Visual math every student should see

A geometric visualization comparing the magnitudes of various means (averages). If you did learn this in school, you had wonderful teachers :)

Send me suggestions by email (address at end of many videos). I may not reply but I do consider all ideas!

If you purchase through these links, I may be compensated for purchases made on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay.

Book ratings are from January 2022.

My Books (worldwide links)

My Books (US links)
Mind Your Decisions: Five Book Compilation
A collection of 5 books:
"The Joy of Game Theory" rated 4.2/5 stars on 224 reviews
"The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias" rated 4/5 stars on 24 reviews
"40 Paradoxes in Logic, Probability, and Game Theory" rated 4.1/5 stars on 38 reviews
"The Best Mental Math Tricks" rated 4.2/5 stars on 76 reviews
"Multiply Numbers By Drawing Lines" rated 4.3/5 stars on 30 reviews

Mind Your Puzzles: Collection Of Volumes 1 To 3
A collection of 3 books:
"Math Puzzles Volume 1" rated 4.4/5 stars on 87 reviews
"Math Puzzles Volume 2" rated 4.1/5 stars on 24 reviews
"Math Puzzles Volume 3" rated 4.2/5 stars on 22 reviews

2017 Shorty Awards Nominee. Mind Your Decisions was nominated in the STEM category (Science, Technology, Engineering, and Math) along with eventual winner Bill Nye; finalists Adam Savage, Dr. Sandra Lee, Simone Giertz, Tim Peake, Unbox Therapy; and other nominees Elon Musk, Gizmoslip, Hope Jahren, Life Noggin, and Nerdwriter.

My Blog

Twitter

Instagram

Merch

Patreon

Press

Send me suggestions by email (address at end of many videos). I may not reply but I do consider all ideas!

If you purchase through these links, I may be compensated for purchases made on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay.

Book ratings are from January 2022.

My Books (worldwide links)

My Books (US links)
Mind Your Decisions: Five Book Compilation
A collection of 5 books:
"The Joy of Game Theory" rated 4.2/5 stars on 224 reviews
"The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias" rated 4/5 stars on 24 reviews
"40 Paradoxes in Logic, Probability, and Game Theory" rated 4.1/5 stars on 38 reviews
"The Best Mental Math Tricks" rated 4.2/5 stars on 76 reviews
"Multiply Numbers By Drawing Lines" rated 4.3/5 stars on 30 reviews

Mind Your Puzzles: Collection Of Volumes 1 To 3
A collection of 3 books:
"Math Puzzles Volume 1" rated 4.4/5 stars on 87 reviews
"Math Puzzles Volume 2" rated 4.1/5 stars on 24 reviews
"Math Puzzles Volume 3" rated 4.2/5 stars on 22 reviews

2017 Shorty Awards Nominee. Mind Your Decisions was nominated in the STEM category (Science, Technology, Engineering, and Math) along with eventual winner Bill Nye; finalists Adam Savage, Dr. Sandra Lee, Simone Giertz, Tim Peake, Unbox Therapy; and other nominees Elon Musk, Gizmoslip, Hope Jahren, Life Noggin, and Nerdwriter.

My Blog

Twitter

Instagram

Merch

Patreon

Press

[vanditstech]

"What do you MEAN " such on point lyrics 😂

[amicisun]

visual tutorials for learning them and applying them to real life circumstances would really help students

[zhinkunakur]

This *SHORT* with all the means visualized and " What you mean " bgm is nothing *SHORT* of a piece of fine Art

[MichaelRothwell]

Beautiful!

For those interested, here is an explanation.

AM (arithmetic mean) = (x+y)/2. As the diameter of the circle is x+y, its radius is (x+y)/2=AM.

GM (geometric mean) = √(xy).
For this, we have to construct some right-angled triangles:
The "left" right-angled triangle is the one with the blue (x) and purple legs.
The "right" right-angled triangle is the one with the orange (y) and purple legs.
The "large" right-angled triangle is the one whose hypotenuse is the diameter (x+y) and has the purple line as a perpendicular (it is right-angled as the diameter of a circle subtends a right-angle at any point on the circumference).
This purple perpendicular divides the large right-angled triangle into the left and right right-angled triangles.
The large right-angled triangle is similar to both the left and right right-angled triangles as it shares an acute angle with each.
So the left and right right-angled triangles are similar.
So x/purple=purple/y, purple²=xy, purple=√(xy)=GM.

RMS (root mean square) satisfies RMS²=(x²+y²)/2, so RMS=√[(x²+y²)/2].
The red line segment is the hypotenuse of a right-angled triangle with legs AM =(x+y)/2 and |x-y|/2 (for example, if x<y, bottom leg=(x+y)/2-x=(y-x)/2).
So red²=[(x+y)/2]²+[|x-y|/2]²
=(x+y)²/4+(x-y)²/4
=¼(x²+2xy+y²)+¼(x²-2xy+y²)
=¼(2x²+2y²)=½(x²+y²).
So red=√[(x²+y²)/2]=RMS.

HM (harmonic mean) satisfies
1/HM=(1/x+1/y)/2, so HM=2/(1/x+1/y).
Now we need to use two more right-angled triangles. The first is the one with the yellow legs (solid and dashed) having the purple line (GM) as hypotenuse. The other is the one whose hypotenuse is the solid yellow line extended to the centre of the circle (a radius), and with legs the purple line (GM) and the segment of length |x-y|/2 (see RMS above).
These two right-angled triangles are similar as they share an acute angle.
So yellow/purple=purple/radius, yellow=purple²/radius=GM²/AM
=xy/[(x+y)/2]=2/(1/x+1/y)=HM.

Note that when x=y the triangles degenerate into line segments, so the diagram and proofs no longer apply.

[code-seven.]

Lmao "what do you mean" after me not understanding anything

[darshraatparsadhraj]

"What do you mean?"
Same here

[jackeown]

I think these would been really cool to see in school, but I think the reason they're not shown is mostly that they only apply with 2 numbers whereas you can take the mean of a larger list and the picture no longer applies. Still I think it'd be great to show students so they get higher dimensional analogies early on! Cool stuff!

[cseraghuveerawankar]

I am really grateful that content creator like you, are making short video in these topics. I am really tired of timepass tik tok and other timepass contents.

[ardwg_]

This is that holding my faith in math. Previously I didn't really into math anymore since algebra came out.. because I can't visualize how it works in graph etc.

[dissmo]

Cool but whys the harmonic mean line there

[chandranisahanone]

The beauty of mathematices. Everyone look math as a subject but for me its my love.

[dariusnuban]

Yeah in France it's called IAG Inégalité aritmetico geometrique

[lakshyadua]

The song relates with the video:
"What do you mean?"

[manamimnm]

This is so insightful. Thank you, sir!

[rajendramisir]

Interesting and insightful visual intuitions of RMS, AM, GM, HM and the geometrical relationships between them. Michael Roth provides an excellent explanation of RMS and the various means in this Shorts comment section. I think Math students will learn more from reading Michael’s explanation and watching this Shorts. Thanks for sharing this Math visual.

[rundmw]

Excellent 😀

Suggested improvement: color the formulas to match the colors in the visualization.

[shivansmishra]

I'm in class 8, it was easy for even me to understand!

[Canda-fhxc]

Creative as usual. We need more challenge word problems.

[awoogagoogaloo]

Next you should visualize the arithmetic geometric mean and other combinations

[kummer]

The song made me laugh so hard. It is exactly what the video shown. These inequalities are fundamental for all branches in mathematics including MEASURE THEORY.