This video explains the Fundamental Theorem of Alegbra and gives an interesting visual proof. The proof is adapted from a similar proof given in 'Galois Theory' by Ian Stewart.
Пікірлер: 24
@spoperty494023 күн бұрын
I've spent an embarrassing number of hours on math youtube explainers, this might be one of the best. Perfectly paced to make me feel smart, simple language throughout, this is what explainers should be like.
@PRIYANSH_SUTHAR25 күн бұрын
Glad to see more and more amazing mathematics channels are created and making amazing content. Almost feels like a revolution on youtube.
@zalibecquerel346326 күн бұрын
I must say, the pacing of your videos is spot on. Slow enough so that I can remember everything, fast enough to keep me engaged. Turns out I'm a "visual learner". I took a couple of postgrad subjects in data science this semester. I would have failed both if not for StatQuest videos. No, really. I could have saved thousands in student loans and dozens of hours by watching youtube videos.
@futnick491724 күн бұрын
Triple BAM!
@copywright563526 күн бұрын
Very nice! Watching the vid rn and it’s always great to see people cover this topic. Especially as a thematic sequel to your previous videos. You push me to do better with mine!
@trevorclinton257326 күн бұрын
I think you forgot to handle the case that a sub 0 is equal to 0. In that case r=0 does not have a winding number of 0 since 0 will be a root.
@TheGrayCuber26 күн бұрын
Yes that is a case that I skipped. In this case 0 is a root, so we've already found a root and don't need the rest of the process
@ingiford17517 күн бұрын
for c in the complex numbers, the equation y=c either has no roots or infinitely many. y=0 has infinite roots, while y=1 has zero roots
@wh1223 күн бұрын
I liked your fonts selection! It'll be good to know.
@TheGrayCuber23 күн бұрын
My primary font is Atkinson Hyperlegible
@marcelob.530026 күн бұрын
Very nice, thanks. I'd recommend including the name of your channel somewhere in your video (beginning or end most usually).
@eveeeon34126 күн бұрын
This video was fantastic, it just left me thinking, maybe it's the mathematical desire for rigour in me, I couldn't help but think at 4:22 do we need to show that the set of roots/factorisation is unique? Like, it feels obvious but something in me wants to explore that a bit more. Also, 7:10 again this feels obvious but I feel like there is an assumption that you can always divide one polynomial into another, like we're trying to show that it has n roots, and in doing so we use the fact we can always divide out a polynomial (one of its roots) and using this by induction, so in a way are we not assuming it has n roots to prove the same? This is not a criticism, its far more of a praise because your video was good enough to get me thinking about these things, and idk if its worth exploring these ideas further in future.
@TheGrayCuber25 күн бұрын
These are good things to point out! Both are important to demostrate for rigor. You can show that (x-r) divides p(x) using a method similar to long division. This gives some constant remainder c such that p(x) = q(x)(x-r) + c. Since r is a root of p, c must be 0. The uniqueness of the n roots can be covered by the demonstration that a polynomial can't have more roots than its degree. There are n roots, but if they weren't unique there would be more than n total, so p would be divisible by their linear product, which would have a higher degree than p
@elshadshirinov163323 күн бұрын
A sub zero is a Mortal Kombat character. This proves everything
@mapwiz-sf5yt25 күн бұрын
Didn't you have to also show that the winding number couldn't decrease (and later increase) as r increased, which would have given our polynomial n+2k roots.....
@TheGrayCuber25 күн бұрын
The change in winding number wasn't to show that there are n roots, it was just to show that there was at least one. We only need the starting and ending winding numbers to be different, not specifically 0 and n
@LogicalQ26 күн бұрын
What if you set the limit to i? r 0 --> i
@TheGrayCuber26 күн бұрын
I like the idea! The way that I graphed it in this video, it would look the same as going 0 -> 1, since iU has all the same points as U. But it would be interesting to find a way to show the difference since 0 -> i would be sliding along itself depending on the path taken
@user-ky5dy5hl4d15 күн бұрын
I thought that a unit circle is not a function.
@TheGrayCuber15 күн бұрын
You're right! The unit circle is not a function. It is a set of numbers, and we're viewing the image of that set through a polynomial
@user-vr2dg5vo3w23 күн бұрын
12x12 blind when?
@CaesarsSalad26 күн бұрын
0 = x - 4 is not linear, it's affine.
@YouTube_username_not_found26 күн бұрын
I think the confusion comes from the fact that equations of the form Ax = b are called linear. And they are called like that because they could be written as f(x) = b where f is a linear map and b is a vector in the output space of f. Addendum: it is from this form ( f(x) = b ) that we deduce stuff about the solvibility of such equations. If b is in the image of f then we have the existence of a solutions If Ker(f) is {0} then we have the uniqueness of solutions. That's why this form is special.
@YouTube_username_not_found26 күн бұрын
In my reply I should have written the equation as Ax + b = 0 (the b here is not the same b in the form f(x) = b ) . This would have been more appropriate to deliver the point I was trying to make, that is, why do they call functions of the form ax + b linear.