Euler’s Pioneering Equation: The most beautiful theorem in mathematics by Robin Wilson (PDF)

Description

In 1988 The Mathematical Intelligencer, a quarterly mathematics journal, carried out a poll to find the most beautiful theorem in mathematics. Twenty-four theorems were listed and readers were invited to award each a ‘score for beauty’. While there were many worthy competitors, the winner was ‘Euler’s equation’. In 2004 Physics World carried out a similar poll of ‘greatest equations’, and found that among physicists Euler’s mathematical result camesecond only to Maxwell’s equations. The Stanford mathematician Keith Devlin reflected the feelings of many in describing it as “like a Shakespearian sonnet that captures the very essence of love, or a painting which brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reachesdown into the very depths of existence”.What is it that makes Euler’s identity, eiπ + 1 = 0, so special?In Euler’s Pioneering Equation Robin Wilson shows how this simple, elegant, and profound formula links together perhaps the five most important numbers in mathematics, each associated with a story in themselves: the number 1, the basis of our counting system; the concept of zero, which was a major development in mathematics, and opened up the idea of negative numbers; π an irrational number, the basis for the measurement of circles; the exponential e, associatedwith exponential growth and logarithms; and the imaginary number i, the square root of -1, the basis of complex numbers. Following a chapter on each of the elements, Robin Wilson discusses how the startling relationship between them was established, including the several near misses to the discovery of the formula.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐One of the previous reviewers criticizes this book for being superficial. However, the book is intended to be an overview for the minimally mathematically sophisticated lay person. It is an ode to very basic number theory and Euler’s equation. Taken as intended, it is an excellent book written from the perspective and with the insight of a retired professional mathematician. I enjoyed the “tangents” discussed in the chapters leading up to the last chapter that focuses on the equation itself. That being said, I too would have preferred a bit more depth, but that doesn’t detract more than one star from what is a quick interesting read.

⭐This book is at its best in supplying historical background for the points it makes. Often, there is enough of a hint of the underlying mathematics that a well-versed reader can supply the rest of the argument or even a proof. I’m not entirely convinced that non-mathematicians will follow as much as the author intended. For example, the term “power series” makes an unheralded appearance. (“Infinite series” were given a brief discussion many pages earlier.) At least for mathematicians it’s a very good summary of some intriguing vistas not all of which were discussed in my classes long ago. For the first time, I can derive one of the formulas for pi. You can even forgive the punny title.

⭐This book is a delight for math lovers, with history, derivations, anecdotes and context for some of the most powerful numbers in math. Very well written in short, rich and humor filled chapters (a feat that’s isn’t easy when discussing calculus and trigonometry).

⭐Wilson does the reader a great service by examining each term in the equation through the lens of math history. I was particularly struck by the “near misses” by Bernoulli and Cotes

⭐See the photos to show why I do not like math E-books. The one photo shows the text font at maximum size showing that the font for the e function does not change. When I first saw this (at a regular sized font) my first thought was that a change of eyeglasses was needed. Little did I know it was not me but the poor formatting on the book.Stop doing it that way!That said the book is interesting, not too technical nor basic.

⭐In my opinion, the most thorough analysis (separation into constituent parts) available to date. As a mathematcian who owns a library of Euler’s works, this book by Wilson is impressive in its “analysis” of Euler’s Equation (and/or Identity). I highly recommend purchasing this book for students of mathematics from the age of 11 to 99. It provides “serious fun” to the understanding of this “most beautiful theorem in mathematics”

⭐Very good info on history of math exploration. From basic need to imagination and beyond. I would recommend for all to replenish their math

⭐A good book and I used it for STEM promo reading list.

⭐The main issue with this book is that the title doesn’t cover the contents at all. The word “pioneering” in the title is I suppose a mathematician’s idea of a pun: it contains the words/letters pi, o (zero), one, “e” and “i” which also occur in Euler’s equation. However, first of all Euler never publicly wrote the equation so to call it Euler’s equation is a bit iffy (the man has done so much that there must be other candidates for this title anyway). More problematically, if the equation is “pioneering”, why does it only get discussed in the final (short) chapter. Why was it pioneering? What impact has it had? Instead the book discusses the history of the component symbols of the equations (albeit not of the + and = signs). In and of itself this is interesting and well-written, but not what the title promises. It’s like writing a book about the impact of Albert Einstein but then only discuss the history of his shoes, the life-story of his tailor, etc. Possibly all very interesting, but not what you promised.

⭐I bought this as a birthday present for my mathematician son-in-law, and borrowed it to read last week. The author unpicks Euler’s equation – which contains five of the most fundamental numbers in mathematics: e, i, pi, 1 and 0 – devoting a chapter to each number, before looking at the equation itself. This is a diverting way to introduce (or remind) the reader about some interesting maths: counting systems, types of numbers (positive, negative, rational, transcendental …), geometry, infinite series and complex numbers.There’s a brisk account of the history of the numbers and their associated mathematical apparatus, emphasising how our understanding has become more refined with the passage of time and the efforts of those who strove in this abstruse field. Surprisingly, this includes some explicit disagreements: when it came to taking the logarthm of a negative number – e.g. ln(-1) – one mathematician (Johann Bernoulli) continued to insist that this had a value of zero, even after Euler deduced (correctly) that it was i times pi (a result which follows from his eponymous equation).I enjoyed this book, which brought forth memories of imperfectly-understood maths lessons at school (and beyond), and made some items more intelligible.

⭐I thought this book would be a lot more interesting and detailled. It mainly describes the history of each component in the equation, and I found it quite disinteresting. I really wanted to enjoy this book but unfortunately this was not the case.

⭐Mathematical poetry. This exquisitely-crafted book has five chapter headings comprising a single character each, then the finale putting them altogether. Only trick missed (second edition, perhaps?) was another pair of chapters on “+” and “=” about which equally-fascinating stories could be told. All in all, though, this book is well worth every single negative e to the i pi penny spent.Dr Chris Palmer, Cambridge UK

⭐I really enjoyed this, I’m not super up on maths – but I guess enough to have been intrigued by the apparent simplicity of the linkage between these three, apparently unrelated constants. Like most books on maths there are a few leaps of logic which maybe are obvious to a mathematician but not to me, but the way the book is structured you don’t feel that you can’t continue without this knowledge

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