0.66666… = 1 (in base 7) @MathVisualProofs

0.66666… = 1 (in base 7) @MathVisualProofs

Uploaded June 2024 | Updated September 2026, 2 weeks ago

This is a short, animated visual proof showing the sum of the infinite geometric series with first term 6/7 and ratio 1/7, which in turn allows us to compute the sum of the series of powers of 1/7 and determine an interesting base 7 representation of 1.

If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!

For a longer, wordless version of this animation with two other proofs, see youtu.be/L-pxcCk3Xi4?si=hCnLn61vVxSOp2b4

Also, check out my playlist on geometric sums/series: youtube.com/playlist?list=PLZh9gzIvXQUsgw8W5TUVDtF0q4jEJ3iaw

This animation is based on a proof by Stephan Berendonk (2020) from the November 2020 issue of The College Mathematics Journal, (doi.org/10.1080/07468342.2020.1830649 p. 385)

#mathshorts #mathvideo #math #calculus #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #geometricsums #series #infinitesums #infiniteseries #geometric #geometricseries #equilateraltriangle

To learn more about animating with manim, check out:
https://manim.community

0.66666… = 1 (in base 7)

$Mediant Inequality: Visual Proof Compilation This video is a compilation of three shorter videos that I have created showing various visual proofs without words of the intriguing mediant inequality, which is related to adding fractions in the wrong way by adding the numerators and the denominators separately. While this is the incorrect way to add fractions as rational numbers, it is related to vector arithmetic and is important for topics like statistics (Simpsons paradox) and number theory (Farey arithmetic). To see the original videos (in shorter form and sometimes with words/explanations), check the links below. Of course there are many other visual proofs of this amazing fact, but these are my favorite seven. Let me know your favorite in the comments! If you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! Here are the original series videos (along with attribution; for more detailed attribution, see the original videos): Area Proof : https://youtu.be/pODylgdI3p8 and https://youtube.com/shorts/-N1GAhkTgbA (attribution: Roger B. Nelsen- see below) Triangle slopes : https://youtu.be/HFGSsPbYc-s and https://youtube.com/shorts/cI92ZID95g4 (attribution: Richard A. Gibbs - see below) Slopes of lines: https://youtu.be/AuxqpiDTCyU and https://youtube.com/shorts/uy6wpTMuPeo (attribution: Li Changming - see below) Area proof: This animation is based on a visual proof by Roger B. Nelsen from the February 1994 issue of Mathematics Magazine (https://doi.org/10.2307/2690552 - page 34). Triangle slopes: This animation is based on a visual proof by Richard A. Gibbs from the June 1990 issue of Mathematics Magazine (https://doi.org/10.2307/2691137 - page 172). Slopes of lines: This animation is based on a visual proof by Li Changming from the January 1988 issue of Mathematics Teacher (https://www.jstor.org/stable/27965677 - page 63). If you like this inequality or others, you can check out my playlists of inequalities: https://youtube.com/playlist?list=PLZh9gzIvXQUuKcFHdtjFkwAVELDxu9nH4&si cVhtnlrrAIQg9f6 #math #inequality #numbertheory #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #fractions #mediant #mathematics #fareyfractions #farey #mathshorts #mathvideo To learn more about animating with manim, check out: https://manim.community$

Square Right Pyramid Volume (height = base) Visually!

$Simpsons Paradox (Visual Proof) In this short video, we show how to think about Simpsons paradox visually in terms of vector arithmetic. In particular we show that ratios can be represented by slopes of lines (or slopes of vectors) and then we can have an interesting slope reversal when adding as you would add proportions or statistics (not usual fraction arithmetic). If you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! The visual proof in this video is based on a proof without words by Jerzy Kocik from the December 2001 issue of Mathematics Magazine ( https://www.jstor.org/stable/2691038 page 399). #mathshorts #mathvideo #math #statistics #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #paradox #simpsonsparadox #simpson #derekjeter #davidjustice #baseball #proportion #vectors #vectoraddition #battingaverage #data To learn more about animating with manim, check out: https://manim.community$

Triangles as translators between Cartesian and polar coordinates

$Adding Fractions Visually This is a short animation showing how to think visually about adding fractions. We show the process only for 2/7+1/5 but a similar idea works for adding any pair of fractions. To buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs Thanks! This is a standard technique for visualizing fraction addition. #mathshorts #mathvideo #math #fractions #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #mathematics #area #visualfractions #addfractions #fractionarithmetic #arithmetic To learn more about animating with manim, check out: https://manim.community$

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0.66666… = 1 (in base 7)