The Fundamental Theorem of Algebra

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.

[@spoperty4940]

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_SUTHAR]

Glad to see more and more amazing mathematics channels are created and making amazing content. Almost feels like a revolution on youtube.

[@zalibecquerel3463]

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.

[@futnick4917]

Triple BAM!

[@copywright5635]

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!

[@trevorclinton2573]

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.

[@TheGrayCuber]

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

[@ingiford175]

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

[@wh12]

I liked your fonts selection! It'll be good to know.

[@TheGrayCuber]

My primary font is Atkinson Hyperlegible

[@marcelob.5300]

Very nice, thanks. I'd recommend including the name of your channel somewhere in your video (beginning or end most usually).

[@eveeeon341]

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.

[@TheGrayCuber]

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

[@elshadshirinov1633]

A sub zero is a Mortal Kombat character. This proves everything

[@mapwiz-sf5yt]

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.....

[@TheGrayCuber]

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

[@LogicalQ]

What if you set the limit to i? r 0 --> i

[@TheGrayCuber]

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-ky5dy5hl4d]

I thought that a unit circle is not a function.

[@TheGrayCuber]

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-vr2dg5vo3w]

12x12 blind when?

[@CaesarsSalad]

0 = x - 4 is not linear, it's affine.

[@YouTube_username_not_found]

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_found]

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.