for 1 here. And 2 is just a different name for 0 here.
So this universe only has two elements, maybe you'll find that a bit borign not sure.
If I misunderstood what you wanted by equation MODULUS then I'm sorry.
@@PedroTricking I meant to say abs(x)=-1
Probably the best field that connects algebra and calculus would be analysis.
A lot of thanks !
La conjecture de Hodge c'est l'ensemble des valeurs de i^ 2 dans l'ensemble même de i Zeta au cube définit la valeur du temps dans l'espace même recroquevillé.
Awesome!!
who else is watching this drunk XDDD
+David Vander Mijnsbrugge LE MEEEEEEEEEEEE XDDDDDD
+David Vander Mijnsbrugge reeeeeeeeeeeee NORMIE REEEEEEEEEEEEE
David Vander Mijnsbrugge
Oh good idea.
At least for me the video is a few seconds out of sync, and it makes it confusing with all the "this line versus this line" commentary not matching the cursor.
Eu tava vendo vídeos de pitbulls, comos cheguei aqui?
This is cool. Take the x and turn it into a I, Y=i to the 3rd minus the i. Now has to the second. Now you y= the the I to the second. Turn y into a I. Now you have i equals i to the second. Take the I one the second , and divide it by 2. Now you have I=I.
Ya It gets really out of sync especially nearing the end. Amazing video though!
Bout to solve that shi
Thx
y=x² + 1(+-/x)(tornado variable I invented)
Child of God
So,
y=x^2 + 1/x
y=x^2 + 1/-x
25:08 😮😳25:11
25:36 there’s two of them ‼️
3 = c degree offense.
🎉
代数方程式の根と複素数平面の可視化
Yo