Animated Fundamental Theorem of Algebra

April 23, 2010

This is what The Fundamental Theorem of Algebra looks like. Click on the above to get it to animate. The polynomial is given by f(z) = z³ – 4z² + 25z – 3000. Fix r>0, and plug in all values of z=x+iy into f such that x²+y²=r². This will give you a single snapshot of the above animation, which shows what happens as r increases. It looks as if there should be some value of r for which the curve crosses through 0, doesn’t it?

The animated .gif is so useful toward the explanation I have in mind that I may decide to write this next essay in .html form instead of as a .pdf. I’ve never tried to put LaTeX into html files before, so I don’t know exactly how well this is going to go. I’m also a bit annoyed that I can’t seem to get the animated .gif to autoplay, and I have to open a new window just for the image. Maybe it’s an annoying WordPress setting.


5 Responses to “Animated Fundamental Theorem of Algebra”

  1. rezniko2 said

    The animation is very good but to fast at the end: the interesting picture when the curve goes through 0 immediately dissapears.
    You can also try to draw it in Flash. It should be not so easy though.

    As far as I know, wordpress works with LaTeX pretty good, but if you want to put formulas in your .html file then it would be a pain. The best way that I know is to make each formula as a .gif and include it as a picture. However, it´ll be not very nice.

  2. […] Euler’s Formula and the Fundamental Theorem of Algebra (work just barely begun) – At the moment, there is only an animation of what happens when you plug in a circle of z-values into a polynomial as you increase the radius. This is one version of the proof, but the reason the animation looks the way it does is Euler’s Formula. I will someday say more. At the moment, I am not sure whether this article should be a .pdf (with no animation) or take the form of a web page. This and some subtleties in how to present the ideas have delayed work on further drafts for now. Possibly related posts: (automatically generated)Humanities HW 4/27Branding papers and articles…O levels so far Posted by Matthew Bond Filed in Uncategorized Leave a Comment » […]

  3. betrudy said

    Perhaps it would be best to just state more then drifting away from the original thought

  4. perringu said

    and that

  5. perringu said

    also we see that as say prof dr mircea orasanu and prof horia orasanu as followed as
    Author Mircea Orasanu
    A fundamental work in the area of snake robots was presented by Hirose [5]. In this work, Hirose considers empirical studies of biological snakes to derive a mathematical approximation of the most common gait pattern among biological snakes, known as lateral undulation. In particular, the shape of a snake conducting lateral undulation can be described by a planar curve (the serpenoid curve) with coordinates in the x-y plane along the curve at arc length s given by
    1 INTRODUCTIBuilding further on this insight, we consider the second-order time derivative of ϕo in the form of a dynamic compensator, which will be used to control the orientation of the robot. In particular, through this control term, we modify the orientation of the robot in accordance with a reference orientationON Deci avem si miscari si in care pot fifolosite anumite ecuatii diferentiale. Atunci pentru THESIS AUTHOR Horia Orasanu
    Studii numerice in cazul In acest caz apar unele probleme care raman inca deschise discutiilor din punct de vedere

    Astfel ca avem in cazurile de mai sus urmeaza ca daca vrem sa gasim informatii despre
    Neolonomie sau constrangeri neolonome ,sau despre fenomenul instabilitatii sau stabilita
    Tii este necesar sa studiem consecintele care se desprind din ecuatia lui Helmholtz

    L( ) =
    + λ{ }

    Acum punctul de intersectie al diagonalelor unui punct fiind in miscare si avand un diame
    Tru al cercului de o raza data ,planul in care un patrat se afla in miscare supuse la legatu
    Ri neolonome ,aici mentionam ca se poate construi un boiler inafara unui cilindru margin
    It de doua emisfere si cu peretii de o subtirime constanta
    Cateva exemple numerice cu caracter experimental au fost obtinute folosind anumite pro
    cedee si care sunt consecinte a celor de mai sus pot fi date
    ( ) = – sin  d

    tan 2 ( / 2) = (k / 2sE)2 = [ (ZZe2 )2 / 4s2E2 ]

    Astfel o discutie sumara asupra tipurilor particulare in categoria carora apar elemente noi
    si unde cateva consideratii sunt necesare pot fi facute. 3rd
    Qj = Fi .

    Printre noile rezultate obtinute si relatiile dintre noile teoreme cerute ,avem acelea care
    Se refera la functiile p armonice ,sau a celor care sunt armonice in mod classic

    Astfel ca avem si in acest caz unele cazuri. Aici reamintim ca in astfel de cazuri functiile
    Armonice si acestea satisfac unele conditii la limita in mod classicsi in anumite cazuri
    Deasemenea avem ca si ca reamintim ca in astfel de cazuri functiile armonice satisfac
    conditii de forma
    Avem si interpretarea fizuca ,legata de cele de mai sus ,si este clasificatain cadrul singula
    Ritatilor locale


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