Four famous proofs without words in 60 seconds!

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).

For longer videos with related animations, see

#math #manim #pythagoreantheorem #pythagorean #triangle #animation #theorem #pww #proofwithoutwords #visualproof #proof #mathshorts #mathvideo #geometricseries #infiniteseries #finitesums #infinitesums #oddsumformula #sumsofintegers #sums #shorts

To learn more about animating with manim, check out:

[holyek]

It's gorgeous! Where was this channel when my calculus teacher terrorized be in HS?

[stapler]

I don't know what this particular soundtrack is from, but it would be interesting if a "proof without words" was accompanied by a Mendelssohn "Lied ohne Worte". 😉

[ChaineYTXF]

superb. the piano music makes it even better. what piece is this?

[jonathanreynolds]

Witchcraft!
Been studying up, am now smart enough to understand this and these videos cement that knowledge in my head so thank you.

[striderpup]

I mean technically it never actually equals 1 (2^-k is always missing) but it’s negligible for nearly all uses.

[rob_olmstead]

What proof is the one about 1 + 3 + 5... ?

[qiangli]

Let me remind of the game: russian block😂

[afj]

a proof without words isn't a proof

[sumdumbmick]

the last one is a non-sequitur, though.

the final point in the square that makes up the upper right corner will never be included in the figure regardless of how big k gets. so the series does not sum to 1, it sums to 1 -L(0), where L(0) is the value you evaluate a limit at as the variable in question approaches 0 from the right.

for instance, f(x)=1/x cannot be evaluated at f(0), because 1/0 is undefinable. however, lim x->0+ f(x) can be evaluated, because now it's being evaluated at L(0). and this evaluates to positive infinity, while lim x->0- f(x) evaluates to negative infinity. so, we have very clear proof of 3 distinct, but adjacent points, -L(0), 0, and L(0).

so, your infinite sum actually converges to 1 -L(0), not 1. and while these values are utterly indistinguishable via arithmetic, they are very definitely different from each other.

[dcmotorkart]

Wow
İt was so amazing.
İ love proof without words✨📐🧮