Q.E.D.: Beauty in Mathematical Proof

Q. E. D. is an abbreviation for the Latin phrase "Quod erat demonstrandum", which means, "what had to be proved." In this book, Polster demonstrates 23 simple "proofs" of fundamental mathematical principles. I enclose the word proof in quotes because they are not always rigorous in the mathematical sense. In some cases they are more in the area of reasonably convincing reasoning. Some examples are: *) Cavalieri's principle *Archimedes' theorem *) The infinitude of primes *) The divergence of the harmonic series *) Slicing a cone by a plane will always give an ellipse *) Formulas for the sums of the first n-th powers. The mathematics is not rigorous, but that is not the intent here. The goal was to give a brief presentation and argument in favor of several fundamental mathematical principles. In my opinion, the author has found the mark, explaining these principles using language within the bounds of the merely interested rather than the learned professional.

If you feel that you have lost the touch of history of mathematics, have lost your creativity into the rigour of formal methods, and need integral calculus to solve simplest of the mathematical problems, this is the book you need. Q.E.D. is a compilation of ancient mathematical problems with unexpectedly short mathematical proofs, which one you know them, are as simple as they can be, yet you may not think of them by yourself. My idea is to train (or re-train) my mind with that creative thought with which you can find elegant proofs to mathematical problems rather than resorting to differential equations at each point. This book is just great on that. I could work myself through half of the book in about two days. So thought-provoking is the content that I ended up proving a few theorems myself that were not included in the book. (Yet I see a simpler proof of one of them later in the book!) I wish this book included five times more material than what it has. I wish to have all of mathematics to be taught in this fashion. Had once encountered a problem from electromagnetism that I could not even start on, finally gave up and continued reading the Feynman lectures on Physics (vol 2) to see the proof. The proof, albeit more complicated than all proofs in this book, Q.E.D., was still unexpectedly simpler. I wish for a book like Q.E.D. that teaches me a lot more mathematics. But this is not to say that Q.E.D. hasn't served the purpose it aimed for.

Great little book! Mathematicians will often tell you that mathematics is beautiful. However, they usually have a hard time conveying the beauty of math to their nonmathematical friends. The author/illustrator has done a great job in capturing this beauty in the form of truly magnificent illustrations of proofs, making Q.E.D. the ideal read for anybody interested in discovering this elusive mathematical beauty for themselves.

I like just about everything about this little book. There are a couple of other books on pictorial proofs out there (The Most Beautiful Mathematical Formulas by Salem et. al. and Proofs without Words by Nelson), but this one is by far the most visually appealing. I particularly like the beautiful etching-like illustrations which, in my opinion, capture the timeless beauty of the various proofs very well. Included in the book is a nice mix of well-known and not so well-known material. For example, many people will know the nifty pizza proof that relates the circumference of the circle with its area, but it is probably quite a pleasant surprise for many that a similar relationship exists between the surface of a sphere and its volume. B.t.w., and if you have also read the other reviews this may surprise you, I really did read most of the book.

Sigh! Unlike the reviewer pond-jumper, I have actually read "QED", in the proper sense. In fact, I have proofread it. Being a colleague of the author, I am hardly an unbiased reviewer, and though I regard "QED" as a gem of a book, the potential buyer needn't accept my opinion. Please peek inside "QED" to judge for yourselves. What I would like to do is to address pond-jumper's criticisms of "QED". To begin, he (it's gotta be a he) objects to the author's characterisation of a mathematical proof. pond-jumper doesn't specify his worry here, but it is fair enough to be concerned or confused over exactly what constitutes a mathematical proof: mathematicians and philosophers have been debating this for thousands of years, and there is plenty of room for disagreement. The approach in "QED" is to avoid the pedantry, to emphasise the clear, intuitive ideas at the heart of some mathematical theorems. As such, the book does not contain completely rigorous proofs, with the last I's and T's dotted and crossed. But the arguments are clear and convincing (and beautiful), AND the arguments are correct: the mathematicians/pedants CAN easily fill in the details if they so wish. There is no sleight of hand in "QED", no "professor's trick". pond-jumper's substantive complaint is that the author's proof that .9999999... = 1 is incorrect. In fact, the author is absolutely correct; I will briefly explain how pond-jumper has led himself astray. Any use of infinity is problematic, prone to confusion, and infinite decimals are no exception: in high school (and, sadly, often at university), the difficulties are simply ignored. Here, the question is, what happens to .999...999... when it is multiplied by 10? The author (correctly) claims the result is 9.999...999..., each 9 moving one place to the left. pond-jumper claims that "a 0 (ZERO, not nine) fills in at the end", giving the result 9.999...9990. This is his mistake: neither a 0 nor a 9 is placed at "the end", because there is no end! That is what the dots after the last 9 indicate, that the pattern goes on forever, without end. It's pond-jumper's leaving off those dots (writing .999...999 instead of .999...999...) which has permitted his error. It surprises many people that .999...999... could equal 1; in fact pond-jumper claims that it is impossible, that "a fundamental premise of mathematics is that no number is equal to any other number". Here, pond-jumper confuses the number 1 with the possible REPRESENTATIONS of that number. For example, the "fraction" 5/5 is the same number 1, even though it looks very different on the page. The same is at least possible for .999...999..., and it is in fact true. Of course, none of what I have written here PROVES that .999...999... equals 1: for that, I urge the potential buyer to peek inside "QED", to see for themselves this (six pages in), and many other, beautiful proofs.