# The Magic of Math. Solving for x and Figuring Out Why

Arthur Benjamin is a mathematics professor at Harvey Mudd College in Claremont, CA. He has made it part of his mission to bring mathematics to the general public. He performs before audiences and gave TED talks. He also wrote a book on how to perform better in mental calculation, and he recorded the video course *The Joy of Mathematics* produced by *The Great Courses*. Elements of this course are used to write this book. Like so many others, he is an admirer of Martin Gardner, and likes puzzles, magic, and mathemagic. Hence probably the title of this book.

The title *The Magic of Math* and the cover picture referring to a magician will make you expect magic tricks with numbers and cards, but do not be mistaken. The message of the book is obviously 'mathematics is fun' with fun-elements omnipresent and there are indeed some suggestions to use mathematical properties (like the casting-out nines check) to surprise your audience, but it becomes in the second half of the book also very course-like in the sense that there are theorems and proofs, all at an elementary level, but still.

Because everything is kept at this introductory mathematical level, there is much in the realm of numbers (meaning natural numbers) to start with, but we also get some algebra, and later geometry and calculus. To some extent, Benjamin follows the historical evolution of mathematics. He starts with numbers and geometry and deals with the properties of numbers much like the ancient Greek did using essentially geometric elements to 'prove' these properties.

The first four chapters are generally dealing with numbers and a bit of algebra. There are number patterns (e.g., triangular and rectangular numbers, i.e., numbers that can be arranged in this geometric form). But there are also many patterns to be discovered in Pascal's triangle. Furthermore the reader is instructed about modulo calculus, Fibonacci numbers, and combinatorics to do all the counting. The algebra is essentially restricted to first and second order equations. There are almost no formal proofs here, but evidence is sometimes given on a geometric basis with graphical arguments.

The sixth chapter introduces 'the magic of proofs' with some elementary examples like a formal proof of the property that "the product of two integers is odd if and only if both numbers are odd". The proof that the square root of 2 is irrational, and the proof that there are infinitely many prime numbers are classics.

Once the reader is familiar with the rigor of a formal proof, it is time to switch to a more axiomatic environment. The most classic rigorous system is provided by Euclid's *The Elements*. So the next chapter introduces some elements of geometry and it has more formal proofs, ending in several variants for the proof of the Pythagoras theorem.

Chapter eight is semi-geometry semi-calculus. It is a discussion of the number pi and how it relates to circular area and circumference but also with mnemonics to memorize its digits and a mock tribute to pi, in the form of a parody of a popular song. The number pi is the best known number that everybody knows about, so it deserves a separate chapter in a popular book on mathematics.

Somewhat less popular, but mathematically equally important are the numbers e and the imaginary unit i. These are however more 'mathematical', meaning that they are further away from people's daily common experience. Therefore the subsequent chapters are more serious lecture-like dealing with trigonometry, the numbers e and i with logarithms and complex numbers, some elements of calculus such as differentiation, and finally some infinite series. The 'fun-element' returns at the end with the proof that the sum of all natural numbers equals −1/12 (an amazing paradox that you may find on the Web in several versions) and magic squares.

The book is richly illustrated and it has many grey boxes, called 'asides', that give some more information, or a proof, or something extra that will appeal to the more advanced readers. These can be skipped without any harm.

From the previous, it will be clear that this is a minimal introduction to the mathematics that one would get at a secondary school level. Is it the mathematics book that I would have loved to I have had then? I doubt it. I did not need that much of show element to be interested. But it might help for others who find more traditional textbooks terribly boring. If, as a teacher, you need to 'force' the math upon some unwilling student, this might be a very helpful alternative.

On the other hand the market for popular science books, and that includes popular mathematics, has never been as big as it is today. So there is great interest for this kind of books. I doubt that the buyers of this kind of books are the secondary school pupils. Perhaps the main target readers are the adults who lost interest during adolescence and regret that later. So they want to catch up, but in a less scholarly way. For this kind of readers this is a marvelous read.

And then there are the ones who already became mathematicians or math teachers. They will not find new mathematical elements and they do not need the motivation anymore, but they can always pick up some of the fun elements. I'm sure some of these will be new even for them. So also for them, there is a reason to enjoy the book.

**Submitted by Adhemar Bultheel |

**2 / Oct / 2015