Fundamental Theorem of Algebra - Numberphile
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when someone wants an intuitive, non-rigorous explanation of the Fundamental Theorem of Algebra or a way to visualize complex functions.
Recommended by 5 people across 5 different months, 2016 to 2024.
What it is
A Numberphile video in which algebraic geometer David Eisenbud gives an intuitive, topological-style argument for why every complex polynomial has a root, using mappings between complex planes and the dominance of the leading term. It is aimed at a general audience and is a follow-up to a part one on odd polynomials.
What people said
Recommends Tom Collinge · 2016-03 · Mathematics Stack Exchange · on “Recommended books that discuss the Fundamental Theorem of Algebra?”
Do watch this 15 minute video. I don't think it is a self standing proof but does offer a somewhat different perspective on things. [this video] (P.S. I'd be interested in a reference that does complete a rigorous proof along these lines: I'd anticipate something topological concerning the shrinking of a complex contour).
Recommends user357980 · 2017-06 · Mathematics Stack Exchange · on “Graphical explanation for non-differentiability of $z \mapsto \bar z$”
A nice way to visual complex functions is thinking is as used here: [this video] where one draws the function as a mapping of points on one complex plane to another. What it means for a complex function $f$ to be differentiable at a point $a_0$ is that it is approximated by $a(z-z_0) + b$ locally (which moves $z_0$ to $0$, then is a dilation/rotation around $0$ then a translation by $b$), where $a = f'(z_0)$ and $b = f(z_0)$. However, when you zoom in you find that $z \mapsto \overline{z}$ always is a reflection so it cannot be approximated like that. This is captured in the Cauchy-Reimann equations and you will not see by inspecting the real and imaginary components separately since it looks lik
Recommends Euro Micelli · 2023-09 · Mathematics Stack Exchange · on “Is there an easier way to prove the Fundamental Theorem of Algebra for polynomials with real coefficients?”
Take a look at this Numberphile video with David Eisenbud which provides, not quite a proof, but a convincing argument that only requires minimal knowledge of complex numbers. [this video]
Recommends David06 · 2023-11 · Mathematics Stack Exchange · on “About an explanation of the Fundamental Theorem of Algebra”
I had asked on this forum for an easy to understand proof of the FTA. Is there an easier way to prove the Fundamental Theorem of Algebra for polynomials with real coefficients? [LINK] One of the explanations, rather than proofs that seem somewhat easier to understand has been the one given in the linked video. [this video] The idea is that given a complex polynomial, for large enough input values (large in terms of absolute value of complex numbers), the output will imitate the leading term of the polynomial, i.e., the leading term will dominate for large values and the outputs will look more and more like the perfect circle we would have if we were dealing with a monomial. Then we gradually star
Recommends Chris Sanders · 2024-01 · Mathematics Stack Exchange · on “What are fun mathematical facts for non-mathematicians?”
You know the Youtube channel Numberphile? Many interviews with professional mathematicians where they introduce their favorite topics to the general public . There's even one trying to explain the topological proof of the Fundamental Theorem of Algebra, if you can imagine it. [this video] EDIT: I'm sure other people here can give dozens of examples. Let me give you one which I didn't invent but which I remember quite well. Imagine you were in a courtroom facing charges of dangerous driving. Would it be enough for the prosecutor to say that the speed limit on the whole road was 100 km/h but two different CCTV cameras showed that you completed a 150-km trip within one hour?
Cites ankit · 2017-06 · Mathematics Stack Exchange · on “Why fundamental theorem of algebra is stated less intuitively in college math?”
Wherever I look on the internet I find the statement of fundamental theorem of algebra like: Every non-constant single-variable polynomial with complex coefficients has at least one complex root. While in school times and in the sources where the target audience are grade school students it is stated like: Every non-zero, single-variable, degree $n$ polynomial with complex coefficients has, counted with multiplicity, exactly $n$ complex roots. Although both of these statements are identical but I can't seem to understand the reason why it is usually presented in the manner of the former definition which appears, at least to me, a little less intuitive than the later one. I guess the reason m