European Mathematical Society - 26-01
https://euro-math-soc.eu/msc-full/26-01
enTrigonometric Delights
https://euro-math-soc.eu/review/trigonometric-delights
<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>In the first two sentences of the preface, Maor writes</p>
<blockquote><p>
This book is neither a textbook of trigonometry —of which there are many— nor a comprehensive history of the subject, of which there are almost none. It is an attempt to present selected topics in trigonometry from a historic point of view and to show their relevance to other sciences.
</p></blockquote>
<p>I could not think of a better characterisation of the book than this. All I can add to this description is to give an idea of which kind of topics were selected and what kind of applications have benefited from these developments.</p>
<p>The successive chapters are organised more or less chronologically, starting with a prologue about the Egyptian Rhind papyrus from around 16-17th century B.C. and ending with Fourier series (18th century). There is of course much attention for the history, but what strikes me in particular, is how much attention is given to the etymology of the mathematical terminology. The origin of the words algorithm and algebra is described in several publications as originating from the Arab author al-Khwarizmi and from al-jabr, which is part of the title of his book, but what is the origin of words such as sine, secant, and many other common mathematical words? Maor carefully pays attention to this. He also shows how trigonometry, which originally was about angles like in pyramid building problems in Egypt, were somewhat made more abstract, in a geometric context of triangles by the Greek, but later, it became more and more part of analysis. The sine and cosine were not only tabulated for computational purposes, but they became functions so that now we see x in sin x as a real or even a complex number, not necessarily corresponding to a physical or geometric angle. The original idea of an angle in degrees or radians in the goniometric unit circle has become somewhat obsolete.</p>
<p>But of course it all starts with angles and chords in planar circles for the Greek, and even earlier in astronomy, which is essentially a three dimensional spherical discipline as practised by Babylonians and almost any civilisation of antiquity. This is the subject of the first two chapters. Then appeared tables of goniometric values of what became our basic goniometric functions. This opens the possibility to introduce algebra (goniometric identities) and gradually also analysis (involving series) into the discipline. This helped considerably to discover (actually re-discover) the heliocentric interpretation of our solar system and to measure our own planet by triangulation and those practical problems in turn stimulated the development of associated trigonometric identities in triangles. But before the heliocentric model, the trajectories of the planets required also more general curves than ther circle like epi- and epo-circles which allow an easy description in terms of trigonometric formulas. Then Maor ventures into a period of proper analysis with the Sine integral and many other relations and series expansions, not in the least for the fascinating number π. These were obtained by master minds such as Gauss and Euler. As complex numbers entered the picture, with Euler's fabulous formula, we are fully involved in complex analysis, conformal maps and ultimately Fourier analysis.</p>
<p>This marvelous survey by Maor of some episodes in the historical evolution of mathematics also allows to sketch some biographies of important mathematicians. There are the "usual suspects" from Greek antiquity (including Zeno whose paradoxes are discussed when infinitesimals from analysis are introduced). Also Regiomontanus (15th C.), François Viète (16th C.), De Moivre (17th C.), Maria Agnesi and her "witch" (18th C.), Jules Lissajous (19th C.), Edmund Landau (20th C.) are discussed in somewhat more detail. aot only the history and mathematicians, also the applications are well documented: astronomy, cartography, spirographs, periodic oscillation, music; and there are detailed mathematical derivations of several trigonometric and other mathematical identities, conformal maps, series converging to π, the solution of the Basel problem by Euler, how Gauss showed that any trigonometric summation formula can be represented geometrically, etc.</p>
<p>All these items are treated requiring only some elementary trigonometric formulas. Some of the standard identities are collected in appendices. In another appendix we find Maor's plea to re-introduce the unit circle and the geometric definitions of the trigonometric functions like cos and sin being projections of the circular point on x- and y-axis, etc. instead of the "New Math" approach. Also Barrow's integration of sec x is moved to an appendix. All the chapters are completed with a section containing notes and references to the sources used. There are many useful mathematical graphs and some grayscale images.</p>
<p>This is an interesting mixture of mathematical history, illustrating the evolution and the usefulness of trigonometry throughout the centuries, and on top of that, it gives some mathematical training by deriving formulas and identities that are easily accessible with only some elementary mathematics knowledge. The book appeared originally in 1998 and is here reprinted in its original form as a volume in the Princeton Science Library. So this is a fortunate occasion to bring this great book back under the attention of a broad audience.</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 a reprint in the Princeton Science Library of the original book from 1998. Maor sketches several episodes on the history of mathematics where especially trigonometry was involved from the Rhind Papyrus to Fourier analysis. The history, the mathematicians, the applications, as well as the derivation of mathematical identities are discussed.</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/eli-maor" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Eli Maor</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-princeton" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press, princeton</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">9780691202198 (pbk), 9780691202204 (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 17.95 (pbk)</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">256</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="https://press.princeton.edu/books/paperback/9780691202198/trigonometric-delights" title="Link to web page">https://press.princeton.edu/books/paperback/9780691202198/trigonometric-delights</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/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a69" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a69</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/26-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">26-01</a></li></ul></span>Sat, 16 May 2020 14:47:19 +0000Adhemar Bultheel50783 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/trigonometric-delights#commentsWhere do numbers come from
https://euro-math-soc.eu/review/where-do-numbers-come
<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>There is a modern trend in calculus courses to start from application examples and practical problem solving, and from there come to some abstraction and theorems. In this book, Körner presents some basic results in a way that is the opposite of this. It is a strict mathematical top-down approach, formulating definitions, properties and theorems with hard proofs starting from a minimal set of axioms. As the title suggests, his guiding idea is to introduce the number systems on a pure axiomatic basis, loosely following the historical evolution. In this sense it is not really a replacement for a calculus course, but rather a complement to it. It does have the structure of a course text, with a sequence of definitions, theorems and proofs that is interrupted with many exercises and challenges for the reader-student.</p>
<p>One would expect that the topics to be covered and the order in which they are introduced are more or less clear. Nevertheless it is somewhat surprising that rational numbers come first in part 1, and natural numbers are a "special case" in part 2, and finally the reals and complex numbers in part 3. The quaternions and polynomials over a field follow as some kind of "encores".</p>
<p>Rationals without integers sounds almost impossible, but here the historical context comes in as a motivation. Originally people counted quantities and that does not really involve numbers in the sense that one added 3 sheep ad 4 sheep to get 7 sheep, but adding 3 sheep to 4 apples was not something to consider. So, to come to abstract counting numbers, the first thing to do in a Greek tradition is to find an axiomatic system for what we call now the positive natural numbers $\mathbb{N}^+$. The Greek had rationals disguised as ratios of lengths or areas but the Indians and Chinese properly considered rationals as numbers. In this book these are introduced as equivalence classes of (numerator, denominator) pairs where numerator and denominator are elements from the previous set with their rules for adding and multiplication. This gives the strictly positive rationals $\mathbb{Q}^+$. Then zero is introduced first as a place holder, and eventually as a number, and this entails the definition of the negative rational numbers and thus besides inverses for multiplication now also inverses for addition will exist. So we now have the structure of $\mathbb{Q}$. This idea of introducing a new mathematical concept as a couple of items from a previous structure with their known composition rules (adding and multiplication) and then identifying them in equivalence classes, is a technique that is used at several places in this book.</p>
<p>In part 2 the natural numbers are derived via the introduction of 1 as the least positive rational whose (reduced) denominator is 1 and then using an induction to generate all the positive integers as part of the rationals, and eventually zero and the negative integers to give $\mathbb{Z}$. Along the way we learn about long division, Bezout's theorem, and prime numbers. Modular arithmetic is introduced via finite fields leading to Fermat's little theorem, coding theory, the Chinese remainder theorem and encryption. The Peano axioms to define the rational numbers $\mathbb{Q}$ as an ordered field is the ultimate conclusion of part 2. It includes some philosophical considerations about the existence of numbers and the idea of an axiomatic approach to mathematics in general. That includes the Russell paradox, Gödel's theorem, and the consistency problem of mathematics.</p>
<p>Historically, mathematics became a profession somewhere in the 17th century. First Körner introduces extensions like $\mathbb{Q}[\sqrt{2}]$ using again the technique of defining addition, multiplication and an order relation on couples of rational numbers. To come to analysis, it requires the fundamental axiom of an intermediate value and hence the existence of a limit for bounded sequences. Equivalence classes of converging sequences are introduced by identifying sequences with the same limit as equivalent. Pointwise operations can be defined for the sequences and the real numbers in the set $\mathbb{R}$ are then identified with these equivalence classes. The complex numbers are then easily introduced as couples of reals (using again the same trick of defining operations for couples of reals) and introducing limits and continuity in $\mathbb{C}$ is relatively easy. This paves the way to define polynomials over a field and to derive the fundamental theorem of algebra. The main reason for considering zeros of polynomials over a field, in particular with integer or rational coefficients, is that this allows to distinguish between the algebraic and the transcendental numbers. Integral domains and quaternions as generalizations of complex numbers are the remaining items with a short discussion.</p>
<p>Clearly this book is probing the fundamentals of mathematical analysis and will be useful as an extra reading for an introductory calculus course. It will certainly satisfy those readers who are looking for abstraction and who want to extract the maximal number of results from the minimal set of axioms. The historical elements on the side are entertaining but not essential. It is not exactly recreational mathematics, but the text is nicely written. It has many explicit proofs and many exercises and invitations to think about a statement. No solutions are provided though. It is an excellent way to get in touch with the foundations of mathematics at a relatively elementary level.</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 book gives an axiomatic approach at a first year university level to number systems and some elements from calculus. It could be a complement for a freshmen's calculus course.</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/tw-k%C3%B6rner" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">T.W. Körner</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/cambridge-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">cambridge 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">978-1-1084-8806-8 (hbk), 978-1-1087-3838-5 (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">£ 59.99 (hbk), £ 24.99 (pbk)</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">268</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/logic-and-foundations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Logic and Foundations</a></li><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="https://www.cambridge.org/be/academic/subjects/mathematics/recreational-mathematics/where-do-numbers-come" title="Link to web page">https://www.cambridge.org/be/academic/subjects/mathematics/recreational-mathematics/where-do-numbers-come</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/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</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/97-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97-01</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/26-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">26-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/26-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">26-03</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/97f30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97F30</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97f40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97F40</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/97f50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97F50</a></li></ul></span>Mon, 25 Nov 2019 09:26:01 +0000Adhemar Bultheel49948 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/where-do-numbers-come#commentsCalculus Simplified
https://euro-math-soc.eu/review/calculus-simplified
<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>Oscar Fernandez is the author of <a target="_blank" href="/review/everyday-calculus-discovering-hidden-math-all-around-us">Everyday Calculus: Discovering the Hidden Math All around Us</a> (2014) and <a target="_blank" href="/review/calculus-happiness-how-mathematical-approach-life-adds-health-wealth-and-love">The Calculus of Happiness: How a Mathematical Approach to Life Adds Up to Health</a> (2017) which are extracurricular texts to illustrate the applications and usefulness of calculus and mathematics in general. With the current book he provides the actual lecture notes, or, as he defines it: a calculus supplement. He starts the book with an extensive commercial, as if he has to justify that he is adding "yet another calculus supplement" to what is already available. His main argument is that he uses the "goldilocks approach", i.e., he provides everything in "just the right amount": just the right level of abstraction, details, insight, intuition, applications, number of pages,... Fernandez characterizes his book as a goldilocks average of a proper calculus text, a calculus supplement, and a calculus teacher. Thus the book provides all the goodies of the usual calculus texts in combination with the help that a teacher would add to it: highlighted text, frames, summaries, take home messages, tips and tricks, many exercises and solutions, and a PUP <a target="_blank" href="https://press.princeton.edu/titles/13351.html">website</a> for interactive content.</p>
<p>All this information is what you usually expect at a publisher's website or on a flyer advertising the book, so it is a bit strange to read it in a book that you already have purchased, but it has the advantage that you are well informed about what and what not to expect, even before you start reading. In fact, "reading" is not the right verb as Fernandez correctly advises the reader to "work through" the book rather than just read it. The level is elementary, somewhere between pre-calculus and first year calculus. The difference between both approaches is static versus dynamic as Fernandez explains: for example pre-calculus just gives the formula for the volume of a sphere (static), while calculus explains the formula as the limiting value as a sum of discs that become infinitesimally thin (dynamic). The infinitesimal concept (something becoming infinitely small without ever being zero) is essential for practical concepts such as instantaneous speed, slope of a curve, and area of a region, which relate to the calculus concepts of limit (the foundation of it all), continuity, derivative, and integral. These are the traditional mathematical topics to be expected but Fernandez manages to cover this in only 109 pages (excluding exercises). The exponential, logarithmic and trigonometric functions are optional, so that everything can be treated using only algebraic functions. At many places a section entitled Transcendental Tales is inserted where the general theory is applied to these transcendental functions. If you skip these sections, then only 89 pages suffice. In fact the main text ends after 158 pages (that includes the exercises). This means that about one third of the main text consists of exercises. The rest of the book is mainly a set of appendices surveying all the material that is supposed to be known in advance (algebra, geometry, functions), answers to exercises, additional applications, bibliography, and index, all together these add an extra 90 pages.</p>
<p>Elementary as the subjects may be, what has been treated has all the rigour that one would expect. There are definitions and there are theorems, but proofs are skipped or hidden in an appendix or in an exercise or it is replaced by several illustrating examples. Examples and applications are main ingredients of the text. Especially optimization as an application of differentiation gets a separate chapter and is rather well elaborated.</p>
<p>I do not think this is suitable for mathematics or engineering students. These definitely need more depth. Unless they are at a pre-calculus level and are so eager that they want to learn more calculus in advance on their own. However, for students that will need some mathematics, and are required to take a calculus course, even if they do not like it, then this book is a nice approach, indeed for the reasons given by the author in his introduction. Many glossy calculus books of up to a thousand pages are a major overkill. The many examples here should stimulate intuition before rigour. Not including the proofs is a practice that has gained popularity, probably not to the liking of mathematicians, but it might help students that are abhorred by the required formal and technical details of a proof. The PUP website is not spectacular but it works nicely and smoothly. There the "dynamic" approach of calculus is lively illustrated by the animations. The text can be "personalized" by skipping for example the sections Transcendental Tales and/or some applications. It will require the guidance of a teacher to help make the proper decisions. Making a selection is however something that one can do with every text. The most interesting property of this book is in my opinion the conciseness (which need to be taken with a grain of salt, since it depends on how much one is prepared to skip, and how many of the exercises are considered to be essential). The abundance of examples instead of proofs is another distinct property, but I believe that also exists in some other texts (or one could just skip the proofs if they are present). Thus I believe there is indeed a wide potential readership for this text since the chosen ones among the students that love mathematics with all its rigour, proofs and technicalities is still a minority with respect to all the students that are submitted to a calculus course because they just need a minimal amount of mathematics.</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 calculus supplement using a goldilocks approach to help the student in his or her transition from pre-calculus to a first year calculus course (limit, continuity, derivative, integral). Most obvious characteristics of the text: concise, many examples and applications, many exercises, avoiding proofs, possibility to avoid transcendental functions (i.e., exp, log, and trigonometric 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/oscar-fernandez" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Oscar Fernandez</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">9780691175393 (pbk), 9780691189413 (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 19.00 (pbk), £ 14.99 (ebk)</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/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</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/13351.html" title="Link to web page">https://press.princeton.edu/titles/13351.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/00a35" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00A35</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/97i10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97I10</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97i40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97I40</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/97i50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97I50</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/26-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">26-01</a></li></ul></span>Mon, 01 Jul 2019 10:43:37 +0000Adhemar Bultheel49489 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/calculus-simplified#comments