European Mathematical Society - ingmar lehmann
https://euro-math-soc.eu/author/ingmar-lehmann
enMathematical Curiosities. A Treasure Trove of Unexpected Entertainments
https://euro-math-soc.eu/review/mathematical-curiosities-treasure-trove-unexpected-entertainments
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
A. Posamentier has has devoted a life to mathematics education and he has written and co-authored many books on the topic at a rate of almost one book per year. When it comes to motivating students for mathematics, one arrives easily at topics that can be classified as popular or recreational mathematics. An early title like <em>Math Wonders to Inspire Teachers and Students</em> (2003) makes clear what I mean. Later he has with his co-authors discussed more coherent subjects like Pi, the Fibonacci numbers, the Pythagorean theorem, the golden ratio. Last year's book with the present co-author I. Lehmann was <a href="http://www.euro-math-soc.eu/review/magnificent-mistakes-mathematics"><em>Magnificent Mistakes in Mathematics</em></a> (2013).</p>
<p>
The present book continues the ideas of <em>Math Charmers: Tantalizing Tidbits for the Mind</em> (2003) and <em>Mathematical Amazements and Surprises: Fascinating Figures and Noteworthy Numbers</em> (2009). Posamentier and Lehmann have collaborated on several previous book projects and form a well-oiled tandem in producing collections like this one. In producing several of the kind of books I just mentioned, I can imagine that as a math lover, you keep a notebook ready at all moments and take a note of whatever mathematical idiosyncratic tidbit one happens to come across. Moreover with the given record and the type of books that the authors have published, I would assume that new ideas and pointers were submitted by enthusiastic readers to feed the notebooks. Their <em>Magnificent Mistakes</em> gives a partial selection of —what I imagine to be— their notebooks. That book was focussing on what can go wrong in computations or when reasoning guided by intuition becomes illogic. Even great historical mathematicians have made mistakes. This book is another selection form this virtually inexhaustible set of mathematical curiosities. So what are these curiosities? They can be mathematical puzzles, or problems that have a counter-intuitive solution, or strange regularities in numbers, that could easily inspire numerologists, etc. In short: anything that amazes a math lover, that makes you raise your eyebrows, or flashes an exciting aha experience. And Posamentier and Lehmann being convinced math lovers let themselves easily be amazed, maybe sometimes more than the reader would admit. Probably math-phobics might have a less exciting “so-what” or “who-cares” experience.</p>
<p>
How did the authors bring some order in what must be a chaos of interesting ideas to put into this collection? They arranged them in five chapters. A first one is about arithmetical curiosities. Of course there are many curious numbers that are particular for various reasons. In fact any number is particular, and if it were not, it would be a curious number because it is an exception to the rule. A reason can always be found up to the absurd like 65 is particular because it is the only number whose first digit is 6 and its second is 5. Numerologists especially are very apt in finding hidden messages behind practicaly any number. Of course here the authors find better reasons to give a number the label of being curious. These are mostly special patterns or arrangements that appear doing computation. It is considered curious to note the arrangement of all 9 digits in the equality 192 + 384 = 576 which involves 3 multiples of 192. Or, if in 16/64 you cancel the 6 in numerator and denominator you get 1/4, the correct answer. But there are also other somewhat more serious number theoretic examples like perfect numbers, lucky numbers, happy numbers, number sequences,... or ways to compute prime numbers or other computational tricks like Babylonian or Russian peasant multiplication. Somewhat playful it is to compute the numbers from 0 to 100 using all possible algebraic operations on only one digit (like 5 = (4*4+4)/4 if the unique digit is 4). Once a curiosity is observed in some example, the authors transfer to the reader the mathematician's attitude to ask if this particular pattern is unique, or are there infinitely many solutions, or how can this be generalized etc.</p>
<p>
The second chapter is a cabinet of geometric curiosities. There are some old Japanese geometric problems called <em>sangakus</em>, there is Kepler's sphere packing problem and its two-dimensional analog, or squaring the square (the problem is to tile a square with squares all of which have integer side lengths but with the least possible repetition of identical squares) and several other problems with circles, quadrilaterals, and triangles.</p>
<p>
While the previous chapter is relatively short, the third one is rather extensive. It is a classical collection of mathematical puzzles of all sorts. Many such collections are available already in the literature. Its title is <em>Curious problems with curious solutions</em>. The problems are not <em>that</em> curious, but the crux most often lies in the less straightforward way in which they can be solved. For example, given the the sum and the product of two numbers, find the sum of their reciprocals. The straightforward way to solve this is solve a quadratic equation to give these numbers, compute their reciprocals and add. However, the sum of the reciprocals is the ratio of their sum over their product so that the result is immediate. This is just one example but there are 81 (mostly more complicated) such problems. Their solutions are also given but separated from the problem formulation to stimulate the reader and prevent that she should be tempted to peak at the solution before she has tried to solve the problem on her own.</p>
<p>
Chapter 4 is again short since it treats a particular subject: How can the arithmetic, geometric and harmonic mean be retrieved using geometrical arguments, i.e., using a right-angled triangle, or rectangles etc. This is a bit more of mathematics and geometry, and a bit less carefree hopping in the mathematical playground.</p>
<p>
The last chapter is devoted to fractions, in particular unit fractions play a central role. The harmonic triangle (an entry in this table is a unit fraction that is the sum of its two unit fractional children) is linked to the Pascal triangle (each entry is and integer, that is the sum of its two integer parents) and other relations illustrating that “there is more in fractions than meets the eye” as the authors conclude this chapter.</p>
<p>
All the mathematics used is very basic and can be appreciated by anybody. The hope is that this will increase the love for mathematics. As much as I hope a book like this will help to pump up its popularity, I have some doubts. These books mainly attract the readers who love solving puzzles, but these are usually the ones who are mathematically oriented already. It might help to make math lessons a bit more interesting and challenging, but many of the problems and examples are somewhat “off the beaten path” and so might not always fit in a course where certain theorems and other chunks of less amusing theory have to be assimilated. Transferring an attitude of mathematical curiosity is certainly laudable. and the problem solving techniques of the third chapter may be helpful in teaching mathematics although some problems and techniques are too “curious” to have a wider applicability. However, it will be true fun reading for anybody with a mathematical mind. You do not need an advanced mathematical education at all. If you did study mathematics you will enjoy it too, but you might find some items a bit too low level, but there are certainly others that are new and/or to you too.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a cabinet of low-level mathematical problems and rarities of all kinds that want to illustrate the fun and playfulness that is possible with elementary mathematics. The problems are of arithmetic, geometric, or logic nature, mostly chosen away from the paved road. Solution methods and outcomes are often unexpected, counter-intuitive, or require `thinking outside the box'.<br />
</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/alfred-s-posamentier" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">alfred s. posamentier</a></li><li class="vocabulary-links field-item odd"><a href="/author/ingmar-lehmann" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">ingmar lehmann</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/prometheus-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">prometheus books</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2014</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-61614-931-4 (pbk) </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">$ 19.95</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">382</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.randomhouse.com/book/236865/" title="Link to web page">http://www.randomhouse.com/book/236865/</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a07" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a07</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li></ul></span>Mon, 25 Aug 2014 07:38:28 +0000Adhemar Bultheel45691 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/mathematical-curiosities-treasure-trove-unexpected-entertainments#commentsMagnificent Mistakes in Mathematics
https://euro-math-soc.eu/review/magnificent-mistakes-mathematics
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
We all learn from our mistakes (and not only the mathematical ones). This is one of the reasons these authors wrote the book. By showing what kind of mistakes can be made in mathematics, and to what absurd conclusions that may lead, it is hoped that the reader understands better the rules of the game and be more careful in jumping to conclusions.</p>
<p>
However, we may learn not only from our own mistakes. There have been many historical mistakes, made by leading mathematicians. The first chapter is a collection of such examples. Pythagoras was mistaken when he thought that nature could be completely explained with natural numbers and their ratios. There have been historical mistakes in the calculation of $\pi$, and many wrong attempts have been made to prove Fermat's last theorem, Goldbach's conjecture, or solve the 4 colour problem, and many other such famous problems. Galileo, Euler, Fermat, Legendre, Poincaré, Einstein, they all made mistakes and often in published papers. Gauss seems to be a glorious exception to this rule. No errors are known in his published papers. This chapter is an enumeration of summaries of these historical errors, although a complete book could be devoted to each of them, why and how the wrong conclusion was made and what kind of research this has started. For example, a wrong calculation of a notorious gambler <em>Chevalier de Mérimé</em> caused him to loose repeatedly. He asked Pascal to explain what seemed to him a paradox, and the correspondence between Pascal and Fermat on this problem can be considered to be the start of probability theory. And we all know that the attempts to prove Fermat's last theorem has given rise to a many new mathematical results.</p>
<p>
The subsequent chapters discuss arithmetical, algebraic, geometrical and statistical mistakes. Here we find many obvious errors that are commonly committed by students like division by zero, or violating the rules of distributivity ($\sqrt{a + b} = \sqrt{a} + \sqrt{b}$ or $\frac{a}{b}+\frac{c}{d}=\frac{a+c}{b+d}$ and the likes). Also jumping too soon to a general conclusion is common. Infinite sums, and working with $\infty$ often leads to false results. Of a somewhat different nature are the rounding errors of digital calculators or computers that may play dirty tricks on us. Sometimes wrong logic may lead to correct answers. Arguments are then needed to convince the student of the low score notwithstanding the fact that the final result was correct. All these are familiar and teachers are desperate when students keep persistently sinning against them. Many examples of mistakes can however be reduced to the same error made under a slightly disguised form. So there is basically a lot of repetition which makes these chapters a bit dull from time to time for readers that are well beyond these rookies mistakes. There are however also errors that are counter intuitive or that have some pitfalls and that are often used in quizzes or to astonish the innocent reader with an apparent paradox. For example suppose the earth is a perfect sphere. Put a rope around the equator and enlarge it by 1 meter. Then keep this longer rope at an equal distance above the surface. Can a mouse pass under the rope? Our intuition says no. However, computation reveals a quite different result. The cat can pursue the mouse at the other hemisphere passing below the rope quite easily. Amusing and puzzling examples are the optical illusions and impossible figures in a geometrical context. A famous geometrical <a href="http://en.wikipedia.org/wiki/Missing_square_puzzle">missing square puzzle</a> showing that 65 = 64 is attributed by the authors to Lewis Carroll, although the principle is much older. It has happened that mistakes were deliberately introduced as a prank. Martin Gardner presented in his April 1975 column of Scientific American a map that would require 5 colours, which turned out to be an April fools joke on his readers. Stories like this and the more recreational pitfalls will keep you reading to the end.</p>
<p>
All the mistakes are discussed, but sometimes this is not really deep, and sometimes it feels like there is still an untold story behind. The broadness of the examples that are covered (the examples mentioned above are just a tiny sample from a vast set) prevents the inclusion of further details, but an author like Martin Gardner for example could have written a full column including the history, background, generalizations and variations on some of the issues, while here it's more like a sequential enumeration, i.e., a catalog, of the same. All historical facts e.g., are essentially restricted to the first chapeter. In brief: the authors present a not always very deep, but a broad and diverse collection of examples of what people can, and unfortunately often will, do wrong when playing with mathematics. May the reader be wiser after finishing this book.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">A. Bultheel</div></div></div><div class="field field-name-field-review-legacy-affiliation field-type-text field-label-inline clearfix"><div class="field-label">Affiliation: </div><div class="field-items"><div class="field-item even">KU Leuven</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
It's mistakes galore. Mistakes of all ages and of all sorts and in all areas of mathematics, logic, and statistics. Historical mistakes made by great mathematicians (yes, they are not immune). Mistakes regularly made by students (the typical beginners mistakes much to the exasperation of their hopeless teachers). Mistakes made on purpose for the fun of it (mistakes with a pun). Easily mistaken results because they are counter intuitive (the puzzling ones). You will certainly meet some that you made yourself, but many others that you did not even think of.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/alfred-s-posamentier" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">alfred s. posamentier</a></li><li class="vocabulary-links field-item odd"><a href="/author/ingmar-lehmann" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">ingmar lehmann</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/prometheus-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">prometheus books</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2013</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-61614-747-1</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">24 USD</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">296</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.prometheusbooks.com/index.php?main_page=product_info&amp;amp;products_id=2180" title="Link to web page">http://www.prometheusbooks.com/index.php?main_page=product_info&products_id=2180</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a05</a></li></ul></span>Tue, 28 Jan 2014 09:17:30 +0000Adhemar Bultheel45543 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/magnificent-mistakes-mathematics#comments