The Joy Of X: A Guided Tour of Math, from One to Infinity

The joy of x is indeed what the author claims it to be in its subtitle: a tour through the enchanting and often intriguing world of mathematics by a wise and selective guide intent on passing over his enthusiasm for the subject regardless of former mathematical training. I must say I have been a fan of Strogatz since I first read his (more technical) Nonlinear Dynamics and Chaos. His lucidity in explaining advanced mathematical concepts made me wish he wrote a book on the more introductory realms of mathematics, and intended for a much broader audience. Soon enough, I heard about his series in the NY times, which clearly indicated his expertise in this arena. And now that it is has been expanded and put out as a hardcover, I made sure I ordered a copy right away! Strogatz focuses not on those who were math wiz-kids in high school. His pace and clarity particularly are meant to encourage those who were even scared of areas of mathematics to try and read this book. As to those who can digest more advanced math, the book still is charming; offering a "snack", to quote Strogatz himself, in any chapter of his work. And this is not a complete book in any-sub area of math, but merely an attempt at revising and rediscovering elementary concepts of the subject. The book is divided into six parts, constructed more or less in a sequence that resembles the way we are (or at least, should be) introduced to elementary mathematics. The first two build up on what numbers mean, their properties, the need for larger number sets, their relationships, and a whirlwind primer to algebra. Strogatz constantly focuses on insight, often digressing into alternative methods to understand concepts, and with a generous supply of figures to support that. He then moves on to Geometry, followed up by a short but extremely illustrative companion to introductory calculus. His examples are interesting and often ingeniously pulled out of daily life. Particularly worth mentioning is the fact that proofs, when presented, are discovered as a child learning math should rather than merely presented, as unfortunately the case is in most introductory textbooks. The penultimate chapter focuses on why statistics and probability should be at the fingertips of anyone today (a point not justified in most education systems today), followed by the extremely interesting final section on the 'frontiers', where topics from prime numbers to differential geometry to the meaning of infinity are touched upon (arguably my favorite section). Who is this book intended for? In my opinion, this work is qualified to be supplementary reading at a high school level. No, this is not a stand alone book in number theory or algebra or calculus or any branch of introductory math, and the author clearly does not intend to make this one. This is a tour, a joyous ride, a display piece that swiftly (half a day in my case, un-put-down-able!) takes you through the intricacies and beauty of mathematics without the terrors of rigor or the banality of (most) textbooks. I would recommend even that every parent of math students attempt to read this, to try and learn (and hopefully enjoy) the beauty of the subject along with their kids. Advanced students of math (like myself) can read this for a tour back into the days when they first meddled with introductory concepts, and see how much easier and more elucidating this could have been. And instructors of math must try this for wonderful pedagogic tools and original ideas that could make passing the tricks on to the next generation so much easier and enjoyable to both parties. PS: For those interested and motivated in more, the 250 or so snippet-notes at the back of the book (sadly not cited systematically through the course of the book except in a handful of occasions) are a treasure trove of information. Keep a log of it along with the chapters you read, and you can unearth a ton of references, links and in many cases deeper insights into the point being conveyed.

A few times I've seen postings on Facebook where people are proud of the fact that they "got through another day without using math". I'm amused but a little sad that they think math is unnecessary in day-to-day life. I wonder if they really didn't use math or did it without thinking of it as math.Or is it true that since they don't have a background in math they just ignore the problems in their lives where math could help Now, I confess I was an English major and ignored math and the sciences; but I've come to undertstand that more math would have been helpful. Steven Strogatz shows us the basic concepts of numbers and math, building from the simple: Sesame Street characters counting fish, to the mind boggling: some infinities are larger than others. We first learn about the power of numbers when we go from calling out "fish, fish, fish" for each fish we see to grouping them together in the abstract idea of "three fish". Numbers are abstract ideas we use to stand in so we can easily measure and compare things. Once we build a set of relationship rules (addition, subtraction) we continue to develop methods of relationships. For example we build fractions as "ratios of integers - hence teir technical name, rational numbers." (p 29). These rules continue to build upon one another and take us through algebra and geometry to calculus. As an example Strogatz demonstrates that adding "all the consecutive odd numbers, starting from 1: The sums above, remarkably, always turn out to be perfect squares" (p10). My biggest takeaway from the book is that when you have a hammer, everything looks like a nail. You can only use the tools in your belt to solve the problems you encounter. And worse if you do use the tools in your belt you may get the wrong answer. Or worse yet; you may have the correct tool set but use them dishonestly to misdirect people - those people like me - who didn't study enough math. An example of that is statistics, where figures lie and liers figure. Most of us have at least a passing understanding of normal distributions (bell curves). They "can be proven to arise whenever a large number of mildly random effects of similar size, all acting independently, are added together. And many things are like that." (p 178). Many, but not all. "[P]lenty of phenomena deviate from this pattern yet still manage to follow a pattern of their own." (p 178). But we are more comfortable with the normal distributions and have the tools (the mean average) to work with them. In Power-law distributions the "modes, medians, and means do not agree because of the skewed, asymmetrical shapes of their L-curves. President Bush made use of this property when he stated that his 2003 tax cuts had saved families an average of $1,586 each. Though that is technically correct, he was conveniently referring to the mean rebate, a figure that averaged in the whopping rebates of hundreds of thousands of dollars received by the richest 0.1 percent of the population. The tail on the far right of the income distribution is known to follow a pwoer law, and in situations like this, the mean is a misleading statistic to use because it's far from typical. Most families, in fat got less that %650. The median was a lot less than the mean." (p. 180) I've been intimidated by calculus but Strogatz does an effective job of making it approachable - you won't learn calculus from the book but you'll get a glimmer of understanding. If we want to find the area of a circle we start by fitting a square inside and calculate its area; then turn it into an 8 sided figure - like slices of a pizza - and calculating its area we get closer yet. And so on as the number of pie slices approaches infinity. Strogatz wraps things up with the theory of infinite sets using the illustration of the Hilbert Hotel which is always full but there is always room for one more. I can't do it justice here but he shows how the infinity of the real numbers between 0 and 1 is bigger than the infinity of whole numbers. Whaaaat? Finally I became acquainted with the "recreational mathemusician" Vi Hart through this book. She is a video illustrator who does some marvelous work demonstrating mathematic concepts. Even if you don't read this book (which you totally should), check out Vi Harts story of Wind and Mr. Ug; a couple of two dimensional beings who live on a transparent Möbius strip.

The Joy ox X is a collection of important mathematical ideas made understandable to the non mathematical reader. It covers a wide range of topics and is split into 6 parts, Numbers, Relationships, Shapes, Change, Data, Frontiers. Each part has several small chapters on selected topics. It is both enjoying to read and understandable given the authors ability to communicate well. I'll go through a few of the kinds of chapters briefly. Despite using numbers all the time, many of the properties of the real number system remain unintuitive for most people who havent been forced to think of the subtle logic often associated with the group. The first part deals with some of these nuances and describes important and understantable aspects of the number system including the properties of negative numbers and things like how operations commute (ie the order of operations can be reversed). The author then goes into using variables and algebra in the section on relationships. The complex numbers are discussed and author discusses one of the most important of formulas, the quadratic formula which gives the solutions to quadratic equations. Also included are how logarithmic and exponential functions arise. The author also discusses geometry and gives a visual proof of the pythagorean theorem. The author also shows some of the logic in simple proofs regarding triangles, like how the interior angles must equal 180 degrees using simple axioms. The author also goes into some of the math associated with conic sections. The first ideas of limiting used by Archimides are included and are the precursor to the next chapter. Change is an overview of ideas in calculus. It discusses calculus's use in finding minimums and maximums. It includes a discussion of the limit of continuous compounding ie the constant e. It discusses some aspects of differential equations as well as some simple ideas associated with maxwells equations. Data gives some nuggets of information in statistics. The author discusses power law distributions and the normal distribution. The author also discusses the ease of thinking about things in frequency terms instead of trying to compute probabilities (sort of like using monte carlo instead of explicit calculation). And the author also discusses how google uses matrix algebra to find best searches. Frontiers is a collection of interesting ideas that use the previous chapters to build on. The author discusses the distribution of primes and the infinity of twin primes. He discusses the group theory involved in rotating mattresses, shortest flights and some properties of infinity. The Jox of X has lots of compact mathematical wisdom for those interested. It is easy to follow and illuminating to read. Not all the chapters are as interesting as one another but it would be hard for almost any reader to not come out after reading it with some new intuition about aspects of math and the world. Definitely worthwhile non-technical read.

I loved, loved, loved this book! It doesn't teach you how to do math, but it really helps you understand how the various equations, formulas, and types of math are used and why. The author's voice throughout is entertaining and friendly. He gives wonderful examples, analogies, and metaphors for the math concepts. I finally understand what logarithms are and what they're used for! I learned how to calculate log equations recently, but I couldn't fathom why you'd need or want to do that. Steve explained it in ways that made sense and using examples I'll remember. The chapters are organized nicely, all terms are explained, and it ends with a lot of resources. But I have to say, I'm disappointed to have reached the end of the book! I want more. This was a fun read, and I learned a lot. It made so much sense of the math I'm learning. I also want to remark I am 50 years old and decided to finally learn math because of my interest in science. I have plenty of math text books and watch online videos to learn *how* to do math. But this was the best book ever to *understand* math and the ways it's used. Steve, please, please, please write another Joy of X. Maybe call it, The Joy of X^. You know you only touched on a few mathematics in this book. I'd love to read another by you, written in the same style, digging into more algebra, geometry, trigonometry, calculus, linear algebra, etc. This was a great book. I highly recommend it. This book is a fun, interesting read even if you don't do math, even if you're not interested in learning more math, and especially if you are doing math.

I am a confessed huge fan of Dr. Strogatz. I also confess, right here in full view of all Amazon math and science fans, that I suffer from professor envy. The students who get the privilege of attending any of Dr. Strogatz's classes are very, very lucky. I got my first taste of his work through The Great Courses DVD series on Chaos. That motivated me to read his books and the books he recommends in the Chaos course. I even played with creating some chaos of my own through some experiments with water and dyes. This book lives up to the description. I want to like math. I want to really get math. I want to be able to knock out formulas like ... err, pies? The claim is that this book will take a person of average intelligence through some pretty difficult math concepts without overwhelming or boring the budding, would-be mathematician. I've heard that before. You pick up the book or DVD course and things go well at first...then it hits, they have to nail you with, "remembering back to the calculus class you took at MIT..." and from there it's impossible to follow. Or, they pull simple examples but never explain where the variables come from. "To solve the problem of how many people are in the mall on a Tuesday at 4:00, we simply solve for X Where x=y+(y-z)*GrekSymbol..." Know what I mean? This book doesn't do that. I takes you through the steps in an easy to follow manner. One great thing about Dr. S, and please excuse how I phrase this, is that he is willing to show less "correct" or less mathematical ways of solving but ways that might be more intuitive to a rookie. Then he'll point the "right" way---without that pedantic tone your high school teacher used. So yes, even I learned from the book. (Meaning I was able to follow the book!) I will need to read it again. And I will really need to continue to work on my skills. But I made it through and had a ball. Dr. Strogatz, thank you so much for contributing to my education in spite of how very busy you must be. I appreciate your work. Chris Reich

In the introduction Strogatz discusses that what he is trying to do is start with 1+1=2 and go from there as far as he can. But he doesn't do that, and really that is what the title implies. Worse though, between one and infinity are a lot of steps and as the last chapter starts to explain some infinities are larger than others. But, then the explanation never really gets there and Set theory is left out in the cold. So, to me that leaves at least one really important and fundamental topic unattended and, although my argument runs that there is always one more topic that could be dealt with, I'll accept that really the goal could not be attained. Still, what is this books greatest downfall is that it is engaging, readable, interesting and good. Not because these are bad things, but because you are left wanting more. And while it is unfair to say a book is poor because it is good, the issue here is that what little there is (~250 pages) is great (90% let's say), but there is so much missing. Really the book never leaves junior math behind. Yes there is calculus and imaginary numbers, but the depth to which this is explained remains fairly theoretical and never descends into math, so much as what and why. Then again who would want to have to work out the math. But I really wanted there to be some hint of things to come. Yes great I remember a lot of math from high school, but we covered more ground. Fine the book is short and high school was 12 years, but then there are no exercises in here. And what is missing, is really two things: Firstly, an ending. Irrespective of any topics that might also be included, what this book lacks is a conclusion. The last chapter ends not with a sensation of topic covered as with a "the bells rung, everyone is running out the door". It feels mid sentence and with the previous chapters it is clear a much better job on this topic could have been done. It is abrupt and shocking. Secondly, even if we accept this book is to remind adults of math, maybe provide some insights missed in school and give a common basis for parents to help their children (provided one can convince the other that school might not give them the full picture), there is so much here that kids would want to know. And again, yes any book on math can be bigger, but that's not what I mean. What I mean is, why is there no mention of some of the math that still remains unproven. Some examples, why it perplexes and what hopes there are for solving it. Another thing in this vein missing, would be meta mathematics. The writing is so clear and precise I think Strogatz could pull it off. Not discuss any of the believes about mathematics that exist (ie it is the same everywhere or no it isn't) but where these discussions come from and what influence they have. Bottom line, this is a great book. It is well written, covers a surprising amount of ground really well (as in the variety of topics is consistently well explained), and it is really easy to understand. But because if this greatness the abrupt end and limited range of content really drag it down. As a reader you can sense the potential, but then it's over. There are links to further readings but there is just so much left that could be covered, that at 250 pages one feels disappointed.

I'm someone who should be good at math and never measures up to that expectation. By this I mean that I'm an engineer and a nerd and science enthusiast, but I found myself firmly in the middle of the pack in my undergrad math/physics courses. When I had The Joy of x recommended to me, I figured it might help me make some connections that I was missing. Math isn't supposed to be difficult––it's supposed to be elucidatory. That's why we invented it, after all. Steven Strogatz's book did a decent job of breezing through our different kinds of math and describing them using language that a reasonably well-read adult should be able to amble through. It doesn't hand you formulas or hand you many practical scenarios for applying mathematics, but it does give a sense of historical context, which I found useful for connecting the disparate areas of math (set theory, statistics, geometry, etc.). I'd recommend The Joy of x to folks who love historical context and who were a bit bored with math in school, but who understand that math is one of the most fundamental parts of our universe (it has even been argued to actually be the essence of our universe by some modern mathematicians). I'm taking a star off because I think some of the subjects were covered unevenly (I'm looking at you, logarithms!) and because I prefer for books of this nature to flow more naturally (it often felt like the author got cut off while waxing poetic about one of the subjects and was forced to just start over, instead of summarizing and connecting the ideas). Still, fun book, especially for anyone who thinks they should love math but can't quite seem to muster that love when it comes time to solve for x.

Supposedly publishers ask authors to begin writing by defining the audience for whom they are writing their book. Later, reviewers delight, de rigueur, in saying who the book was "intended for". When this book starts with arithmetic and gets to topics like group theory, we might wonder. But he is NOT trying to teach us any of these things. He is telling us, I think, why these things REALLY are so interesting, why he loves them, and why we should too. And he has done a remarkable job of establishing a relatively uniform level of understanding (or is appreciation the word) despite varying levels of mathematical sophistication that WOULD be necessary, IF one were inclined to take on an extensive study. Perhaps his MO was to put himself in the position of being asked at a party "What is X all about?" knowing that the questioner really wanted to know, was an intelligent person, had limited time, and really would be derailed by any patronizing snags in any descriptions proffered. Everything here is thoughtful and respectful. So it may have been intended by some involved in the publication as basically an introduction, but it is hard for me to envision anyone who would not find this interesting, if not compelling. Will a professional mathematician of engineer be bored? Not likely. I had never thought of "fish, fish, fish, fish, fish, fish" as being different from 6 fish - he is making a point about numbers, not about arithmetic. Later, seemingly more profound ideas emerges, such as: "Whenever a state of featureless equilibrium loses stability - for whatever reason, and by whatever physical, biological, or chemical process - the pattern that appears first is a sine wave, or combination of them." In some sense, the sine wave and fish comments are equally deep. So I envision Strogatz as enthusiastically sitting you down to tell you some things that he finds interesting, and imagining that you would too. Perhaps his narratives are accordingly described as infectious. Reading this, you have run into a person who delights in telling you interesting things - the kind of person who is sometimes a pest - if he really did grab you off the street and begin his spiel. But after all, YOU picked up the book. Oh - did I mention that you really should pick up this book?