Top Four Visual Proofs of Mathematical Theorems

In this short, we show animations of four of the most famous proofs without words: the formula for the sum of the first n integers; the pythagorean theorem using negative space/sliding rectangles/Chou pei suan ching; the formula for the sum of the first n odd integers; and the infinite geometric series of positive powers of 1/2 (using a rectangle/square dissection of a unit area square). We include brief justifications for these visual proofs.

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For a slightly different version of this short without words, see:
https://youtube.com/shorts/eG72HxNceEs

For longer videos with related animations, see
Summing Integers: https://youtu.be/eHbtc50-qXo
Pythagorean Theorem: https://youtu.be/o2mbN452ywA (and old version: https://youtu.be/yfGtbNgcrQ8)
Summing Odds: https://youtu.be/I4A2T55k0cg
Geometric Series: https://youtu.be/JteQEN1XPyc

The first proof was known to the ancient greeks (cited by Martin Gardner), the second proof is adapted from the Chou pei suan ching (around 200 BCE according to Roger Nelsen), the third is attributed to Nicomachus of Gerasa by Roger Nelsen, and the final one is attributed to Warren Page (from the September 1981 issue of Mathematics Magazine, page 201 - https://www.jstor.org/stable/2689632 ). The first three can all be found in Roger Nelsen's first compendium, "Proofs Without Words: Exercises in Visual Thinking: https://bookstore.ams.org/view?ProductCode=CLRM/1 .

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