Uploaded May 2026 | Updated September 2026, 2 weeks ago
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 (Simpson's 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: buymeacoffee.com/VisualProofs. Thanks!
Here are the original series videos (along with attribution; for more detailed attribution, see the original videos):
Area Proof : youtu.be/pODylgdI3p8 and youtube.com/shorts/-N1GAhkTgbA (attribution: Roger B. Nelsen- see below)
Triangle slopes : youtu.be/HFGSsPbYc-s and youtube.com/shorts/cI92ZID95g4 (attribution: Richard A. Gibbs - see below)
Slopes of lines: youtu.be/AuxqpiDTCyU and 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 (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 (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 (jstor.org/stable/27965677 - page 63).
If you like this inequality or others, you can check out my playlists of inequalities:
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
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 (Simpson's 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: buymeacoffee.com/VisualProofs. Thanks!
Here are the original series videos (along with attribution; for more detailed attribution, see the original videos):
Area Proof : youtu.be/pODylgdI3p8 and youtube.com/shorts/-N1GAhkTgbA (attribution: Roger B. Nelsen- see below)
Triangle slopes : youtu.be/HFGSsPbYc-s and youtube.com/shorts/cI92ZID95g4 (attribution: Richard A. Gibbs - see below)
Slopes of lines: youtu.be/AuxqpiDTCyU and 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 (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 (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 (jstor.org/stable/27965677 - page 63).
If you like this inequality or others, you can check out my playlists of inequalities:
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
