*Algebraic Inequalities: New Vistas*

Titu Andreescu and Mark Saul

MSRI Mathematical Circles Library

Volume: 19; 2016; 124 pp; Softcover

https://bookstore.ams.org/mcl-19

If you like algebra, then doing the work in

*Algebraic*

*Inequalities: New Vistas*will make you better at it. If you're not sure if you like algebra, then a better plan might be to start a book club devoted to

*Algebraic Inequalities*, with perhaps an expert to moderate the group. As it happens, such an arrangement forms one of the main purposes of the book. It's part of a series called "Mathematical Circles Library," co-published by the Mathematical Sciences Research Institute (MSRI) and the American Mathematical Society (AMS). What's a Mathematical Circle? From the MSRI website:

Mathematical circles are gatherings that give mathematicians the opportunity to share their deep understanding and appreciation for the subject with school students [[and teachers—JZ]] who are looking for new challenges in mathematics. Using a variety of approaches such as problem-solving, individual and group exploration of noncurricular topics, reading guidance, competitions, and just plain fun, the circles help students acquire useful mathematical ideas and techniques.Books in the Mathematical Circles Library provide grist for such a gathering, in the form of "collections of solved problems, pedagogically sound expositions, discussions of experiences in math teaching, and practical books for organizers of mathematical circles." See the website for all twenty-two volumes (and counting).

*Algebraic Inequalities*is coauthored by Titu Andreescu, a professor of mathematics, and Mark Saul, a longtime colleague who serves as Executive Director of the Julia Robinson Mathematics Festival (https://www.jrmf.org/). Their book contains a wealth of problems, including many gems. Here's one that I enjoyed, from a chapter about algebraic symmetry:

Solve simultaneously:

x+ [y] + {z} = 1.1

{x} +y+ [z] = 2.2

[x] + {y} +z= 3.3

where [

*x*] means "the greatest integer not exceeding

*x*" and {

*x*} means "the fractional part of

*x*," that is, {

*x*} =

*x*− [

*x*].

True to the description on the book's back cover, "The exposition is lean. Most of the learning occurs as the student engages in the problems posed in each chapter." I'd phrase that more strongly:

*. One cannot improve at math by passively reading. Ditto for physics by the way, as I said in the introduction to my textbook,*

**do the problems***Force and Motion: An Illustrated Guide to Newton's Laws*:

Readers have told me that they were surprised by the brevity of the chapters in

*Force and Motion*. I say what needs to be said, and then it's time for the reader to get to work. The exposition in

*Algebraic Inequalities*is even leaner; it can afford to be, since (by comparison with Newton's Laws) the content itself is leaner.

"Algebraic Inequalities: New Vistas" is the kind of title that could earn you funny looks from your fellow subway riders, but within mathematics this topic is anything but esoteric: algebraic inequalities are prevalent in advanced math and science. For instance, the Heisenberg uncertainty relation Δ

*x*Δ

*p*≥ ℏ/2 is an inequality, one that's often proved using the Cauchy-Schwarz inequality—Chapter 9 in

*Algebraic Inequalities*.

Reading this book made me nostalgic for the inequalities I'd proved in the past, when proving things was part of my job description. There have even been inequalities on this blog—most recently,

*a*+

*b*− Sqrt[

*a*

^{2}+

*b*

^{2}−

*ab*] < Sqrt[

*ab*] in the Cutting a Triangle problem. Could I have proved this inequality more easily using the tools described in

*Algebraic Inequalities*?

Going back further, Curious Cuboids features an inequality in three variables and also uses the Cauchy-Schwarz inequality and the inequality of arithmetic and geometric means (Chapters 2 and 3 in

*Algebraic Inequalities*).

A major reason why inequalities recur on Zimblog is that optimization does. Students who've taken calculus often come away thinking that optimization is fundamentally about taking a derivative; but fundamentally, optimization is better thought of as proving an inequality of one kind or another. Among the vistas which

*Algebraic Inequalities: New Vistas*opens up for the student is the perspective that algebra alone can be a powerful approach to optimization.