Arithmetic Mean-Root Mean Square Inequality (visual proof)

This is a short, animated visual proof demonstrating the arithmetic mean - root mean square inequality using two adjacent squares.

For other visual proofs of inequalities like this see:

To learn more about animating with manim, check out:

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Music in this episode:
Procession -- Geographer

[curtiswfranks]

Not an obvious proof method to me (although I suspect that all generalized means might be expressible in a nice graphical way), but totally intuitive once I have seen it. This is how I will remember this. Elegance itself.

[Ninja]

My teacher taught us to prove this inequality using the cauchy schrawz inequality, as that method can be used to generalise the result when we have n terms instead of only 2. Still good video

[darogajee]

Nth root of any number how to visualize that..

[pkskyutube]

This is really beautiful, but when I tried applying this geometry to prove or demonstrate simple arithmetic means, I got completely lost. And also, I'd like to see how this extends to more than one sample, a, b, c, ... Amazingly I searched YouTube and found no geometric proof of the Arithmetic Mean anywhere, only proofs of the various mean inequalities and no proofs extended to more that one sample.