The Shape of Inequalities
andreinc.netMy favorite bit of trivia is related to the following game:
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 !!
Also fun: The Arithmetic-Geometric Mean can be used to calculate Pi! (Most usefully, the AGM of 1 and sqrt(1/2).)
I was confused by this because pi is not between 1 and sqrt(1/2). But I found https://www.cs.miami.edu/home/burt/manuscripts/gaussagm/agma... and it clarifies that this AGM is used in the course of approximating pi not that it’s sufficient on its own to do so.
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.
Weird, in Canada (at least some provinces) I think that's a pretty standard part of both high school and undergraduate maths.
The relationships between the functions are pretty standardly taught, but their derivation from the right triangles on the unit circle less so (other than sin and cos).
The inequality stuff is just typical math olympiad material. Needed for solving olympiad problems, but doesn't matter that much for your overall math education from college and beyond.
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.
This I didn’t know!
I think this is not quite right as stated there because the root mean square (quadratic mean) is always positive or 0 while the arithmetic mean can be negative, making it smaller. I guess the inequality only holds for positive numbers.
That's actually one argument for not calling the root mean square a "mean", because a mean should arguably have the property that it is always a number between the largest and smallest value. But the RMS of two negative numbers is positive. (On the other hand, the median would qualify as a mean in this sense, even though it is not a "power mean".)
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.
My favorite geometric proof of an inequality is the one I read on Terry Tao's blog. Interestingly, it's not presented as a geometric proof, but it is very much one: if you have two vectors x, y, you just shrink the longer one and grow the shorter one until they reach the same size, without changing the LHS and the RHS of the inequality. Then you expand the norms of ||x - y||^2>=0 and ||x + y||^2>=0 and see -||x||^2 - ||y||^2 <= 2<x,y> <= ||x||^2 + ||y||^2, and since ||x||=||y|| you get the result.
The animated visuals are very cool, but I desperately want to turn them off in order to understand what they depict and reason about it geometrically. A pause button would be greatly appreciated.
That's actually a good advice. Does a separate, "static" screenshot also work?
Sure, that would work just as well. Plus, then you get to pick a "good" placement instead of making the user try to find one.
Ok. I will add them tomorrow. Also I will fix the first animation (b doesn't grow there as it should).
Fixed.
The book "When Less is More: Visualizing Basic Inequalities" by Claudi Alsina might also be of interest.
The AM/GM inequality is why the world switched from (what in the UK is called) the Retail Price Index (AM) to the Consumer Price Index (GM).
The first chart is super confusing. The OP line is changing size as the circles move, yet (a-b)/2 is a constant.
I've fixed that.
Oh, these are really nice Andrei! Thanks for posting them.