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The Mathematics Lover’s Companion: Masterpieces for Everyone
Twenty-three mathematical masterpieces for exploration and enlightenment
“A mind-broadening experience.”—Paul J. Campbell, Mathematics Magazine
How can a shape have more than one dimension but fewer than two? What is the best way to elect public officials when more than two candidates are vying for the office? Is it possible for a highly accurate medical test to give mostly incorrect results? Can you tile your floor with regular pentagons? How can you use only the first digit of sales numbers to determine if your accountant is lying? Can mathematics give insights into free will?
Edward Scheinerman, an accomplished mathematician and enthusiastic educator, answers all these questions and more in this book, a collection of mathematical masterworks. In bite-sized chapters that require only high school algebra, he invites readers to try their hands at solving mathematical puzzles and provides an engaging and friendly tour of numbers, shapes, and uncertainty. The result is an unforgettable introduction to the fundamentals and pleasures of thinking mathematically.
“A mind-broadening experience.”—Paul J. Campbell, Mathematics Magazine
How can a shape have more than one dimension but fewer than two? What is the best way to elect public officials when more than two candidates are vying for the office? Is it possible for a highly accurate medical test to give mostly incorrect results? Can you tile your floor with regular pentagons? How can you use only the first digit of sales numbers to determine if your accountant is lying? Can mathematics give insights into free will?
Edward Scheinerman, an accomplished mathematician and enthusiastic educator, answers all these questions and more in this book, a collection of mathematical masterworks. In bite-sized chapters that require only high school algebra, he invites readers to try their hands at solving mathematical puzzles and provides an engaging and friendly tour of numbers, shapes, and uncertainty. The result is an unforgettable introduction to the fundamentals and pleasures of thinking mathematically.
296 pages, Hardcover
Published March 21, 2017
36 people are currently reading
520 people want to read
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Displaying 1 - 15 of 15 reviews
May 11, 2017
The worrying thing about this title is that I'm not sure I am a maths lover. I find some parts of mathematics interesting - infinity and probability, for example - but a lot of it is just a means to an end for me. The good news is that, even if you are like me, there's a lot to like here, though you may find yourself skipping through some parts.
Edward Scheinerman takes us through 23 mathematical areas, so should you find a particular one doesn't work for you, it's easy enough to move onto another that does. Sometimes it wasn't the obvious ones that intrigued - where I found the section on infinity, for example, a little underwhelming, I really enjoyed the section on factorials. The book opens with prime numbers, which while not the most exciting of its contents, gives the reader a solid introduction to the level of mathematical thought they will be dealing with. It's enough to get the brain working - this isn't a pure fun read and you have to think - but not so challenging that you feel obliged to give up.
Along the way, Scheinerman is enthusiastic and encouraging with a light, informative style. Each page has side bars (meaning there's a lot of white space), which contain occasional comments and asides. I found these rather irritating for two reasons. In part because it really breaks up the reading process - if it's worth saying, say it in the main text - and partly because (in good Fermat fashion) there's not a lot of room so, for example, when we are told the origin of the RSA algorithm in the side bar, there's space to say it's named after Rivest, Shamir and Adelman but not to say that Cocks came up with it before them.
Occasionally, as often seems the case with mathematicians, the author seemed to be in a slightly different world. He says that the angle trisection problem is more famous that squaring the circle - which seems very unlikely - and though he notes that pi day is usually considered to be 14 March (when written as 3/14) he doesn't point out it makes much more sense in the non-US world for it to be the 22 July (22/7).
A typical section for me was the one the constant e (like pi, a number that crops up in nature and is valuable in a number of mathematical applications). There were parts of the section that I found really interesting: I'd never really seen the point of e before, the compound interest example was an eye-opener and there's the beautiful eiπ= -1. The two other examples, though, I did have to skip as they were a little dull.
My favourite part was at the end - the sections on uncertainty, including non-transitive dice (where you can have a series of dice, each of which can beat one of the others) and equivalent poker hands, Bayesian statistics, how to have a fair election and a fascinating game (Newcomb's paradox) - where it seems that you should choose what's not best for you to come out best - were all great. It would have been even better if the election section had used terms like 'first past the post' and 'single transferable vote' to make a clearer parallel with real election systems - and the Newcomb's paradox section should have made more of the difficulty of predicting an individual's choice - but these are small concerns.
So will anyone love all of it? Probably not. If you truly do love maths, you'll know a lot of this already. If you aren't sure about your relationship with the field, the book won't all work for you - but that bits that do should be enough to show that mathematics can make an entertaining and stimulating companion.
Edward Scheinerman takes us through 23 mathematical areas, so should you find a particular one doesn't work for you, it's easy enough to move onto another that does. Sometimes it wasn't the obvious ones that intrigued - where I found the section on infinity, for example, a little underwhelming, I really enjoyed the section on factorials. The book opens with prime numbers, which while not the most exciting of its contents, gives the reader a solid introduction to the level of mathematical thought they will be dealing with. It's enough to get the brain working - this isn't a pure fun read and you have to think - but not so challenging that you feel obliged to give up.
Along the way, Scheinerman is enthusiastic and encouraging with a light, informative style. Each page has side bars (meaning there's a lot of white space), which contain occasional comments and asides. I found these rather irritating for two reasons. In part because it really breaks up the reading process - if it's worth saying, say it in the main text - and partly because (in good Fermat fashion) there's not a lot of room so, for example, when we are told the origin of the RSA algorithm in the side bar, there's space to say it's named after Rivest, Shamir and Adelman but not to say that Cocks came up with it before them.
Occasionally, as often seems the case with mathematicians, the author seemed to be in a slightly different world. He says that the angle trisection problem is more famous that squaring the circle - which seems very unlikely - and though he notes that pi day is usually considered to be 14 March (when written as 3/14) he doesn't point out it makes much more sense in the non-US world for it to be the 22 July (22/7).
A typical section for me was the one the constant e (like pi, a number that crops up in nature and is valuable in a number of mathematical applications). There were parts of the section that I found really interesting: I'd never really seen the point of e before, the compound interest example was an eye-opener and there's the beautiful eiπ= -1. The two other examples, though, I did have to skip as they were a little dull.
My favourite part was at the end - the sections on uncertainty, including non-transitive dice (where you can have a series of dice, each of which can beat one of the others) and equivalent poker hands, Bayesian statistics, how to have a fair election and a fascinating game (Newcomb's paradox) - where it seems that you should choose what's not best for you to come out best - were all great. It would have been even better if the election section had used terms like 'first past the post' and 'single transferable vote' to make a clearer parallel with real election systems - and the Newcomb's paradox section should have made more of the difficulty of predicting an individual's choice - but these are small concerns.
So will anyone love all of it? Probably not. If you truly do love maths, you'll know a lot of this already. If you aren't sure about your relationship with the field, the book won't all work for you - but that bits that do should be enough to show that mathematics can make an entertaining and stimulating companion.
April 10, 2019
Very enjoyable. Who knew there are 6 ways to define the center of a triangle? (Well, preobably much younger me knew that sometime during the Nixon presidency). Lots of good stuff on basic number theory, geometry, and probability. The section on fractal dimensions was really well done, using the Sierpinski triangle as a starting point.
Nothing in this book requires more than elementary algebra and very basic geometry, but the results (most of them with proofs) are fundamentally interesting. I liked it, a lot.
Nothing in this book requires more than elementary algebra and very basic geometry, but the results (most of them with proofs) are fundamentally interesting. I liked it, a lot.
January 28, 2023
It’s a shame that math is one of the most hated subjects in school. There are so many beautiful concepts that enough people don’t appreciate. Numbers and shapes have beauty to them.
August 23, 2017
If you have ever wondered why mathematicians get so excited about maths, this book might be a good place to start.
I was missing maths a bit over the holidays, so thought I would try this book. Most of the chapters I already knew something about, but there were often new little titbits that I had not met before, or different ways of looking at things, so it kept my interest up throughout.
A wide range of mathematical areas are covered, with sections on Number, Shape and Uncertainty, so there really is something for everyone. While some of the topics are quite complex, and many require at least A-level maths to fully understand on your own, most chapters could be further explained and simplified through an intermediary, and coped with then by even year 9 to 11 students. The book would be an invaluable asset to maths teachers looking for ways to extend their pupils interest in mathematics beyond the curriculum. I would have certainly liked to have had it when I was still teaching.
My favourite section was on shape, and I was particularly pleased with the chapter on Fractals. I am about to start an MSc module on Fractals this year, and the chapter will explain to my husband much more clearly than I could, what I will be trying to learn.
There is so much beauty and wonder in mathematics, and this book makes a good attempt to put that across to the lay person.
I was missing maths a bit over the holidays, so thought I would try this book. Most of the chapters I already knew something about, but there were often new little titbits that I had not met before, or different ways of looking at things, so it kept my interest up throughout.
A wide range of mathematical areas are covered, with sections on Number, Shape and Uncertainty, so there really is something for everyone. While some of the topics are quite complex, and many require at least A-level maths to fully understand on your own, most chapters could be further explained and simplified through an intermediary, and coped with then by even year 9 to 11 students. The book would be an invaluable asset to maths teachers looking for ways to extend their pupils interest in mathematics beyond the curriculum. I would have certainly liked to have had it when I was still teaching.
My favourite section was on shape, and I was particularly pleased with the chapter on Fractals. I am about to start an MSc module on Fractals this year, and the chapter will explain to my husband much more clearly than I could, what I will be trying to learn.
There is so much beauty and wonder in mathematics, and this book makes a good attempt to put that across to the lay person.
July 22, 2018
Scheinerman, a professor of applied math, takes us through 23 mathematical areas, chosen (according to the author) because they are not well known to non-mathematicians, do not rely on college-level math, focus on proofs (particularly proof by contradiction), involve an element of surprise, and have a practical application. His stated goal is to share some of the beauty of mathematics, and to some extent, he succeeds. However, as with most books that cover a wide range of topics, some chapters are more engaging than others. For example, his chapter on pi explains clearly how one can estimate its value by inscribing and circumscribing n-gons with greater values of n. Equally clear is his explanation for why we define 0! = 1 and his exploration of several proofs of the Pythagorean theorem, including one by former US President James Garfield. Occasionally, however, explanations are less clear, and of the last chapters, for example, the ones of nontransitive dice, chaos, and social choice are good, while the chapter on medical probability is not, and the chapter on Newcomb’s paradox is mixed, often getting a bit dense. In any event, for readers with some interest in math, a reasonably entertaining volume.
November 11, 2018
Nice book, especially the bit about number theory. I enjoyed the chapter on Benford’s Law, which is as counterintuitive as they come.
The later chapters on Geometry and Uncertainty seem a lot less inspired. These read more like “mentionings” of interesting things, than like “discoverings” of same. Hence the 3 stars.
The later chapters on Geometry and Uncertainty seem a lot less inspired. These read more like “mentionings” of interesting things, than like “discoverings” of same. Hence the 3 stars.
January 4, 2020
Emocjonalna negatywna deklaracja kogokolwiek w stosunku matematyki jest dla mnie czymś zdumiewającym. Jeśli jednak ktoś mimo to czuje do niej ‘brak mięty’, to opiniowana książka jest wprost wymarzona dla niego. Matematyka uczy myślenia, choć to banał, to wciąż mocne stwierdzenie. W tym szczególnym przypadku myślenia, chodzi o budowanie struktur, których użyteczne bogactwo daje się wywieść z zaledwie kilku pierwotnych pojęć. Tak zbudowany jest gmach królowej nauk. Jestem przekonany, że posiadacz szkolnej traumy, po lekturze opiniowanej książki, pozbędzie się jej bezboleśnie.
Profesor matematyki Edward Scheinerman w „Przewodniku miłośnika matematyki. Arcydzieła dla każdego” pięknie opisał wybrane pojęcia, obiekty i sposoby analizy problemów, które są proste technicznie do wytłumaczenia, a jednocześnie każdego zapoznającego się z ich siłą wprawiają w zdumienie. Zagadnienia pogrupowane są w trzy bloki – o liczbach, o kształtach i o niepewności.
Po raz kolejny zastanawiam się jak zachęcać do sięgania po takie książki? Może tak. Czy wiecie, że są różne typy nieskończoności? Czy wiecie, że poza obiektami jednowymiarowymi i dwuwymiarowymi są takie, o wymiarach pośrednich, np. 1,8927… Czy wiecie, że liczby opisujące różne wielkości w przyrodzie (dowolne wielkości stanowiące wystarczająco liczny zbiór) zaczynają się cyframi tak, że istnieje między nimi bardzo jednoznaczna analityczna relacja częstości występowania? I ostatecznie – czy wiecie, że wytłumaczenie tych nietrywialnych ustaleń matematyki jest naprawdę proste i zrozumiałe dla każdego?! Scheinerman nie spieszy się, wszystko wyjaśnia dokładnie, dopowiada z innej strony, podaje pomocne grafiki czy tabelki z przykładami. Wszystko, by nie zostawić w procesie analizy zjawiska nikogo. Jaki jest sens liczby ‘pi’, ‘e’ czy cel wprowadzenia jednostki urojonej, w czym tkwi siła liczb pierwszych? – to wszystko też niespotykanie klarownie jest tu podane.
Jakie są praktyczne zdobycze po lekturze? Przede wszystkim satysfakcja, że dotknęło się matematyki od fascynującej strony, nawet w kilku miejscach tej współczesnej. Poza tym czytelnik dowiaduje się (albo sobie przypomina) o pięknie relacji tłumaczących związki między figurami czy liczbami, co przydaje się do budowania wyobraźni przestrzennej czy ułatwiania sobie życia w natłoku liczb. Poza tym warto chyba wiedzieć, czemu szyfry RSA na razie są bezpieczne (str. 39-40). Albo, że jeśli w medycznym teście na rzadką chorobę w którym wynik jest poprawny w 98% przypadków, a na którą zapada 0,1% populacji, dowiemy się, iż akurat u nas wynik jest pozytywny, to prawdopodobieństwo, że faktycznie dopadła nas ta choroba wynosi zaledwie 4,7% (str. 282-284)!! Jak mając do poukładania alfabetycznie 1000 prac studenckich, zrobić to optymalnie (str. 167-173)?
Ile wystarczy mieć książek na regale, by liczba możliwych kolejności ich ustawienia była większa od liczby cząstek elementarnych we Wszechświecie? Podpowiedź w książce (a sam rachunek to zaledwie kilka symboli).
Polecam „Przewodnik miłośnika matematyki” z pełnym przekonaniem. Opowieść zaczyna się od stwierdzenia: ‘były sobie liczby naturalne 1,2,3,…’ i kończy matematycznym dowodem na istnienie wolnej woli różniącej ludzi od maszyn (dyskusyjna jest dla mnie jednak semantyczna strona sformułowania tezy, bo nie ma jednej definicji ‘wolnej woli’, ale sam wywód autora - piękny).
Cały wielki świat przygody z wyobraźnią zamknięty w niewielkiej książce.
ŚWIETNE - 8/10
Profesor matematyki Edward Scheinerman w „Przewodniku miłośnika matematyki. Arcydzieła dla każdego” pięknie opisał wybrane pojęcia, obiekty i sposoby analizy problemów, które są proste technicznie do wytłumaczenia, a jednocześnie każdego zapoznającego się z ich siłą wprawiają w zdumienie. Zagadnienia pogrupowane są w trzy bloki – o liczbach, o kształtach i o niepewności.
Po raz kolejny zastanawiam się jak zachęcać do sięgania po takie książki? Może tak. Czy wiecie, że są różne typy nieskończoności? Czy wiecie, że poza obiektami jednowymiarowymi i dwuwymiarowymi są takie, o wymiarach pośrednich, np. 1,8927… Czy wiecie, że liczby opisujące różne wielkości w przyrodzie (dowolne wielkości stanowiące wystarczająco liczny zbiór) zaczynają się cyframi tak, że istnieje między nimi bardzo jednoznaczna analityczna relacja częstości występowania? I ostatecznie – czy wiecie, że wytłumaczenie tych nietrywialnych ustaleń matematyki jest naprawdę proste i zrozumiałe dla każdego?! Scheinerman nie spieszy się, wszystko wyjaśnia dokładnie, dopowiada z innej strony, podaje pomocne grafiki czy tabelki z przykładami. Wszystko, by nie zostawić w procesie analizy zjawiska nikogo. Jaki jest sens liczby ‘pi’, ‘e’ czy cel wprowadzenia jednostki urojonej, w czym tkwi siła liczb pierwszych? – to wszystko też niespotykanie klarownie jest tu podane.
Jakie są praktyczne zdobycze po lekturze? Przede wszystkim satysfakcja, że dotknęło się matematyki od fascynującej strony, nawet w kilku miejscach tej współczesnej. Poza tym czytelnik dowiaduje się (albo sobie przypomina) o pięknie relacji tłumaczących związki między figurami czy liczbami, co przydaje się do budowania wyobraźni przestrzennej czy ułatwiania sobie życia w natłoku liczb. Poza tym warto chyba wiedzieć, czemu szyfry RSA na razie są bezpieczne (str. 39-40). Albo, że jeśli w medycznym teście na rzadką chorobę w którym wynik jest poprawny w 98% przypadków, a na którą zapada 0,1% populacji, dowiemy się, iż akurat u nas wynik jest pozytywny, to prawdopodobieństwo, że faktycznie dopadła nas ta choroba wynosi zaledwie 4,7% (str. 282-284)!! Jak mając do poukładania alfabetycznie 1000 prac studenckich, zrobić to optymalnie (str. 167-173)?
Ile wystarczy mieć książek na regale, by liczba możliwych kolejności ich ustawienia była większa od liczby cząstek elementarnych we Wszechświecie? Podpowiedź w książce (a sam rachunek to zaledwie kilka symboli).
Polecam „Przewodnik miłośnika matematyki” z pełnym przekonaniem. Opowieść zaczyna się od stwierdzenia: ‘były sobie liczby naturalne 1,2,3,…’ i kończy matematycznym dowodem na istnienie wolnej woli różniącej ludzi od maszyn (dyskusyjna jest dla mnie jednak semantyczna strona sformułowania tezy, bo nie ma jednej definicji ‘wolnej woli’, ale sam wywód autora - piękny).
Cały wielki świat przygody z wyobraźnią zamknięty w niewielkiej książce.
ŚWIETNE - 8/10
September 4, 2021
Perhaps how you approach popular math books is very much dependent on where math is in your life. Are you good at it? Did you like it in school, at least in elementary school? Did you study it as a part of your career training? Do you use it now? Is it your profession? Do you wish you knew more? Assuming that you are pretty good at it and you liked it at least at some point, there is a place for popular math books. If you wish you knew more, then there is place for a certain type, one that is both enjoyable, a pleasant read and fun, but also one that covers reasonably rigorously some ideas with which you are not familiar.
For TMLC, I’d give it a B+ for being enjoyable. There are few stories, parables, amusing use-cases etc. For reminding one why one ever like math (and a review of things you should never forget but almost certainly have), it gets an A. And for presenting a “greatest hits” of many different branches of math, it gets an A+.
So you don’t remember Euclid’s algorithm? Or you should know it but somehow can’t remember exactly how it works? Perfect. Arithmetic fundamentals? Check. New things and connections with Fibonacci numbers? Plus. Reading it a second time I skipped many pages, but there was a surprising amount that it was good to review. And a few completely new things, like hyperbolic geometry.
If there is any negative, it’s that it is not quite either of two models for this kind of book. The first, of which “The grapes of math” is the best example, is simply a blast to read. The second, of which “fractals, a very short introduction” is a good example, really means to teach you something particular. In TMLC, the fractal part is adequate, but doesn’t really get across the intuition behind it. That’s probably not the point of this book, which is more of a “greatest hits” – which is perhaps both its strength and its weakness.
For TMLC, I’d give it a B+ for being enjoyable. There are few stories, parables, amusing use-cases etc. For reminding one why one ever like math (and a review of things you should never forget but almost certainly have), it gets an A. And for presenting a “greatest hits” of many different branches of math, it gets an A+.
So you don’t remember Euclid’s algorithm? Or you should know it but somehow can’t remember exactly how it works? Perfect. Arithmetic fundamentals? Check. New things and connections with Fibonacci numbers? Plus. Reading it a second time I skipped many pages, but there was a surprising amount that it was good to review. And a few completely new things, like hyperbolic geometry.
If there is any negative, it’s that it is not quite either of two models for this kind of book. The first, of which “The grapes of math” is the best example, is simply a blast to read. The second, of which “fractals, a very short introduction” is a good example, really means to teach you something particular. In TMLC, the fractal part is adequate, but doesn’t really get across the intuition behind it. That’s probably not the point of this book, which is more of a “greatest hits” – which is perhaps both its strength and its weakness.
September 22, 2022
This book had many of the usual suspects when it comes to whirlwind tours of mathematics for a general audience - e, pi, i, the joys of geometry (Euclidean and non-Euclidean), paradoxes of voting, and counterintuitive results from joint and conditional probability. What sets it apart is how astonishingly well these things were explained.
The author doesn't skip steps or use jargon without first defining the terms. The puzzles and exercises left to the reader serve to reinforce understanding rather than simply reduce the page count while leaving the reader unenlightened. Be sure to try solving them.
If you love math, read this book. If you hate math, read this book. Either way, you will get fresh insights and a deep appreciation for these topics, perhaps for the first time.
The author doesn't skip steps or use jargon without first defining the terms. The puzzles and exercises left to the reader serve to reinforce understanding rather than simply reduce the page count while leaving the reader unenlightened. Be sure to try solving them.
If you love math, read this book. If you hate math, read this book. Either way, you will get fresh insights and a deep appreciation for these topics, perhaps for the first time.
May 9, 2018
I enjoyed reading this book. Though I did not under stand the math in the book as much as I want to. At least I am trying to learn about math now instead of saying to myself. Oh, math I can't do math and then running. I liked the chapter on primes. I also liked that he showed how a mathematician might make a proof. I think he did a proof in ever chapter. Fun to read. I thought that was interesting since I had no idea how a mathematician makes a proof. Now I just need to get better at math. A tough task for me. I love math but, don't understand math so well.
January 7, 2020
Simple and accessible exposition about some interesting math topics. A great resource for math curious kids. Books like this are a great way to get kids excited about math as the book poses math problems and concepts as puzzles. Being able to discuss the concepts and problems with others will help amplify the learning. Ideal gift for math curious kids and adults alike!
November 28, 2024
Most chapters are far from masterpieces. Instead, they are straightforward and dull, with the most exciting aspects of mathematics left out. Also, most of the material in the book has been written to death. However, the author clearly knows mathematics well, writes without error, and has given us a few stimulating chapters.
June 2, 2020
Pitched pretty much perfectly for me, easy enough to follow most of the time but never patronising. Love the messiness and seemingly counter intuitive results that can emerge when you pick on certain mathematical threads. Great fun.
May 14, 2019
Of the many books I have read of this genre, this is the BEST!
February 5, 2018
This book was wonderfully written and enlightening. With it only requiring High School Math the book is also really accessible to the common man. I enjoyed the little portions discussing Euler's Number and how they use Benford's Law.
Displaying 1 - 15 of 15 reviews