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.