Arithmetic mean vs Geometric mean | inequality among means | visual proof

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  • เผยแพร่เมื่อ 26 ม.ค. 2025

ความคิดเห็น • 138

  • @Mathologer
    @Mathologer 6 ปีที่แล้ว +281

    Very nice as usual. Your videos really deserve to be known more widely :)

    • @ThinkTwiceLtu
      @ThinkTwiceLtu  6 ปีที่แล้ว +17

      Mathologer thank you :)

    • @trobin
      @trobin 6 ปีที่แล้ว +1

      Mathologer speak of the devil

    • @浩彬朱
      @浩彬朱 6 ปีที่แล้ว

      Yeah,it is very easy for me to understand.

    • @kaustubha7371
      @kaustubha7371 6 ปีที่แล้ว +2

      Two of my most favourite channels in one comment section.

    • @brogcooper25
      @brogcooper25 5 ปีที่แล้ว

      You as well.

  • @iorek4149
    @iorek4149 6 ปีที่แล้ว +14

    This is one of the most under-viewed, under-rated videos I've found to date. Well done.

    • @ThinkTwiceLtu
      @ThinkTwiceLtu  6 ปีที่แล้ว

      thanks a lot. glad you enjoyed it

  • @aldolunabueno2634
    @aldolunabueno2634 6 ปีที่แล้ว +11

    I love your content, and I feel identified with it. Some time ago I discovered "Proofs without words", and was completely fascinated. See a figure for five minutes, and then, like a lightning bolt, realize how obvious it was. Most people will never use the mathematics that goes the most to add and multiply. If this were taught in schools, at least then they could tell how beautiful it is. I hope you keep growing, and that you have the support you deserve for this job.

  • @diligar
    @diligar 6 ปีที่แล้ว +4

    This is such a beautiful way to show that inequality! I’m absolutely blown away

  • @kozimjobs7000
    @kozimjobs7000 หลายเดือนก่อน

    Best Video tutorial I have ever seen, no looong story, no add, no words, just let people to understand

  • @atomiccompiler9495
    @atomiccompiler9495 6 ปีที่แล้ว +3

    I love your videos, they are really the bestest best over the whole youtube, and nothing gives me goosebumps like when I see your notifications. Please don't ever stop ! ♥

    • @ThinkTwiceLtu
      @ThinkTwiceLtu  6 ปีที่แล้ว

      haha appreciate that :) I'll try to upload as frequently as I can

    • @atomiccompiler9495
      @atomiccompiler9495 6 ปีที่แล้ว

      Think Twice Thanks :D

  • @plominecraft8852
    @plominecraft8852 6 ปีที่แล้ว +4

    Awesome video man!
    Keep the hard work :D

  • @ashutoshmahapatra537
    @ashutoshmahapatra537 4 ปีที่แล้ว +1

    Your videos are my definition of perfect.

  • @AlexandrBorschchev
    @AlexandrBorschchev 4 ปีที่แล้ว

    this is the most creative and unique math channel ever hands down. just out of the world talent, you can't find any other channel that does this.

  • @satyambhatia9427
    @satyambhatia9427 6 ปีที่แล้ว

    You're doing a great job. Keep it up. Your channel surely deserve more followers.

  • @NarutoShippudenIntro
    @NarutoShippudenIntro 6 ปีที่แล้ว +11

    Ayy he's back lovely content

  • @KnThSelf2ThSelfBTrue
    @KnThSelf2ThSelfBTrue 6 ปีที่แล้ว +1

    Here from 3b1b! Really solid visual explanations. Very intuitive.

  • @channel395jfjsifkf
    @channel395jfjsifkf 2 ปีที่แล้ว

    Such a great video and visualization. ❤️🙏

  • @vpambs1pt
    @vpambs1pt 6 ปีที่แล้ว +2

    Nice to know you're "back" ^.^

    • @ThinkTwiceLtu
      @ThinkTwiceLtu  6 ปีที่แล้ว +1

      nice to know you're still here haha

    • @vpambs1pt
      @vpambs1pt 6 ปีที่แล้ว

      What's the use of the geometric mean?
      nice video, as always, good intuiton and music!

  • @nasim09021975
    @nasim09021975 5 ปีที่แล้ว +1

    What a superb geometrical proof! 🙂

  • @yamansanghavi
    @yamansanghavi 6 ปีที่แล้ว +3

    Perfect as always.

  • @bwayagnesarchives
    @bwayagnesarchives 6 ปีที่แล้ว

    I love these content! Keep it up

  • @fawadmirza.
    @fawadmirza. 6 ปีที่แล้ว +1

    Awesome as always

  • @prakharshukla8086
    @prakharshukla8086 5 หลายเดือนก่อน +1

    I don't understand why are we making this semicircle can somebody explain?

    • @eliotvarda
      @eliotvarda หลายเดือนก่อน

      Him making a semicircle is not the point of crucial importance. The point of crucial importance is if everything he stated was true, which i think it is. But in any case, he used semicircle as a way of presenting the proof, an analogy for easier understanding could be: you can choose to represent pythagorean theorem through making a square out of each side, add the two smaller ones and see that it is equal to the area of the square of the largest side(hypothenuse) if the right angled triangle, or you can do the same with half circles of which sides of triangle are perimeters, result is the same, ...

  • @hpp6116
    @hpp6116 6 ปีที่แล้ว

    Beautiful animation!

  • @JoaoVitor-go2ph
    @JoaoVitor-go2ph ปีที่แล้ว

    bro I love what you do

  • @brogcooper25
    @brogcooper25 5 ปีที่แล้ว +2

    Everything about this is wonderful.

  • @michalnemecek3575
    @michalnemecek3575 6 ปีที่แล้ว +1

    Nice visualization of the geometric mean. BTW how do you visualize the geometric mean of 3 numbers?

  • @moonwatcher2001
    @moonwatcher2001 4 ปีที่แล้ว

    awesome!!! when harmonic mean?

  • @txikitofandango
    @txikitofandango 4 ปีที่แล้ว +1

    Great animation! It's very clear that h must always be less than or equal to r, but how would you prove that rigorously?

  • @aleksandersabak
    @aleksandersabak 4 ปีที่แล้ว +1

    How about the unloved harmonic mean? 2/(1/a + 1/b)? I kow it's allways smaller or equal to the geometric mean, but is there a nice graphical proof like this?

  • @Dr.Waffles
    @Dr.Waffles 6 ปีที่แล้ว

    That smooth animation woah

  • @ShaileshDagar
    @ShaileshDagar 5 ปีที่แล้ว

    Simply loved it!

  • @RG-pr5xx
    @RG-pr5xx 6 ปีที่แล้ว +2

    Wait.. can someone explain how the 2 triangles had all equal angles?

    • @Extraordinary10s
      @Extraordinary10s 5 ปีที่แล้ว

      Rajdeep Gadgil This is due to the fact that as the triangle is drawn in a semicircle, with its two ends at the end of the diameter, then it is a right angled triangle, that is, one of its angles is 90 degrees.

  • @Zaspar
    @Zaspar 6 ปีที่แล้ว +10

    How do you know the triangles are similar?

    • @2000maichel
      @2000maichel 6 ปีที่แล้ว +5

      the top angle is 90

    • @toferg.8264
      @toferg.8264 6 ปีที่แล้ว +9

      This confused me for a moment too.
      Yes the top corner (of the combined triangle) is 90 degrees. And triangles always sum up to 180 degrees.
      Therefore in the bottom right (your right) of the combined triangle, is some portion of the _remaining_ 90.
      Therefore the bottom left (your left) of the combined triangle, is the _other_ portion of the _remaining_ 90 degrees.
      One hint the video gave, was by aligning the two triangles the same.
      The smaller triangle has the _other_ portion of the _remaining_ 90 degrees of the combined triangle. The smaller triangle is also a right triangle. Therefore, its corner that makes up the combined triangle's right angle, is also the original portion of the _remaining_ 90 degrees.
      Therefore the triangles have the same corners.
      Therefore the proportions of their sides are the same.

    • @kaustubha7371
      @kaustubha7371 6 ปีที่แล้ว

      Same common Hypotenuse

    • @ultimatewierdness
      @ultimatewierdness 5 ปีที่แล้ว

      I'm a bit late to the party, but a more thorough explanation would be to use what's known as the inscribed angle theorem. It states that the inward angle towards two points on the circle from another point on the circle is half of the outward angle if the third point was instead located in the middle if the circle. It's a bit hard to put into words, so look it up yourself. Since we're dealing with a half-circle, the second outward angle would be 180 degrees (forming a line across the diameter), thus the inward (inscribed) angle would be 90 degrees. This can then be combined with the combined angle of the triangle being 180 degrees to show that they are similar (see previous comment).

    • @AlexandrBorschchev
      @AlexandrBorschchev 4 ปีที่แล้ว

      look up thale's theorem.

  • @Magnasium038
    @Magnasium038 6 ปีที่แล้ว

    This is a great visualization! I'm not sure if your style is to teach or guide, but I felt the ratios at similar triangles thing went too fast

  • @brogcooper25
    @brogcooper25 5 ปีที่แล้ว

    What is this genre of music called and what else is similar? It's not something I would normally listen to but I really like it.

  • @NoNTr1v1aL
    @NoNTr1v1aL 6 ปีที่แล้ว +3

    That was awesome! The music is too. What is its name?

    • @ThinkTwiceLtu
      @ThinkTwiceLtu  6 ปีที่แล้ว +6

      thanks man. I honestly just found this song on my pc and I don't rly know the name... If anyone finds out let me know, i will include it in the description.

    • @Simmimmi99
      @Simmimmi99 6 ปีที่แล้ว +5

      "so far away" by outlax

  • @FatihKarakurt
    @FatihKarakurt 6 ปีที่แล้ว

    Visuals are great, I think music may need more attention!

  • @pushkarshukla1229
    @pushkarshukla1229 4 ปีที่แล้ว

    Beautiful video

  • @JoaoVitor-go2ph
    @JoaoVitor-go2ph 2 ปีที่แล้ว

    please come back

  • @anserali551
    @anserali551 6 ปีที่แล้ว +1

    how h/b = a/h??

  • @jkoh93
    @jkoh93 6 ปีที่แล้ว

    what if 1 number is -ve? then lhs could be +ve or -ve but rhs would always be imaginary. i think abs(rhs) would be larger than abs(lhs)
    and if both numbers are -ve? lhs would always be -ve and rhs always +ve. but abs both side will satisfy the original inequality

  • @farhankarim28
    @farhankarim28 6 ปีที่แล้ว

    Cool, what program for this animation?

  • @EchoHeo
    @EchoHeo 6 ปีที่แล้ว

    I love the aesthetics of these videos

  • @-Kerstin
    @-Kerstin 5 ปีที่แล้ว

    Awesome, thanks

  • @ameremortal5715
    @ameremortal5715 6 ปีที่แล้ว

    You rock man how do u get these intuitions

  • @brogcooper25
    @brogcooper25 5 ปีที่แล้ว

    What is this type of music called? Links to anything similar?

  • @farhadghaseri2013
    @farhadghaseri2013 3 ปีที่แล้ว

    incredibly great

  • @dragoncurveenthusiast
    @dragoncurveenthusiast 6 ปีที่แล้ว +3

    Math is so beautiful!

  • @lamalamalex
    @lamalamalex 6 ปีที่แล้ว +1

    I don’t get it

  • @JorgetePanete
    @JorgetePanete 6 ปีที่แล้ว +1

    everything about this is perfect

  • @ThePharphis
    @ThePharphis 6 ปีที่แล้ว

    before watching: Is it because a square is the maximal way to get an area rather than a rectangle?
    edit: nope, your proof is much more visually appealing. I'm thinking of how to approximate square roots, and modeling them as rectangles is the easiest way to do so (which is why I thought of the comparison). The math also agrees

    • @pedronunes3063
      @pedronunes3063 6 ปีที่แล้ว

      Say that the perimeter is 2p.
      And S=r^2. Any retangle has the area (r+k)(r-k) that equal to r^2 - k^2. So the maximum area is when k=0

    • @pedronunes3063
      @pedronunes3063 6 ปีที่แล้ว

      I think it should be easy to proof visually.

  • @bhavyajainnd
    @bhavyajainnd 6 ปีที่แล้ว

    What an amazing analogy. I know an arithmatic proof but this is way prettier

  • @moisesmoises5046
    @moisesmoises5046 6 ปีที่แล้ว

    Now how do we add the harmonic mean?

  • @bananaforscale1283
    @bananaforscale1283 4 ปีที่แล้ว

    You didn't do it for negative numbers :(

  • @emreozkarasn9226
    @emreozkarasn9226 ปีที่แล้ว

    This is very usefull thank you

  • @satviksharma1146
    @satviksharma1146 6 ปีที่แล้ว

    Nailed it

  • @石安-p3e
    @石安-p3e 5 ปีที่แล้ว

    amazing!

  • @stephenstark313
    @stephenstark313 3 ปีที่แล้ว

    its damn beautiful...

  • @arifahmad-7261
    @arifahmad-7261 3 ปีที่แล้ว +1

    Cool

  • @tomburns5231
    @tomburns5231 6 ปีที่แล้ว +1

    Another nice animation, but without explanations these aren't very useful for beginners.

  • @joesiu4972
    @joesiu4972 6 ปีที่แล้ว

    very nice my friend

  • @timmychanson
    @timmychanson 6 ปีที่แล้ว

    what about harmonic mean?

  • @V_V-V_V
    @V_V-V_V 6 ปีที่แล้ว +5

    This also shows that AM = GM only if a = b :)

    • @captainsnake8515
      @captainsnake8515 5 ปีที่แล้ว

      Uhhh... what? It clearly does not only show that.

    • @bernd8608
      @bernd8608 5 ปีที่แล้ว

      @@captainsnake8515
      He said it does, he doesn't said it ONLY does. Do you even English?

  • @vkilgore11
    @vkilgore11 6 ปีที่แล้ว

    Always nice.

  • @travellcriner6849
    @travellcriner6849 6 ปีที่แล้ว

    Amazing.

  • @theguythatmakesyoumad3834
    @theguythatmakesyoumad3834 6 ปีที่แล้ว

    This only applies if a+b is equal or greater than 2

  • @garogarabed6196
    @garogarabed6196 6 ปีที่แล้ว

    those who disliked thought it meant "download"

  • @szturmix
    @szturmix 6 ปีที่แล้ว

    Brilliant! More stuff like that

    • @ThinkTwiceLtu
      @ThinkTwiceLtu  6 ปีที่แล้ว

      Jakub Kałuża I'll try my best

  • @محمدحارثبھائی
    @محمدحارثبھائی ปีที่แล้ว

    ..... and where is harmonic mean?

  • @erikamayumi2134
    @erikamayumi2134 6 ปีที่แล้ว

    nice!

  • @pavelsapehin4308
    @pavelsapehin4308 4 ปีที่แล้ว

    The explanation is great, though the music a bit "Silent Hill"-ish :)

  • @chiletv2936
    @chiletv2936 6 ปีที่แล้ว

    amazing

  • @MagicGonads
    @MagicGonads 6 ปีที่แล้ว

    How would one extend this to the general case? (sum of items over n is greater or equal to the 'nth' root of the product of the items where n is the number of items)

    • @ThinkTwiceLtu
      @ThinkTwiceLtu  6 ปีที่แล้ว

      hmm im not sure. let's say we have the diameter split into 3 parts a,b and c. would h be a square root of abc?

    • @MagicGonads
      @MagicGonads 6 ปีที่แล้ว +1

      We don't want a square root of abc, we want a cube root of abc

    • @ThinkTwiceLtu
      @ThinkTwiceLtu  6 ปีที่แล้ว

      ya sorry that's what i mean haha.. so would h be a cube root of abc?

    • @MagicGonads
      @MagicGonads 6 ปีที่แล้ว

      I have no idea, btw I've been working on this
      www.desmos.com/calculator/8zexv8woxa
      The next thing I wanna do is attempt to add ghosts and use the powerup to influence the ghost """AI""" using a summation for the ghost position over time, then I'm gonna add a maze builder type thing so the background matches pacman.
      The last project I completed was
      www.desmos.com/calculator/ikwwzexjpp

    • @ThinkTwiceLtu
      @ThinkTwiceLtu  6 ปีที่แล้ว

      interesting. tbh i have no idea what the second project it about..

  • @rasoulkhoshravan5912
    @rasoulkhoshravan5912 4 ปีที่แล้ว

    very nice. keep going.

  • @mayyousayhd5967
    @mayyousayhd5967 6 ปีที่แล้ว +1

    Mind blown

  • @rishikeshchoudhury7661
    @rishikeshchoudhury7661 หลายเดือนก่อน

    Lovely

  • @kruan2661
    @kruan2661 ปีที่แล้ว

    supperb!

  • @mahmoudel-bahnasawi5283
    @mahmoudel-bahnasawi5283 4 ปีที่แล้ว

    Awesome ✴️✴️

  • @ozgurozmen3378
    @ozgurozmen3378 4 ปีที่แล้ว

    nice video

  • @rishabhdhiman9422
    @rishabhdhiman9422 6 ปีที่แล้ว

    Lol, I still don't get why geometric mean is called geometric mean. It isn't as if it is the only mean with geometric significance.

  • @mohitjani4975
    @mohitjani4975 6 ปีที่แล้ว

    From 3b1b amazing recommendation it was

  • @elliottmcollins
    @elliottmcollins 6 ปีที่แล้ว +1

    What the what is happening at 0:35?

    • @ThinkTwiceLtu
      @ThinkTwiceLtu  6 ปีที่แล้ว +4

      the triangles are similar, so the ratios of sides must be equal. it's all just to show that h = sqrt(ab)

    • @juanfranciscoaleman6938
      @juanfranciscoaleman6938 6 ปีที่แล้ว

      Think Twice sorry for the dumb question but how can you tell they are equal?

    • @juanfranciscoaleman6938
      @juanfranciscoaleman6938 6 ปีที่แล้ว

      Oh nevermind

    • @elliottmcollins
      @elliottmcollins 6 ปีที่แล้ว

      Glad you got it, Juan, but I still don't understand why the triangles are similar.

    • @juanfranciscoaleman6938
      @juanfranciscoaleman6938 6 ปีที่แล้ว +2

      Elliott Collins It took me quite some time, but this is how I figured it out: To be simmilar the triangles need to have the same angles. Lets call the angle on the down left of the triangle "A" and the angle on the down right "B" . At 0:22 imagine the "h" line doesnt exist. In that triangle you can see that the upper angle is 90 degress. That means that A+B has to be 90, because 90 plus the 90 on the upper angle is 180.
      Now we can see that the "h" line breaks the 90 dgs upper angle. But now we see that both triangles now have a 90 degrees angle. Like A and B hasnt changed and A+B=90, AND in the left triangle we see angles A and 90, so what is the other angle:
      A+90+x(the unknown angle)= 180
      A+x=180-90
      A+x=90. >>>>>A+B=90
      So that means x=B
      So the angles of the left triangle are A, 90 and B
      In the right triangle we do the same
      B+90+y(the unknown upper right angle)= 180
      B+y=180-90
      B+y=90 >>>>> B+A=90
      So y=A
      So the angles on the right triangle are also A, B and 90. In conclussion, the triangles are simmilar.
      Sorry if Its not clear, I am not very good at explaining.

  • @valorkurzatx
    @valorkurzatx 5 ปีที่แล้ว

    My Mind: Booooooooooom