The Shape of Inequalities

Start with 2 numbers, a and b and calculate HM and GM Now you have 2 numbers again, so you can play the game again with the new values Every step brings the results together, one from above, the other from below, sandwiching the value in the limit. That value is called Geometric-Harmonic Mean

This works for all 3 pairs of means (HM-GM, GM-AM, HM-AM). The fun fact I was talking about is about the last combination: playing the game with two "extremal" means, the AM and HM, the value they converge to is GM !!

Re: The Shape of Inequalities

#5

Post by dhosek » Thu, Mar 19, 2026, 4:48 PM UTC

There’s a whole pile of math like this that kind of lies in this nether land between more advanced than you’ll get in most high school math¹ but less advanced than you’ll get in most college high school math that I was only ever exposed to when I took the classes for my teaching credential. One of my favorite was how cos/sin, tan/cot and sec/csc all can be derived from right triangles on a unit circle with the first setting the hypotenuse to the radius, the second with a vertical side tangent to the circle at x = ±1 and the third with the horizontal side tangent to the circle at y = ±1 (you can use similarity and Pythagoras to get all the standard identities like tan = sin/cos, etc.)

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1. I kind of did a speed run through high school math, taking essentially 5+ years of math in three years, so it’s likely that I ended up missing/glossing over stuff that people who were learning at a more rational pace did learn, although I think some of my teachers were too intimidated by me to try actually teaching me, much to my detriment.

Re: The Shape of Inequalities

#6

Post by epgui » Thu, Mar 19, 2026, 5:19 PM UTC

dhosek wrote: post #5

There’s a whole pile of math like this that kind of lies in this nether land between more advanced than you’ll get in most high school math¹ but less advanced than you’ll get in most college high school math that I was only ever exposed to when I took the classes for my teaching credential. One of my favorite was how cos/sin, tan/cot and sec/csc all can be derived from right triangles on a unit circle with the first…

Weird, in Canada (at least some provinces) I think that's a pretty standard part of both high school and undergraduate maths.

Re: The Shape of Inequalities

#7

Post by Sniffnoy » Thu, Mar 19, 2026, 5:40 PM UTC

In case people aren't aware, the inequality of these specific four means is a special case of the more general power mean inequality: https://en.wikipedia.org/wiki/Generalized_mean#Generalized_m...

Re: The Shape of Inequalities

#8

Post by abnry » Thu, Mar 19, 2026, 5:45 PM UTC

Sniffnoy wrote: post #7

In case people aren't aware, the inequality of these specific four means is a special case of the more general power mean inequality: https://en.wikipedia.org/wiki/Generalized_mean#Generalized_m...

Which IIRC are all a consequence of Jensen's inequality.

Re: The Shape of Inequalities

#9

Post by foldU » Thu, Mar 19, 2026, 5:58 PM UTC

The geometric representation of AM/GM is very cool, but the first animation seems wrong to me, it should be varying the value of `b`, not the location of the circle, for it to make sense, no?

Re: The Shape of Inequalities

#10

Post by nomemory » Thu, Mar 19, 2026, 6:10 PM UTC

foldU wrote: post #9

The geometric representation of AM/GM is very cool, but the first animation seems wrong to me, it should be varying the value of `b`, not the location of the circle, for it to make sense, no?

Thanks for spotting this. I've mixed two ideas. Need to comeback to it. The smaller circle has to increase its size as b grows. As it is now it works because o triangle degeneration.