# 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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