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

hi~
It's quite hard to upload videos from china because of KZbin being banned, but I'll be back home in two/three months and will be uploading more frequently (i will try at least).
Same thing with Patreon..
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Support my animations on:
/ think_twice
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Any further questions or ideas:
Email - [e-mail]
Twitter - / thinktwice2580
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Programs used:
- Cinema 4D
- Adobe Premiere Pro
- Processing
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Music:
"so far away" by outlax

[@Mathologer]

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

[@ThinkTwiceLtu]

Mathologer thank you :)

[@trobin]

Mathologer speak of the devil

[@user-ie2sv1ri8k]

Yeah,it is very easy for me to understand.

[@kaustubha7371]

Two of my most favourite channels in one comment section.

[@brogcooper25]

You as well.

[@iorek4149]

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

[@ThinkTwiceLtu]

thanks a lot. glad you enjoyed it

[@diligar]

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

[@aldolunabueno2634]

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.

[@plominecraft8852]

Awesome video man! Keep the hard work :D

[@ThinkTwiceLtu]

thank you~

[@KnThSelf2ThSelfBTrue]

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

[@satyambhatia9427]

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

[@ThinkTwiceLtu]

Thank you~

[@atomiccompiler9495]

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]

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

[@atomiccompiler9495]

Think Twice Thanks :D

[@AlexandrBorschchev]

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.

[@voice_from_pizza]

Such a great video and visualization. ❤️🙏

[@vpambs1pt]

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

[@ThinkTwiceLtu]

nice to know you're still here haha

[@vpambs1pt]

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

[@bwayagnes2452]

I love these content! Keep it up

[@yamansanghavi]

Perfect as always.

[@ashutoshmahapatra537]

Your videos are my definition of perfect.

[@fawadmirza.]

Awesome as always

[@ThinkTwiceLtu]

thanks a lot

[@ShaileshDagar]

Simply loved it!

[@hpp6116]

Beautiful animation!

[@ThinkTwiceLtu]

HPP thank you(:

[@nasim09021975]

What a superb geometrical proof! 🙂

[@NarutoShippudenIntro]

Ayy he's back lovely content

[@ThinkTwiceLtu]

what's up tomac

[@NarutoShippudenIntro]

Doing good my dude still loving the videos! China doing you good?

[@ThinkTwiceLtu]

ya it's quite nice

[@Dr.Waffles]

That smooth animation woah

[@brogcooper25]

Everything about this is wonderful.

[@JoaoVitor-go2ph]

bro I love what you do

[@-Kerstin]

Awesome, thanks

[@pushkarshukla1229]

Beautiful video

[@user-xt3hg6kx2j]

amazing！

[@farhadghaseri2013]

incredibly great

[@travellcriner6849]

Amazing.

[@JorgetePanete]

everything about this is perfect

[@erikamayumi2134]

nice!

[@satviksharma1146]

Nailed it

[@txikitofandango]

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

[@vkilgore11]

Always nice.

[@ThinkTwiceLtu]

gud

[@chiletv2936]

amazing

[@Magnasium038]

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

[@emreozkarasn9226]

This is very usefull thank you

[@arifahmad-7261]

Cool

[@stephenstark313]

its damn beautiful...

[@szturmix]

Brilliant! More stuff like that

[@ThinkTwiceLtu]

Jakub Kałuża I'll try my best

[@moonwatcher2001]

awesome!!! when harmonic mean?

[@mayyousayhd5967]

Mind blown

[@kruan2661]

supperb!

[@michalnemecek3575]

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

[@High_Priest_Jonko]

Prove (a1*a2*...*an)^1/n

[@FatihKarakurt]

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

[@aleksandersabak]

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?

[@ameremortal5715]

You rock man how do u get these intuitions

[@NoNTr1v1aL]

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

[@ThinkTwiceLtu]

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]

"so far away" by outlax

[@mahmoudel-bahnasawi5283]

Awesome ✴️✴️

[@joesiu4972]

very nice my friend

[@ThinkTwiceLtu]

thank you my friend

[@ozgurozmen3378]

nice video

[@brogcooper25]

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.

[@rasoulkhoshravan5912]

very nice. keep going.

[@EchoHeo]

I love the aesthetics of these videos

[@ThinkTwiceLtu]

Thank you~

[@dragoncurveenthusiast]

Math is so beautiful!

[@pavelsapehin4308]

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

[@farhankarim28]

Cool, what program for this animation?

[@bhavyajainnd]

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

[@mohitjani4975]

From 3b1b amazing recommendation it was

[@V_V-V_V]

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

[@captainsnake8515]

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

[@bernd8608]

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

[@brogcooper25]

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

[@Zaspar]

How do you know the triangles are similar?

[@2000maichel]

the top angle is 90

[@toferg.8264]

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]

Same common Hypotenuse

[@ultimatewierdness]

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]

look up thale's theorem.

[@jkoh93]

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

[@MagicGonads]

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]

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]

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

[@ThinkTwiceLtu]

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

[@MagicGonads]

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]

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

[@moisesmoises5046]

Now how do we add the harmonic mean?

[@RG-pr5xx]

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

[@Extraordinary10s]

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.

[@theguythatmakesyoumad3834]

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

[@JoaoVitor-go2ph]

please come back

[@ThePharphis]

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]

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]

I think it should be easy to proof visually.

[@timmychanson]

what about harmonic mean?

[@valorkurzatx]

My Mind: Booooooooooom

[@garogarabed6196]

those who disliked thought it meant "download"

[@anserali551]

how h/b = a/h??

[@tomburns5231]

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

[@bananaforscale1283]

You didn't do it for negative numbers :(

[@lamalamalex]

I don’t get it

[@user-be4ib8jt6l]

..... and where is harmonic mean?

[@rishabhdhiman9422]

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.

[@elliottmcollins]

What the what is happening at 0:35?

[@ThinkTwiceLtu]

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

[@juanfranciscoaleman6938]

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

[@juanfranciscoaleman6938]

Oh nevermind

[@elliottmcollins]

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

[@juanfranciscoaleman6938]

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.

[@atilla5584]

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