Proving with an Interactive App

Source: RJ Lipton website, May 2019
<see the app here:  https://duetosymmetry.com/tool/polynomial-roots-toy/>

an app with animation conveying the essence of a mathematical proof? This means more than “proofs in pictures” or “proofs without words”—the animation and interactivity are crucial.

a novel, I think, proof that uses an app. Stein has written the app and it is here. He explains how to use it. I strongly suggest that you try this yourself.

To get a feel for all this, drag the {a_{0}} coefficient to {-1} and the {a_{1}} coefficient to {1/2}. You should have two real roots in root space (one at {\approx -1.28}, the other at {\approx 0.78}). Let’s call {r_{1}} the negative root, and {r_{2}} the positive root. Now move the coefficient {a_{0}}around in a small loop (i.e. move it around a little bit, and then return it to {-1} where it started). Note that the roots move continuously, and then return to their original positions. Next, move {a_{0}} in a big loop (big enough that it orbits around {r_{2}}). Something funny happens: the roots {r_{1}} and {r_{2}} switch places. 

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