European Mathematical Society - paul j. nahin
https://euro-math-soc.eu/author/paul-j-nahin
enHot molecules, cold electrons
https://euro-math-soc.eu/review/hot-molecules-cold-electrons
<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>Paul Nahin is well known as a popular science writer. Some twenty books he has published since he started at the end of the previous century with a biography of Oliver Heaviside. Most of his books are dealing with topics involving physics, but there is always keen attention given to mathematics. For example he authored books on explicit mathematical topics like <em>An imaginary tale. The story of √-1</em> (1998) and <em>Dr. Euler's fabulous formula</em> (2006).</p>
<p>The present book is like a mathematical textbook for engineering or science students in which all the derivations are given. Nahin uses an historical approach to introduce Fourier analysis, derive the heat equation, and solve it for different geometries and boundary conditions. When applied to a cooling sphere, this illustrates how William Thomson (Lord Kelvin) estimated the age of our planet by computing how a molten sphere cools down to a sphere with a solid crust (that explains the hot molecules of the title). When the equation is solved for a long cable, it explains how electrons travel through the transatlantic submarine telegraph cable (hence the cold electrons).</p>
<p>So there are a lot of formulas and derivations, but it is not a course as it would be written in modern times. It is taken out of a regular university curriculum and it assumes only the basic calculus from a course at a first year science, engineering, or mathematics level. Fourier series and the Fourier transform are developed from basic principles. Nowadays, the heat equation can be solved efficiently using for example Laplace transforms, but Nahin prefers to use essentially the mathematics available to Fourier who solved it in the time domain. Every step is explained to the smallest details. Sometimes the approach is using an engineering style of mathematics. This means that Nahin is just using an insight from the underlying physics to propose a certain method or to justify a certain solution. Infinite sums and integrals are interchanged, postponing to when the eventual result is obtained whether this makes sense or not. The square root of minus 1 is however denoted by the mathematical standard i, and not by j as is customized by the engineering community to distinguish it from electric current which is also indicated by i or I.</p>
<p>This "engineering mathematics" is also what Fourier applied. His original report on the solution of the heat equation in 1807 was criticized by Lagrange and Laplace because he used his formally obtained infinite sums as if they were ordinary functions. It is not until his "new mathematics" was better understood, ten years later that he was taken seriously and was accepted as a member of the French Academy of Science. Chapter 1 is an eye opener to the sort of mathematics that Fourier introduced. It is for example shown how Fourier obtained $\frac{\pi}{4}=\sum_{k=0}^\infty (−1)^k\frac{\cos(2k+1)x}{2k+1}$. This is well known for $x=0$ (Leibniz formula), but there are many other values of $x$ for which this is also true, much to the surprise of Fourier's contemporaries.</p>
<p>In Chapter 2, the Fourier series are derived and it is shown that they are optimal approximations in a least squares sense. Convergence is not proved. Nahin asks the reader to "accept that our mathematician colleagues have, indeed, established its truth". In this way Fourier series, the Parseval identity, Dirichlet's integral, and the Fourier transform are introduced.</p>
<p>Chapter 3 derives the heat equation $\frac{\partial u}{\partial t}=k(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2})$ from first principles. When the medium is a long radiating cable, it is essentially one dimensional and a simple solution is found as a decaying exponential assuming a constant energy loss per unit length, not depending on time. The solution of the equation for different geometries and different physical boundary conditions is discussed in the next chapter. It starts with a cooling problem of an infinite slab with finite thickness ($0\le x \le L$) using a separation of variables ($u(t,x)=f(t)g(x)$) as Johann Bernoulli did. This results in an infinite series with terms of the form $\exp(−ak^2t)\sin(bkx)$ which has to satisfy the boundary conditions. Next, the spherical problem is solved. Assuming isotropy for a sphere, it becomes one-dimensional in the radius $r$. This problem was solved by Lord Kelvin when he applied it to a cooling Earth, which however drastically underestimated its existence to 98 million years because he did not know about radioactive decay or tectonic plates. Next is the solution in a semi-infinite medium with infinite thickness. This is the first case of the slab where the thickness $L$ goes to $\infty$. This is an occasion to show how the Fourier series used for finite $L$ migrates into the Fourier transform when $L\to\infty$. The heat equation is also solved for other cases like a circular ring and an insulated sphere These were also discussed by Fourier in his <em>Théorie analytique de la chaleur</em> (1822), although the last one did not result in a Fourier series.</p>
<p>Chapter 5 starts with a crash course on electrical circuits: resistors, capacitors, inductors and Kirchoff's laws and describing the behaviour of electrons in an electrical field. And lo and behold, the electrons in a one-dimensional semi-infinite induction-free telegraph cable behave according to the heat equation, again an ingenious insight of Lord Kelvin. Solving that equation was a theoretical achievement, producing the cable and letting it sink to the bottom of the ocean was a risky and adventurous enterprise. In this book, that technological adventure is only lingering in the background. A nice account of this adventure can be found for example in the book <em><a target="_blank" href="/review/mind-play-how-claude-shannon-invented-information-age">A Mind at Play: How Claude Shannon Invented the Information Age</a></em> by J. Soni and R. Goodman (2017).</p>
<p>Heaviside also features in the last chapter discussing the evolution after the 1866 Atlantic cable was realized. He added the inductance to the heat equation which turns it into a wave equation (actually the telegrapher's equation describing traveling waves in transmission lines, smartly solved by d'Alembert). That removes the assumed instantaneous action at a distance in the heat equation, which was causing a diffusion of the signal. The parameters of the cable can be controlled to remove that effect and this improved the usefulness of the cable considerably. Nahin ends by discussing the computation of how an arbitrary signal is transmitted. The diffusion however destroys the information during the transmission. This is illustrated by a matlab program that computes this deformation. The short code is given so that you can try it out yourself. The example shows that the signal is unrecognizable, it can still work though for a binary signal since the only information that one needs to detect is whether or not a bit is zero or one. We can also read how Heaviside explained the asymmetry of the transmission time: a message sent from England took longer than a message sent to England.</p>
<p>The sources used by Nahin, and some additional historical notes are listed at the end of the book, organized per chapter. There is no separate bibliography but there is an index that includes references to these notes. He has also one appendix about Leibniz's formula, i.e., how to compute the derivative of an integral if the boundaries of the integral are varying.</p>
<p>The book confirms what is already known from his previous books: Nahin knows how to write a book mixing physics and (a lot of) mathematics and (still) make it readable for a (relatively) broad public (with only some basic mathematical knowledge). The mathematics in this book certainly take the leading role like it does in lecture notes about the solution of differential equations. Nahin takes his time to explain everything and derive things from the very basics. When the mathematics become too involved or advanced, he uses intuition and asks the reader to accept and believe the result. The hard core mathematical mind may have some problems with his "engineering approach", but it works perfectly well for a first introduction. Anyway, from the historical perspective, this approach was used by the people who originally developed the theory.</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>Nahin introduces us through an historical approach to Fourier series, Fourier transforms, and how Fourier used this to solve the heat equation. Lord Kelvin used the heat equation to model the cooling of the Earth and hence estimate its age and he, and others, solved essentially the same equation to model the flow of electrons in the transatlantic telegraph cable.</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/paul-j-nahin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">paul j. nahin</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/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</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">2020</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">9780691191720 (hbk)</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">USD 24.95 (hbk)</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">232</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><li class="vocabulary-links field-item odd"><a href="/imu/partial-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Partial Differential Equations</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="https://press.princeton.edu/books/hardcover/9780691191720/hot-molecules-cold-electrons" title="Link to web page">https://press.princeton.edu/books/hardcover/9780691191720/hot-molecules-cold-electrons</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/35-partial-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35 Partial differential equations</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/35k05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35K05</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/42a16" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42A16</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/35s30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35S30</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/35l05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35L05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/35k57" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35K57</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/94c05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">94C05</a></li></ul></span>Tue, 26 May 2020 16:18:52 +0000Adhemar Bultheel50806 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/hot-molecules-cold-electrons#commentsHow to Fall Slower Than Gravity
https://euro-math-soc.eu/review/how-fall-slower-gravity
<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>
Paul Nahin is known for his many books written to popularize mathematics, but readers familiar with his work, know that there is always quite some mathematics involved, and it is not always the simplest of problems or computations that he describes. This book is a sequel to <a href="/review/praise-simple-physics-science-and-mathematics-behind-everyday-questions" target="_blank"> <em>In praise of simple physics</em></a> in which he introduces, like in this one, problems from physics with solutions. The MacGuffin for this book, as Nahin writes in the preface, is a letter published in the <em>Boston Globe</em> in which it is contested that in an exam for college placement it is required to know about quadratic equations. The title was <em>Who needs to know this stuff?</em>. Instead of writing an angry answer, Nahin wrote this book to illustrate that mathematics and physics in combination with basic laws of physics can solve real life problems. As a motto for the book, it opens with a problem of Lord Rayleigh from the 1876 mathematical tripos in Cambridge. Those who excelled in these exams had a bright future, whatever they chose as their profession. I doubt that this book will convince the authors of the letter in the <em>Globe</em> that quadratic equations are useful to know for everybody, but if the reader has taken calculus and physics courses at an introductory level, and is intrigued by the power of mathematical physics, then this book will give nice examples of what is possible and it has some challenges for the reader too. This is to illustrate that anyone who had these elementary courses (which is about anybody whatever his or her later profession turns out to be) should in principle be able to solve such problems. Several examples are already proposed in the long preface, which sets the tone for the rest of the book.</p>
<p>
The bulk of the book consists of 26 problems (with variations). Some of these are classic and have appeared elsewhere. If they are simple exercises as in introductory courses, then the reader is mainly on his own to solve the problem. For the more demanding problems, Nahin goes through a discussion leading towards a solution and at the end leaves some challenges for the reader. Such a challenge can be a variation of what has been discussed or a step that has been skipped or a similar problem, or it is asked to write a computer program to check some of the numbers numerically. Usually the challenge is simpler than what he has gone through already. An extensive discussion of the solutions to these challenges are found in the second part. Not all solutions can be found analytically, so sometimes a short matlab code is included if necessary. These programs are not always written in the most elegant programming style, but most importantly they are quite readable and they just do what they need to do.</p>
<p>
To give an idea about what kind of problems are discussed, here are some examples.</p>
<ul>
<li>
The first one is a classic problem of launching a projectile over a wall. This typically involves a parabolic trajectory and thus the quadratic equation pops up.</li>
<li>
The second is a fun problem, seemingly impossible to solve: before noon, snow starts falling at a constant rate and a snowplow starts clearing a long road at noon at a constant volume per hour. The second hour it travels half as far as in the first hour. When did it start to snow? Hard to believe, but Nahin gives an analytic solution that results in the exact moment (up to the second) that it started to snow.</li>
<li>
There are some problems involving Monte-Carlo simulation because an analytic solution is infeasible, and hence programming a simulation is required.</li>
<li>
On the other hand, there are problems related to combinatorics where straightforward programming is excluded because numbers become too large (unless extended precision is used), and so these require the analytic derivation of asymptotic formulas.</li>
<li>
A classic more involved example is to find the tangential speed and time when someone falls off a slippery log (assumed to be a perfect cylinder).</li>
<li>
A 1967 paper of Nahin is recycled in a discussion of NASTYGLASS. That is theoretical glass that acts as a filter cutting off all electrical power below a certain threshold but leaving intact what is larger. Looking at a nice picture through this glass is supposed to make it ugly (hence the name).</li>
<li>
The problem described by the book's title is about the physics of a raindrop that is accumulating mass as it falls though a humid environment. In its simplest form it will accelerate at only one quarter of the gravitational constant.</li>
<li>
As we progress in the book, the problems become more involved with longer elaborations by Nahin. Some earlier problems return like rocket launch but now launching underwater, and it is explained how <em>Enola Gay</em> could launch the atomic bomb and escape the blast.</li>
<li>
It is shown how to compute ζ(6) using only undergraduate mathematics assuming Fourier series and the Dirac impulse are known (which he supposes to be available at the end of the undergraduate level).</li>
<li>
He also connects ζ(s) to prime numbers and cryptography. This connection can be verified experimentally by computing with a simulation the probability that two (or more) randomly selected numbers are coprime.</li>
<li>
After an excursion via cubic and quartic equations, in the last problem, the quadratic equation turns up again in a model to detect a fault in an undersea cable and in a Wheatstone test bridge.</li>
</ul>
<p>
In some appendices, extra material is provided about continued fractions, and the problem by Lord Rayleigh mentioned in the beginning of the book.</p>
<p>
</p>
<p>
The different problems can be considered independently in most cases, although there are cross references for some. What Nahin offers is a bit of an unusual mixture of explanations and analysis of physical phenomena and some related problems for the reader. Much is in the style of his previous book <em>In simple praise of simple physics</em>. Whether we should classify all the discussed problems as "practical" in the sense that people are confronted with these in everyday life is highly disputable, but they all do involve basic laws from physics (although somewhat simplified) and it is illustrated (at this elementary level) how mathematics helps a physicist to solve such problems. As Nahin writes in his preface: "Millions of students are enrolled worldwide in calculus and physics courses, and the majority of them will not become mathematical physicists, but this does not mean that they cannot enjoy the power of mathematics making sense of a physical world".</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>
As a sequel to his book <em>In praise of simple physics</em>, Nahin discusses 27 more problems from physics that can be solved using relatively simple mathematics and some elementary physical laws. He leaves some challenges for the reader for which he gives solutions in the second part of the book. The title of the book refers to the problem describing a raindrop collecting more mass as it falls through a humid fog.</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/paul-j-nahin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">paul j. nahin</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/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</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">2019</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">978-0-6911-7691-8 (hbk); 978-0-6911-8502-6 (ebk)</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">USD 27.95 (hbk)</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">320</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/mathematical-physics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematical Physics</a></li><li class="vocabulary-links field-item odd"><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="https://press.princeton.edu/titles/13262.html" title="Link to web page">https://press.princeton.edu/titles/13262.html</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/00a79" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a79</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/65z05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65Z05</a></li></ul></span>Wed, 14 Aug 2019 13:42:26 +0000Adhemar Bultheel49641 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/how-fall-slower-gravity#commentsIn Praise of Simple Physics: The Science and Mathematics behind Everyday Questions
https://euro-math-soc.eu/review/praise-simple-physics-science-and-mathematics-behind-everyday-questions
<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>
Paul Nahin has written sixteen popular math books so far. They deal with different topics: biographies, mathematical history, logic, games, probability, computation, philosophy, etc. About half of them have reviews in the EMS book review database. For the present book he collected a number of cases of physical problems that are 'behind everyday questions' as the subtitle says. Not that this kind of questions are essential to survive from day to day, but they could indeed pop up in a casual conversation or they could be raised by a curious somebody during a socializing discussion. What is meant is that you do not need to be working at CERN or be a particle physicist to wonder about the problems that, usually in a simplified but rigorous form, are discussed by Nahin in this book. They could for example be exercises in a physics course or be illustrations in a mathematics course. The physics required are elementary such as Archimedes' principle, Ohm's law, Newton's laws of motion, conservation laws of energy and momentum, calculating the center of mass and the momentum of a system. But their meaning and definition are recalled when needed. On the mathematical side some algebra, trigonometry, vectors, and calculus are used. With Einstein and Zemansky in mind "the problems are kept as simple as possible, but not so simple that they are simply wrong". The book also wants to illustrate that G.H. Hardy's quote "[Knowledge of] a little... physics... has no value at all in ordinary life" is definitely wrong and that mathematics is not "just a bunch of theorems, proofs and boring multiplication tables".</p>
<p>
Nahin starts his book with a chapter that list some short teaser problems. Some are solved other are solved in later chapters. We find also the solution of a problem formulated in note 14 of the preface (which actually should be note 15, small typo). Every chapter ends in a section with endnotes. Such an endnote can give a reference or a short bio of a person, or some history of the problem, or some extra comments about the physics or mathematics. And then there follow 22 more chapters with a bunch of problems that are solved. Some examples: If you see a traffic light switch from green to orange, should you hit the break or hit the accelerator? Is it the Moon or or the Sun that exerts the greatest gravitational force on Earth and hence is responsible for the tides? And while tides come into the discussion, how much energy can be extracted from moving water, and how much from moving air? Pressing ecologically relevant questions! With a small energy input, a sequence of monotonically larger dominos can be tumbled over, how large is the energy amplification for this process? At what altitude should a geosynchronous satellite be brought in order to stay there by gravitation only? Why is the sky dark at night? This is a classic. If you throw some volume out of your boat, floating on a pond, will the water in the pond raise or not? How to hit a target that is uphill from you with a canon ball? How can you travel in a vacuum tube from New York to Melbourne in 44 minutes? What are the physics of a ski jump, a Tarzan swing and a bungee jump? How many different experiments can you perform at home to measure gravity? And there are many more problems like this. I am sure that at least one of these questions, and probably more than one, must have caught your attention or has started you thinking. The full solution (at least in a simplified form) can be found in the book.</p>
<p>
The last chapter is slightly different. It analysis a problem formulated by Newton and for which he gave a wrong solution. It is different in the sense that it is not 'practical' but it is rather a thought experiment. The problem is: How long does it take for two spheres, each with the mass of the Earth, that are a quarter of an inch apart, to touch each other when only subject to gravitational force. Newton gave no computations but claimed that it would take more than a month. Nahin gives the computations and finds it would take only 336 seconds.</p>
<p>
Each of the chapters do indeed involve only elementary physics and are worked out in all detail. Often the problems have some historical background which is carefully researched and communicated. There are of course a lot of formulas and computations, which leaves relatively little room for storytelling in between, but the introductions have some humor to make it more lively. Nahin gives much attention to the proper units and checking that all quantities do have the proper dimensions (dimensions in a physical sense that is). This is something that mathematicians (or mathematics students) are not so familiar with. So Nahin is right in emphasizing this aspect in all his computations. Unfortunately, the book is written for the reader who is not used to the mks-system but in daily life thinks and reasons in inches, feet, miles, pounds and the likes. Since Nahin's intention is to consider problems from the reader's 'everyday's environment', he uses these English units to give a better feeling of what the results really mean. So there is a lot of unfortunate unit conversion going on. But anyway, like in his other books, he knows how to catch the attention of his reader. You will not regret buying any of his books, and I am sure after reading it, you will pick up this one to check again on one of his models and his solution methods.</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>
Using only elementary physics, Nahin solves many fascinating problems, and answers questions that one can formulate by just looking around in your dayly life and being curious. The book illustrates marvelously that with some elementary mathematics and simple physics one may solve problems that are well above the obvious and may lead sometimes to surprising results.</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/paul-j-nahin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">paul j. nahin</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/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</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">2016</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">9780691166933 (hbk)</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">USD 29.95 (hbk)</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">272</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://press.princeton.edu/titles/10690.html" title="Link to web page">http://press.princeton.edu/titles/10690.html</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/00a79" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a79</a></li></ul></span>Tue, 05 Jul 2016 13:49:48 +0000Adhemar Bultheel47038 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/praise-simple-physics-science-and-mathematics-behind-everyday-questions#commentsAn Imaginary Tale: The Story of √-1
https://euro-math-soc.eu/review/imaginary-tale-story-%E2%88%9A-1
<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>
This is a reprint in the <em>New Princeton Science Library</em> of a classic. The series brings reprints in cheap paperback and eBook format of classics, written by major scientists and makes them available for a new generation of the broad public. The series includes not only math books but covers a broader area, although there are several mathematics classics in the catalog written by E. Maor, J. Havil, and R. Rucker, but also J. Napier, A. Einstein, O. Toeplitz, R. Feynman, S. Hawking, R. Penrose, W. Heisenberg, etc. So if you missed out on some of the original editions, or were not even born at that time, this is a chance to get one of these more recent reprints. Another classic, reprinted at about the same time is Maor's <a href="/review/e-story-number"><em>e: the story of a number</em></a> which follows a similar idea that may have inspired the current author.</p>
<p>
The current reprint is of the first paperback edition of 2007 which is an updated version of the original from 1998. Paul Nahin is an electrical engineer who wrote several successful popular science books. His first one was a biography of Heaviside, and this book about complex numbers (it contains even an introduction to complex functions) was his second. Several other were to follow, some of which have been reviewed in this EMS database: <a href="/review/chases-and-escapes-mathematics-pursuit-and-evasion"><em>Chases and Escapes. The Mathematics of Pursuit and Evasion</em></a> (2007), <a href="/review/digital-dice-computational-solutions-practical-probability-problems"><em>Digital Dice. Computational Solutions to Practical Probability Problems</em></a> (2008), <a href="/review/number-crunching"><em>Number Crunching</em></a> (2011), <a href="/review/logician-and-engineer-how-george-boole-and-claude-shannon-created-information-age"><em>The Logician and the Engineer. How George Boole and Claude Shannon Created the Information Age</em></a> (2012), <a href="/review/holy-sci-fi-where-science-fiction-and-religion-intersect"><em>Holy Sci-Fi! Where Science Fiction and Religion Intersect</em></a> (2014), Of course complex numbers and functions are important tools in electrical engineering. The book has a strong historical component of course, but, unlike the book by Maor about the history of the number e, this book has much more mathematics in it. Hence it requires some mathematical affinity to understand much of what is presented here. It requires the knowledge of advanced secondary school or even freshman's university level, in particular when it turns into an introductory course on complex functions in the trailing chapter. Nevertheless, Nahin avoids a textbook structure of definitions, theorems and proofs, but keeps the level of a casual account, cheering up the reader with some witty remarks now and then.</p>
<p>
The historical background starts with the solution of the cubic equation and the search for a formula that gives its roots, which was a hot topic in the 16th century. Knowing such a formula was a strong weapon for `mathematicians' that made a living as (human) computers, so it was important not to share it with competitors. Obviously this includes the well known story of del Ferro who knew how to solve the cubics and who told it to Antonio Fior before he died. Niccolo Fontana, better known as Tartaglia, also knew how to solve them. It came to a public duel between the latter two to solve the most equations in a given time. Tartaglia won much to Fior's surprise. Cardano who was a well respected mathematician in those days, stalked Tartaglia to tell him the magic formula, and after much pressure, Tartaglia eventually told Cardano, but made him promise to keep it a secret. However Cardano could consult the letters by del Ferro and considered his promise to Tartaglia not valid anymore and published the result anyway, which resulted in a vigorous priority fight.</p>
<p>
Solving cubic equations is important for the history of complex numbers because the square roots in the relatively complex formulas could give complex conjugate solutions. However, even though the square roots of negative numbers made no sense to them, it turned out that when computations were performed as if these were genuine numbers, this could lead to valid results, which was most puzzling at first. The formulas are known as Cardano's but historically this is clearly a mistake. They were re-discovered a couple of times by others (e.g. Leibniz).</p>
<p>
Since in antiquity (think of the Greek) many computations were done by geometric constructions. Descartes, Wallis, Newton, and others thought about a construction of the square root with compass and ruler. In geometric constructions it is difficult to give a meaning to a negative number when it concerns the length of a line segment or the area of a polygon. It was not before Bombelli had the idea of drawing a line with marks for the numbers (which we now call the real line) that negative numbers finally made sense. It was even more staggering in those days to make geometric sense of an imaginary number. Some possible interpretation was that a line intersected another one outside an certain interval, which made the intersection `imaginary'.</p>
<p>
Of course, as we know, the proper interpretation of a complex number should be a point in the complex plane. That idea and the modulus-argument representation of the complex numbers with a complex exponential was first given by the Norwegian Caspar Wessel in 1797. He was not even a professional mathematician and succeeded where many great minds had failed before him. His finding went unnoticed though, until much later. Argand and Buée came to the same solution about a decade later and published their findings almost simultaneously which started another row between them. Again these names do not sound very familiar to us. Argand was a French amateur mathematician and Buée was a French priest who published a confusing, almost mystical paper on the subject. Just like Wessel's, their results faded away and were only re-discovered much later. Of course once the complex plane is accepted, one gets the goniometric representation, the formulas of De Moivre, the multiplication with i corresponds to a rotation over 90 degrees, etc. This is close to the vector interpretation by Hamilton, who finally gave a formal definition of the complex number field where a complex number was a couple of reals with vector addition and scaling and with a particular way of multiplying the vectors.</p>
<p>
Once the complex playground has been fixed in the first three chapters, the next three chapters deal with applications of complex numbers. Since the complex numbers are like vectors in a plane, some geometric problems can be easily solved with complex arithmetic. A less known theorem of Cotes and a puzzle problem from Gamow's book <em>One, Two, Three... Infinity</em> are given as examples. Other applications discussed include the `imaginary' time axis in the space-time indefinite metric of relativity theory, the maximal distance of a random walk with decreasing step sizes, Kepler's laws, and electrical circuits. More mathematics is found in the chapter on Euler and the famous Euler formula (exp(ix) + 1 = 0) but also infinite series, infinite products, the calculation of i to the power i, and even the gamma and zeta functions. The final chapter is an introduction to complex functions, derivatives, contour integration, Cauchy integral theorem, Green's theorem, analytic and harmonic functions.</p>
<p>
Concerning the structure of the text, I can mention that it is occasionally interrupted by `boxes' that discuss some topic closely related to the surrounding text, but that is not essential and could be skipped without a problem. On the other hand, some technical material is moved to appendices. The reprint is the unaltered version of 2007. That means that no additions or corrections are added since and the little defects that remained are still there. For example, the strange looking capital gamma symbol on page 177 (it is at least a different font from the surrounding pages) is still there. The preface of 2007 does explain what corrections and additions were made on that occasion. This book is not recommended if you are allergic to formulas, but if you want to peek behind the formulas and theorems in a textbook (a textbook is more `to the point' and hence necessarily `duller'), this is a the book that I recommend to read. You will definitely enjoy it. In fact it clearly reflects the the joy and delight that the author experienced when he was confronted with complex analysis during his engineering studies. </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 paperback reprint in the New Princeton Science Library of the bestselling original from 1998. It tells the story of the square root of –1, and that includes complex numbers, how they came about and what they can be used for. At the end there is even a brief introduction to complex functions. </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/paul-j-nahin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">paul j. nahin</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/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</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">2016</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">9780691169248 (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">£11.95</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">296</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/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li><li class="vocabulary-links field-item odd"><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://press.princeton.edu/titles/9259.html" title="Link to web page">http://press.princeton.edu/titles/9259.html</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/01-history-and-biography" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01 History and biography</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/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</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/01a40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A40</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a45" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A45</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/01a50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A50</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a55" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A55</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/30-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">30-03</a></li></ul></span>Wed, 27 Apr 2016 09:27:50 +0000Adhemar Bultheel46900 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/imaginary-tale-story-%E2%88%9A-1#commentsHoly Sci-Fi!: Where Science Fiction and Religion Intersect
https://euro-math-soc.eu/review/holy-sci-fi-where-science-fiction-and-religion-intersect
<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>
Springer Verlag has started a new series <em>Science and Fiction</em> that wants to bring popular science (in relation with literature) as well as novels (with a strong science plot). This sounds very much like Science Fiction (which it is in several cases) but, as far as I understand, the "science" is supposed to be dominant and the "adventure" that is often predominant in SF literature is pushed to a second level.</p>
<p>
The eight titles in the series currently listed on the Springer website includes books with speculative essays about space travel, genetic manipulation, etc, but also novels. These novels have appendices with a discussion of the underlying science aspect which can be psychology, medicine, biology, physics, etc. The book by Nahin is not a novel but explores how religion was dealt with in SF literature. Paul Nahin is best known among mathematicians for his popular books dealing with mathematics or probability (<em>An Imaginary Tale, Dr. Euler's Fabulous Formula, Duelling Idiots, Digital Dice, Number Crunching</em>,..., mostly published with Princeton University Press). However mathematical physics and SF are never far away with Nahin. It may be less known that he also has committed some SF short stories, but more prominent exponents of his love for science and SF is his 1993 book on <em>Time machines</em> and in 1997 he wrote one on <em>Time travel</em>. So one might expect this book to be in the same vein. If you are primarily expecting a lot of physics and mathematics, you will be disappointed though. The main goal, as the subtitle of the book says, is to highlight the religious aspects, questions, and dilemmas that one may encounter in SF literature. The relation is quite obvious if one starts thinking about what is "out there", when we humans are placed in a cosmological context, or what if other, non-humanoid civilisations exist? Did God create them? With a soul? Do humans have a soul anyway and what does that mean? So there is a lot of philosophy and metaphysics going on in this book. If there are no answers to all the questions, at least it makes you think about them. Many SF authors did tackle such questions in their work. Nahin is obviously well read in SF literature and there are many summaries and references, not only to SF novels but even more so to a long list of short stories. Many of them are classics from the early days of SF. But Nahin is still Nahin and he cannot deny his background. So there is still some science and mathematics, often in the beginning of the chapters.</p>
<p>
In an introductory chapter Nahin declares himself as an agnostic to hedge against criticism and strongly claims not having the intention to be offensive at any moment. He also gives some appetizers for topics to come, like in a SF novel by C. Sagan where a secret message from God is given because somewhere in the sequence of digits of pi, there is a sequence of zeros and ones, that when printed on a matrix printer as a square array, it forms the picture of a circle. Not all examples are as funny, but it gives an idea.</p>
<p>
Next, a sketch is given of the start of SF in pulp magazines at the end of the 19th century, although there were novels that preceded (H.G. Wells, Jules Verne and others). There is Voltaire's <em>Micromégas</em> and even Kepler wrote about a trip to the moon in <em>Somnium</em> that was published posthumously. Nahin summarizes several examples of early SF stories having a strong "religious" component.</p>
<p>
Then the subjects are discussed in a more systematic way in the subsequent chapters. For example, the title of chapter 3 is "Time, space, God's omnipresence, and free will". The <em>Time machine</em> by H.G. Wells was about the first using time travel, and he was certainly not the last. Since Einstein, we know that time and space are not independent and that at least in theory, time travel to the past is possible. However, it involves travelling faster than light (FTL). It has not been done, but what if practical restrictions could be solved. Exploring the consequences of these "what ifs" is what SF is all about. We certainly would end up with paradoxes. If at time t we end up in situation A, can I go back in time to change the initial conditions to not end up in A. Well no! because we ended up in A, otherwise I would not have gone back. Does that eliminate free will then? Does the finite speed of light prevents God's omnipresence? All these themes were discussed in several SF stories, of which Nahin gives several examples.</p>
<p>
In the next chapter ponders on the question whether intelligent robots can become "human" or even God. The beginning lies in the theoretical developments by Norbert Wiener and Alan Turing. If a machines passes the Turing test, it can not be distinguished from a human, and numerous are the SF stories where a supercomputer takes over the power and pretends to be God or rather the Devil or at least it pretends to be an almighty human usually with bad intentions. If it is not human, then what is the difference? Is it lacking a soul? Then what is this soul that makes the difference?</p>
<p>
The next topic is space travel, and encounters with aliens, close encounters or by radio contact. Given the astronomical amount of possible earth-like planets and the time the universe exists, the probability that advanced civilizations exist is quite high. However, we did not have contact. <em>The Great Silence</em> still rules. Why? Where are they? Or where is the physical evidence that we have been visited? That is the <em>Fermi paradox</em>. SF writer Stanislaw Lem (one of Nahin's favourites since he is cited a lot) states somewhere that it is remarkable that mathematics, the abstract product of our intellect, is able to describe our physical observations. However, the match is only local, in our "bubble" of physical laws. With our growing understanding and the ability of mathematics to describe reality, our bubble grows larger. When these bubbles of different physical laws meet each other, we get a Big Bang. The <em>Great Silence</em> is a consequence of the impossibility to communicate between bubbles. What we believe to be impossible now (e.g. time travel) is because we have not understood enough the proper laws of physics, or at least mathematics was not able to describe physics well enough in all its details. Nevertheless, answering the "what if" question, alien encounters is an often returning issue in SF literature and movies.</p>
<p>
Once more in the next chapter Nahin returns to time travel. Isaac Asimov, one of the best known SF authors rejects it, just because the paradoxes make it impossible. Also Hawking doesn't believe it is possible, although Kurt Gödel has shown that it is theoretically possible according to present knowledge of physics. But Hawking conjectures that improved physical knowledge will show it impossible which he calls the <em>Chronology Protection Conjecture</em>. It is actually a reformulation of <em>Niven's Law</em> after the SF writer Larry Niven who formulated it 20 years earlier. The topic of this chapter is whether we, or whether God, can change the past. For example could we go back and prevent the crucifixion of Jesus? Would we not go against God's will because it is exactly his intention to die on the cross for the salvation of humankind. Perhaps the crowd in those days consisted mainly of time travelling tourists who wanted to attend "the real thing"?</p>
<p>
The last chapter discusses another what-if question: What if God revealed himself? Aldous Huxley in <em>Point Counter Point</em> has a character mathematically prove the existence of God using zero times infinity is finite, translated as an almighty can create something from nothing. Many people, among which SF authors, have discussed this topic of which Nahin relates. The bit string in the digits of pi that I mentioned earlier is just one example.</p>
<p>
In an appendix some game-theoretic paradox is explained and in others a number of short stories are (re)printed. The book is also available in e-form, each chapter having a different DOI. That seems to come at the expense of a less nice typesetting for the printed version. There are many references included as footnotes. These do not affect the fixed text height so that is okay. There are a few illustrations that do not seem to float properly. They are inserted following a particular sentence and if they do not fit at the bottom, they are moved to the next page possibly leaving a large blank space at the previous page as on p.184. There is even a large blank space without an obvious reason (p. 130). Much more frequent inserts are quotes which are left and right indented and typeset in a smaller font but on p.117, it happens to be in a tiny footnote size font. When reading an e-version these flaws might not be so obvious, but it is somewhat disturbing in a printed version. Of course, this does not diminish the intrinsic value of the contents which clearly illustrates that science and fiction really meet in SF and when deeper and fundamental questions are at stake, then philosophy, and yes indeed, also religion, in not far off.</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>
Paul Nahin is the author of several books on popular mathematics and physics. He is also a lover of science fiction and author of a number of SF stories. In this book he explores the relation between science fiction literature and religion, connecting the three topics: science, fiction, and religion.</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/paul-j-nahin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">paul j. nahin</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/springer-verlag-0" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">springer verlag</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">978-1-4939-0617-8 (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">21,19 €</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">242</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.springer.com/978-1-4939-0617-8" title="Link to web page">http://www.springer.com/978-1-4939-0617-8</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/00a99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a99</a></li></ul></span>Thu, 12 Jun 2014 09:21:48 +0000Adhemar Bultheel45567 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/holy-sci-fi-where-science-fiction-and-religion-intersect#commentsThe Logician and the Engineer. How George Boole and Claude Shannon Created the Information Age
https://euro-math-soc.eu/review/logician-and-engineer-how-george-boole-and-claude-shannon-created-information-age
<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>
The logician George Boole (1815-1894) and the (electrical) engineer Claude Shannon (1916-2001) are briefly introduced with two short biographies because, as the subtitle of the book says "they created the information age". Their work was indeed influential for the development of digital computers and the "basics of the evolution of (digital) computers" is what this book is about. Boole's masterpiece was entitled <em>An investigation of the laws of thought</em> (1854) and Shannon's main work was <em>A mathematical theory of communication</em> (1948). The author, being an electrical engineer himself, chooses these scientists as the protagonists of the early information age in view of the main evolution that he sketches in this book: from elementary circuits to (abstract) computing devices.</p>
<p>
Seeing this evolution with the eye of an electrical engineer, explains why a great deal of the attention goes initially to Boolean calculus and to the design of logical switching circuits which is how the Boolean algebra can be implemented. Throughout the book, the deeper mathematics and the physics at the level of electrons are avoided. The circuits are merely graphical schemes of which it is known that they can be realized by hardware, but the hardware itself stays in the background.</p>
<p>
Some elements of probability theory are introduced because Boole discussed conditional probability, which was done earlier by Bayes, and also Shannon was interested because he needed it to study circuits with noisy (he calls them "crummy") relays. After that chapter, Boole is fading a bit to the background and slightly more advanced topics are discussed like Shannon's information entropy, the capacity of a channel, and error correcting codes. In the latter Shannon has also done something, which makes him a precursor of what Hamming did later. From there on, Nahin takes off to the design of (abstract) computational devices. First the principles of sequential state machines. Even some circuits for their realization are introduced. Then the tale moves on to Turing machines grasping the occasion to discuss countable and uncountable sets. I wonder why a short biography of Alan Turing is not included along with the ones of Boole and Shannon. At the same level as the logician and the engineer, he could have been the programmer or the cryptologist. That might not have done unjust in view of the concept of this book as a whole. The final step taken is a discussion about the physical constraints of what is possible with traditional digital computing. For example how much energy one would have to dissipate as heat when one would be able to simulate a human brain with traditional circuits. That clearly explains the boundaries of what is possible and what is not, even in an ideal situation. The only alternative is to look for a completely different model for our computing devices. A quantum computer might be the alternative we have to look for. So the principle of qubits is explained. If quantum computing is futuristic, then why not think about time-travel. Traditional or quantum computers, can be set to work and after long computationsin the future and after obtaining the result, give us via time-travel the answer right now.</p>
<p>
Although the book is technical, it is always easily understandable for anyone (for those who need it, some basic rules for electrical circuits are collected in a short appendix). It is not only understandable but also pleasantly bantering and at occasions even facetious. For example, an epilogue on a fictitious machine trying to translate and simplify the totally unnatural language used in legal texts. Nahin has quite some reputation in writing books about popular mathematics as testified by his 12 previous books. But it is also instructive because along the road, the reader is clearly explained many of the concepts that is probably only vaguely familiar to a broad public like for example public key cryptography, Moore's law, lab-burn, uncertainty principle, bilking paradox, etc. It remains to mention that each chapter ends with a rather extensive section with "Notes and references" that give indeed references of some nuances of the main text.</p>
<p>
So let me round up remarking that the title should not give the impression of a story for children as a similarity with Edward Lear's poem <em>The owl and the pussy-cat</em> might suggest. Neither should the title create the expectation that the book is about the live and work of Boole and Shannon. Of course, there is the short biography, and the ideas explained in this book are direct consequences of what these fathers of information theory did. However the contents is about circuits and computing devices, and not a (romanticized or not) extensive biography of the two men.</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>
Starting with the short biography of Boole and Shannon, the book moves to a gentle introduction to elements and concepts of Boolean algebra, logical switching circuits, sequential-state digital machines, Turing machines, and even the principles of quantum computers.</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/paul-j-nahin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">paul j. nahin</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/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</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">2012</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">97-069-1151-00-7 </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">248</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://press.princeton.edu/titles/9819.html" title="Link to web page">http://press.princeton.edu/titles/9819.html</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/94-information-and-communication-circuits" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">94 Information and communication, circuits</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/94-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">94-01</a></li></ul></span>Wed, 05 Dec 2012 06:22:59 +0000Adhemar Bultheel45479 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/logician-and-engineer-how-george-boole-and-claude-shannon-created-information-age#commentsNumber-crunching
https://euro-math-soc.eu/review/number-crunching
<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>This is an entertaining book which comprises a variety of mathematical problems, mostly related to calculations, and addressed to students in a undergraduate level in either Mathematics, Physics or Engineering degrees. The common argumentative line of all topics in the book is the use of Matlab programs which solve or simulate the problems at hand.</p>
<p>The main topics included are:<br />
- The study of the Laplace equation in a region in the plane, mainly through the problem of finding the temperature of a plate given the distribution of temperatures of the border. This is solved via Fourier analysis, by discretizing, and also by a Monte-Carlo technique.<br />
- Some problems on dynamical mechanical systems, mostly the study of several configurations of hanging masses (with hooks), studying how the systems develop and issues related to periodicity.<br />
- The three-body problem, studying the stability of the system starting from different configurations, and giving a theoretical analysis and also numerical simulations. This chapter includes nice historical remarks on the problem.<br />
- Analysis of electric circuits, with a theoretical analysis, and also using computer simulations of circuits.</p>
<p>Some other fun problems are included in the book, most of them aimed to show how computers may help in finding or guessing a solution. Also several fictiional stories appear scattered along the book. Finally, each chapter includes a collection of challenge problems for the interested reader (with solutions appearing at the end of the book).</p>
<p>Altogether, this is a brain stimulating read, recommended for a general audience of mathematically minded people with interest in physics or in the use or programming to solve mathematical problems.</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">Vicente Muñoz</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">Universidad Complutense de Madrid</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 an entertaining book which comprises a variety of mathematical problems, mostly related to calculations, and addressed to students in a undergraduate level in either Mathematics, Physics or Engineering degrees. The common argumentative line of all topics in the book is the use of Matlab programs which solve or simulate the problems, to help finding or guessing a solution.</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/paul-j-nahin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">paul j. nahin</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/princeton-univ-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton univ. press</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">2011</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">978-0-691-14425-2</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">$29.95</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://press.princeton.edu/titles/9530.html" title="Link to web page">http://press.princeton.edu/titles/9530.html</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/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li></ul></span>Mon, 21 Nov 2011 20:47:33 +0000Anonymous45428 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/number-crunching#comments