Mathematical Visual Proofs
Mathematical Visual Proofs
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Arithmetic Mean-Geometric Mean Visual Proof Compilation
This video is a compilation of seven shorter videos that I have created showing various visual proofs without words of the famous AM-GM inequality, which states that the arithmetic mean of two numbers (a+b)/2 is greater than or equal to the geometric mean of the two numbers square root of a times b.
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: www.buymeacoffee.com/VisualProofs. Thanks!
Here are the original series videos (along with attribution; for more detailed attribution, see the original videos):
0:00 Introduction
0:05 Folding Proof : ua-cam.com/video/IJGwvvQdfgg/v-deo.html and ua-cam.com/users/shortsYAGjrfpHq-Q (attribution: Yukio Kobayashi)
0:40 Circle with Triangles : ua-cam.com/video/HFGSsPbYc-s/v-deo.html (attribution: Roger B. Nelsen)
1:07 Sliding Triangles: ua-cam.com/video/SEPkdPWKdO0/v-deo.html (attribution: Claudi Alsina and Roger B. Nelsen)
1:52 Tangent Circles : ua-cam.com/video/jNx05gXSblE/v-deo.html (attribution : Roland H. Eddy)
2:38 Eight Kites : ua-cam.com/video/02yM5LqMWUs/v-deo.html (attribution: Ayoub B. Ayoub)
3:27 Tatami : ua-cam.com/video/62G9fak1vyk/v-deo.html and ua-cam.com/users/shortsif800mpoJaY(attribution: Doris Schattschneider)
4:05 Semicircle : ua-cam.com/video/Pt8IX_Q5U1A/v-deo.html and ua-cam.com/users/shortsjB0OZYA_EI0 (attribution: Charles D. Gallant)
4:49 Ending citation
If you like this inequality or others, you can check out my playlists of inequalities and AM-GM inequality:
Inequalities: ua-cam.com/play/PLZh9gzIvXQUuKcFHdtjFkwAVELDxu9nH4.html&si=_cVhtnlrrAIQg9f6
AM-GM : ua-cam.com/play/PLZh9gzIvXQUuoRR4kTd628cWdFQ1BDyS6.html&si=zZYGTtPAsp0jXr42
#math #amgminequality #area #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #mathshorts #mathvideo #inequality #inequalities
To learn more about animating with manim, check out:
manim.community
Переглядів: 2 515

Відео

Sum Geometric Series with Ratio 1/2 in Rectangles (visual proofs)
Переглядів 3,2 тис.День тому
In this video, we show an animated version of a recent proof without words that finds the sum of the geometric series with first term 1/root(2) 1/2 and ratio 1/2 using the unit square. Then we show how to find other geometric series sums with ratio 1/2 using different sizes of rectangles. Careful, the final bonus question might require a different technique than rotating :) If you like this vid...
A Factorial Sum Produces the Factorial Number System (visual proof)
Переглядів 10 тис.14 днів тому
This is a short, animated visual proof demonstrating a finite sum involving products of factorials. The proof exploits the classic inductive proof of the formula in question. As a bonus, we discuss the factorial number system and show how the formula can be used to count up in this system. If you like this video, consider subscribing to the channel or consider buying me a coffee: www.buymeacoff...
Terdragon from the Chaos Game (visual fractal construction from iterated function system)
Переглядів 2,2 тис.21 день тому
In this video, we show how to use a random process of iteratively applying three affine transformations in the real plane to generate a 3-rep-tile known as the Terragon. What happens if you do something similar with different affine linear transformations? Check out these videos for related constructions: ua-cam.com/video/rLxx6b-oJGo/v-deo.html ua-cam.com/video/NBI_5GdvqUo/v-deo.html ua-cam.com...
Root 2 is Irrational from Isosceles Triangle (visual proof)
Переглядів 19 тис.Місяць тому
In this short, we use a famous argument by Tom Apostol to prove that the square root of two is irrational by infinite descent using a right isosceles triangle. We also go a bit further and show how this proof hints at the number theoretic construction of the convergents of the square root of two, which are the best rational approximations of root 2. If you like this video, consider subscribing ...
Three Geometric Series in an Equilateral Triangle (visual proof without words)
Переглядів 12 тис.Місяць тому
Three Geometric Series in an Equilateral Triangle (visual proof without words)
Which is larger: A^ B or B^A? (two visual proofs)
Переглядів 35 тис.Місяць тому
Which is larger: A^ B or B^A? (two visual proofs)
Levy Dragon L-System Visualization (65535 steps!)
Переглядів 2,3 тис.Місяць тому
Levy Dragon L-System Visualization (65535 steps!)
Viviani's Theorem III (visual proof without words via area)
Переглядів 5 тис.Місяць тому
Viviani's Theorem III (visual proof without words via area)
Exact Trig Values for Multiples of 18 Degrees Geometrically (visual proof)
Переглядів 8 тис.2 місяці тому
Exact Trig Values for Multiples of 18 Degrees Geometrically (visual proof)
Jordan's Inequality Visual Proof
Переглядів 5 тис.2 місяці тому
Jordan's Inequality Visual Proof
Infinitely Many Infinite Sums Geometrically (visual proof)
Переглядів 3,6 тис.2 місяці тому
Infinitely Many Infinite Sums Geometrically (visual proof)
Sum of Odd Cubes is Triangular (visual proof without words)
Переглядів 3,8 тис.2 місяці тому
Sum of Odd Cubes is Triangular (visual proof without words)
Area of a Regular Dodecagon | Proofs without Words Levels 1 to 4
Переглядів 3,6 тис.3 місяці тому
Area of a Regular Dodecagon | Proofs without Words Levels 1 to 4
Tri-Nums Sums! (visual proof IV)
Переглядів 2,5 тис.3 місяці тому
Tri-Nums Sums! (visual proof IV)
Five Pi Formulas in "Pi/2 minutes" (Pi Day visual proofs)
Переглядів 6 тис.3 місяці тому
Five Pi Formulas in "Pi/2 minutes" (Pi Day visual proofs)
Consecutive Squares as Up/Down Sum of Odds
Переглядів 2,8 тис.3 місяці тому
Consecutive Squares as Up/Down Sum of Odds
Sum of Cubes as Sum of Odds and more (visual proof VI)
Переглядів 3 тис.3 місяці тому
Sum of Cubes as Sum of Odds and more (visual proof VI)
Infinite Geometric Series in a Square (visual proof without words) III
Переглядів 2,9 тис.4 місяці тому
Infinite Geometric Series in a Square (visual proof without words) III
Integrating by Weierstrass Substitution (visual proof)
Переглядів 7 тис.4 місяці тому
Integrating by Weierstrass Substitution (visual proof)
Unfolding the Year of the Dragon: 2024 Lunar New Year Special
Переглядів 2 тис.4 місяці тому
Unfolding the Year of the Dragon: 2024 Lunar New Year Special
Picture me Differentiating (visual calculus)
Переглядів 47 тис.4 місяці тому
Picture me Differentiating (visual calculus)
3D Sum of Triangular Numbers (visual proof without words III)
Переглядів 2,6 тис.5 місяців тому
3D Sum of Triangular Numbers (visual proof without words III)
Infinite Sum of Powers of -1/7 in a Square (visual proof without words)
Переглядів 3,2 тис.5 місяців тому
Infinite Sum of Powers of -1/7 in a Square (visual proof without words)
A proof to remember: Double Angle Formulas I (visual proof)
Переглядів 13 тис.5 місяців тому
A proof to remember: Double Angle Formulas I (visual proof)
A Factorial Series Geometric Dissection (visual proof)
Переглядів 4,7 тис.5 місяців тому
A Factorial Series Geometric Dissection (visual proof)
Sum of Squares Visual Proofs: From Level 1 to Level 5
Переглядів 5 тис.6 місяців тому
Sum of Squares Visual Proofs: From Level 1 to Level 5
Recreational Math Christmas Tree Problem (with visual proof)
Переглядів 3 тис.6 місяців тому
Recreational Math Christmas Tree Problem (with visual proof)
Geometry of the Twelve Days of X-mas (visual proof)
Переглядів 2,5 тис.6 місяців тому
Geometry of the Twelve Days of X-mas (visual proof)
Pythagorean Theorem XII (visual proof)
Переглядів 7 тис.6 місяців тому
Pythagorean Theorem XII (visual proof)

КОМЕНТАРІ

  • @matiasgarciacasas558
    @matiasgarciacasas558 14 годин тому

    Much more quickly: since (x-1)^2 ≥ 0 for all x, x^2-2x+1 ≥ 0 -> x^2+1 ≥ 2x If you divide both sides by x the inequality is preserved as long as x>0 and you end up with x+1/x > 2

  • @yahyagd5637
    @yahyagd5637 16 годин тому

    How ironic that the dude who died on monday has the same name as someone who despises mondays

  • @luna_the_star
    @luna_the_star 16 годин тому

    🥐

  • @Wiesto1
    @Wiesto1 18 годин тому

    I dont see a Dragon

  • @Wiesto1
    @Wiesto1 18 годин тому

    7 is very weard

  • @Comeoniwantaccount
    @Comeoniwantaccount 18 годин тому

    Me: tree centimeter times for centimeter, divide by 2 since its a treeangle, not a reltangle, area is 6²

  • @vennstudios9885
    @vennstudios9885 18 годин тому

    Basically Summation to infinity n=0 of A×(-r)^n § is stand in for summation because no symbol in keypad :( Or just Aק-r^n

  • @vennstudios9885
    @vennstudios9885 19 годин тому

    Wait so 4 points has 2×(√2)² soo maybe which us 4 chords So if we back track The chord would be a triangle with a base of 1+|cos120| or 3/2 and a height of √3/2 so Pythagorean theorem it comes out as 9/4+3/4 which comes down to 3sq Then √3 Then realization kicks in I'm not smart enough to conclude anything with this Google is my trusted navigator

  • @canyoupoop
    @canyoupoop 19 годин тому

    _n_ th roots of unity

  • @Croissantification
    @Croissantification 20 годин тому

    I did , its hell

  • @L_M12wastaken
    @L_M12wastaken 20 годин тому

    Maybe I could, if it were physically possible to draw a perfect circle.

  • @joaoneves4150
    @joaoneves4150 22 години тому

    ua-cam.com/video/8kYkciD9VjU/v-deo.html ?

  • @brendankoelsch4461
    @brendankoelsch4461 23 години тому

    The Chanel name covers it, but for anyone who’s wondering the bottom number says “chord length product”

  • @GenericInternetter
    @GenericInternetter День тому

    Huh???

  • @catmacopter8545
    @catmacopter8545 День тому

    I wonder if there is a proof for this using roots of unity! Each chord would have length of abs(1-[root]).

  • @GovindShukla-cc6ul
    @GovindShukla-cc6ul День тому

    Best content creator 😎❤

  • @YouTube_username_not_found
    @YouTube_username_not_found День тому

    Grant's liitle theorem!

  • @user-yh9kk6kz3b
    @user-yh9kk6kz3b День тому

    😎🆒🆒

  • @muffinmainia4340
    @muffinmainia4340 День тому

    I’d love to see an explanation/proof for this!

  • @leungnamnam8372
    @leungnamnam8372 День тому

    Boo

  • @reecec626
    @reecec626 День тому

    I hadn't seen this proof!

  • @Nothingx303
    @Nothingx303 День тому

    Why this is so satisfying and interesting 😊😢

  • @InfinityCS77
    @InfinityCS77 День тому

    sec^2(theta)= 1^2 +tan^2(theta) by the pythagorean theorem. same for cosec and cot

  • @playOLDmusic
    @playOLDmusic День тому

    How is the smaller circles 1 and also the inital circle 1?

  • @CGR89
    @CGR89 День тому

    Yeah this is definitely useful to know and has infinite practical applications in the real world.

  • @augustday9483
    @augustday9483 День тому

    Looks like a nautilus shell.

  • @kkupsky6321
    @kkupsky6321 День тому

    This is how the Greeks invented bikini bottoms and SpongeBob…

  • @JacobBarrowlikecursive
    @JacobBarrowlikecursive День тому

    Average calamity mod boss attack:

  • @69_184
    @69_184 День тому

    the roblox stairs

  • @mauisstepsis5524
    @mauisstepsis5524 День тому

    Amazing video, cool music is a bonus too!

  • @Pronuxish
    @Pronuxish День тому

    0 be like: Am I joke to you?

  • @mrquackadoodlemoo
    @mrquackadoodlemoo День тому

    So...... 6+1=7?

  • @CatDogDailyPosts
    @CatDogDailyPosts День тому

    What is the real-life use case for this?

  • @yahyagd5637
    @yahyagd5637 День тому

    This sounds like alien talk to me bro

  • @turtlgamez
    @turtlgamez День тому

    I don’t understand some of it but it looks like it makes sense

  • @thegamesuniverse308
    @thegamesuniverse308 День тому

    Amazing, great explanation. Thank you. ❤

  • @realshavez
    @realshavez День тому

    Bro please Give Your video Source code 😢😢

  • @rkidy
    @rkidy День тому

    If you have a^b and b^a, the one with the base closer to e will be bigger. 3^2 > 2^3, 10^100 > 100^10

  • @breakdown3317
    @breakdown3317 День тому

    It fails because even though it approaches the size, it never reaches it.

  • @moscripts
    @moscripts День тому

    This is beautifu! and delightful to watch! Pls what software/tool do you use for your animations?

    • @MathVisualProofs
      @MathVisualProofs День тому

      I use manim for these (python library from 3blue1brown)

  • @rajkotalovic9784
    @rajkotalovic9784 День тому

    Za mene je ovo mnogo komplikovano, nista ne razumem. Svaka cast vama koji razumete!

  • @natashok4346
    @natashok4346 День тому

    Hey, diddle diddle, the median is middle... 😜

  • @xjuhox
    @xjuhox День тому

    How about using the geometrical multiplying as a basis of proof? That is, if we have a,b > 0, we can construct the product a*b.

  • @The-Professional-Dumbass
    @The-Professional-Dumbass 2 дні тому

    SNAIL

  • @appybane8481
    @appybane8481 2 дні тому

    Many of them are cool. However, some of them are just proving that (a+b)^2=(a-b)^2+4ab then simplify.

    • @MathVisualProofs
      @MathVisualProofs День тому

      Yes. That is essentially the proof of am-gm so you expect visual proofs to reflect that but geometrically.

  • @20icosahedron20
    @20icosahedron20 2 дні тому

    Really nice proofs 👌🏻

  • @melchorkinghilarioesquillo8899

    Elegant ❣️

  • @Dhanika112
    @Dhanika112 2 дні тому

    Can you make a video on this concept on why x +1/x range. Is( -infinite,-2] and [2, infinit)

    • @MathVisualProofs
      @MathVisualProofs 2 дні тому

      These essentially prove that. I have four shorts investigating that variation of these.

    • @Dhanika112
      @Dhanika112 День тому

      @@MathVisualProofs 😊okay , I am going to watch those

  • @Dhanika112
    @Dhanika112 2 дні тому

    I love this concept, it is going on now in my maths coaching

  • @kkupsky6321
    @kkupsky6321 2 дні тому

    You be square. Too cool for school.