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A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics
An award-winning science writer introduces us to mathematics using the extraordinary equation that unites five of mathematics' most important numbers
Bertrand Russell wrote that mathematics can exalt "as surely as poetry." This is especially true of one ei(pi) + 1 = 0, the brainchild of Leonhard Euler, the Mozart of mathematics. More than two centuries after Euler's death, it is still regarded as a conceptual diamond of unsurpassed beauty. Called Euler's identity or God's equation, it includes just five numbers but represents an astonishing revelation of hidden connections. It ties together everything from basic arithmetic to compound interest, the circumference of a circle, trigonometry, calculus, and even infinity. In David Stipp's hands, Euler's identity formula becomes a contemplative stroll through the glories of mathematics. The result is an ode to this magical field.
Bertrand Russell wrote that mathematics can exalt "as surely as poetry." This is especially true of one ei(pi) + 1 = 0, the brainchild of Leonhard Euler, the Mozart of mathematics. More than two centuries after Euler's death, it is still regarded as a conceptual diamond of unsurpassed beauty. Called Euler's identity or God's equation, it includes just five numbers but represents an astonishing revelation of hidden connections. It ties together everything from basic arithmetic to compound interest, the circumference of a circle, trigonometry, calculus, and even infinity. In David Stipp's hands, Euler's identity formula becomes a contemplative stroll through the glories of mathematics. The result is an ode to this magical field.
240 pages, Hardcover
Published November 7, 2017
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October 28, 2021
The God Equation
Turning the infinite into the finite (and back again) is no mean feat. It is what Christians claim God did with Jesus. It is also just one of the things that the equation produced by the Swiss mathematician, Leonhard Euler, in the late 18th century does in mathematics. Or to put it another way, Euler’s equation shows how things that we know are real are actually manifestations of something inconceivably alien, not simply infinite but also entirely other. I think it would only diminish the significance of Euler’s triumph to even state it in mathematical terms to non-mathematicians. So here’s a descriptive rather than analytical summary of Stipp’s highly accessible book.
The ancient Greeks learned how to deal with the idea of infinity by treating it as unreal and illusory, as something which didn’t really exist outside our heads. Since we couldn’t see it, touch it or feel it, infinity wasn’t, as today’s jargon has it, ‘a thing.’ Euler didn’t prove that infinity was a real thing (that was Georg Cantor a century later); but he did show that the infinite went into the construction of real things. Thus whatever the infinite was, it wasn’t only in our heads. Infinity may be irrational, that is beyond our thought of concrete objects, but it is still real as shown in Euler’s equation.
The fact of something real called infinity is relatively easy to deal with in the scheme of things, however. An infinite amount or number or degree of something is always in relation to something we already know about - people, distances, speeds, or indeed numbers - just many, many more or much, much larger; or alternatively much much smaller with many, many more fine divisions. The infinity we talk about, therefore, is merely a collection of these things that we are familiar with. Infinity is the sum, as it were, of these real things.
But things get considerably stranger when we leave the domain of the senses. Even quantum physics may seem to make relative sense. Euler’s equation introduces an entirely novel ‘substance’ into our thinking about what constitutes reality. This substance is something no one has ever seen. It is the dark matter and dark energy of mathematics. The substance is composed of what mathematicians call transcendental numbers.
Transcendental numbers, like π, can only be expressed as an unpredictably infinite set of decimals (technically speaking they never converge). It is their unpredictability that makes them so strange - in terms of how many there might be, where they might be found, and their full identities. Transcendentals are certainly numbers but they can’t be expressed as numbers other than as themselves - not as fractions, formulas, or combinations of other numbers.
In simple terms, transcendental numbers don’t follow any of the normal mathematical laws, even those of arithmetic. They stand in splendid isolation. We know they exist but we don’t know how many there are or what they’re made of. Yet Euler’s equation shows that they too are a constituent of reality. In fact they are so important that they are the rough equivalent of the Higgs boson in particle physics - in a sense promoting the existence of the numbers we measure for things we can feel and see in everyday life - from circles to the mysteries of alternating current.
Some object to this treatment of numbers as if they themselves are real things. They call this view ‘Platonist’ and criticise it as mystically religious. But even if one adopts the view that numbers are all in one’s head tout court, the implication is that mathematics is how our minds consistently (and effectively) work in dealing with the world.
Whether the world is ‘really’ mathematical or only appears that way because of how our minds work would therefore seem a meaningless distinction. All of science, indeed all of thought, is as much a discovery of the reality of ourselves as it is of the universe. And what we discover are patterns that have a remarkable aesthetic attraction. That is to say, we find beauty.
Finally, there are those most mysterious of all numbers, the so-called imaginary numbers. Everyone agrees that this is an inapt description but we’re stuck with the term. A better term than ‘imaginary’ would probably be ‘impossible.’ This is the number i, which is the square root of negative 1. On the face of it, such an entity indeed appears impossible. There is no number, neither a negative nor a positive, which can be the square root of a negative number since whatever number is used will always yield a positive value.
And yet not only does i exist in mathematics, it is of widespread relevance in mechanical, electrical, electronic, and computer engineering, to name just a few of its applications. Infinity may be irrational; transcendentals may be irrational and unseeable; but the imaginary number i is not just irrational, and unseeable, it is entirely incomprehensible. And yet it too is shown by Euler to be a fundamental constituent of the world.
Benjamin Peirce, The 19th century American mathematician (and father of the first great American philosopher, Charles Sanders Peirce) summed up the import of the Euler equation rather succinctly. “It is absolutely paradoxical,” Stipp quotes from Peirce’s published lectures, “We cannot understand it, and we don’t know what it means, but we have proved it, and therefore we know it must be the truth.” I have never encountered a better or more inspiring claim to divine revelation.
Postscript: for a further literary interpretation of the Euler formula, see: https://www.goodreads.com/review/show...
Further postscript: https://www.wired.co.uk/article/googl...
Turning the infinite into the finite (and back again) is no mean feat. It is what Christians claim God did with Jesus. It is also just one of the things that the equation produced by the Swiss mathematician, Leonhard Euler, in the late 18th century does in mathematics. Or to put it another way, Euler’s equation shows how things that we know are real are actually manifestations of something inconceivably alien, not simply infinite but also entirely other. I think it would only diminish the significance of Euler’s triumph to even state it in mathematical terms to non-mathematicians. So here’s a descriptive rather than analytical summary of Stipp’s highly accessible book.
The ancient Greeks learned how to deal with the idea of infinity by treating it as unreal and illusory, as something which didn’t really exist outside our heads. Since we couldn’t see it, touch it or feel it, infinity wasn’t, as today’s jargon has it, ‘a thing.’ Euler didn’t prove that infinity was a real thing (that was Georg Cantor a century later); but he did show that the infinite went into the construction of real things. Thus whatever the infinite was, it wasn’t only in our heads. Infinity may be irrational, that is beyond our thought of concrete objects, but it is still real as shown in Euler’s equation.
The fact of something real called infinity is relatively easy to deal with in the scheme of things, however. An infinite amount or number or degree of something is always in relation to something we already know about - people, distances, speeds, or indeed numbers - just many, many more or much, much larger; or alternatively much much smaller with many, many more fine divisions. The infinity we talk about, therefore, is merely a collection of these things that we are familiar with. Infinity is the sum, as it were, of these real things.
But things get considerably stranger when we leave the domain of the senses. Even quantum physics may seem to make relative sense. Euler’s equation introduces an entirely novel ‘substance’ into our thinking about what constitutes reality. This substance is something no one has ever seen. It is the dark matter and dark energy of mathematics. The substance is composed of what mathematicians call transcendental numbers.
Transcendental numbers, like π, can only be expressed as an unpredictably infinite set of decimals (technically speaking they never converge). It is their unpredictability that makes them so strange - in terms of how many there might be, where they might be found, and their full identities. Transcendentals are certainly numbers but they can’t be expressed as numbers other than as themselves - not as fractions, formulas, or combinations of other numbers.
In simple terms, transcendental numbers don’t follow any of the normal mathematical laws, even those of arithmetic. They stand in splendid isolation. We know they exist but we don’t know how many there are or what they’re made of. Yet Euler’s equation shows that they too are a constituent of reality. In fact they are so important that they are the rough equivalent of the Higgs boson in particle physics - in a sense promoting the existence of the numbers we measure for things we can feel and see in everyday life - from circles to the mysteries of alternating current.
Some object to this treatment of numbers as if they themselves are real things. They call this view ‘Platonist’ and criticise it as mystically religious. But even if one adopts the view that numbers are all in one’s head tout court, the implication is that mathematics is how our minds consistently (and effectively) work in dealing with the world.
Whether the world is ‘really’ mathematical or only appears that way because of how our minds work would therefore seem a meaningless distinction. All of science, indeed all of thought, is as much a discovery of the reality of ourselves as it is of the universe. And what we discover are patterns that have a remarkable aesthetic attraction. That is to say, we find beauty.
Finally, there are those most mysterious of all numbers, the so-called imaginary numbers. Everyone agrees that this is an inapt description but we’re stuck with the term. A better term than ‘imaginary’ would probably be ‘impossible.’ This is the number i, which is the square root of negative 1. On the face of it, such an entity indeed appears impossible. There is no number, neither a negative nor a positive, which can be the square root of a negative number since whatever number is used will always yield a positive value.
And yet not only does i exist in mathematics, it is of widespread relevance in mechanical, electrical, electronic, and computer engineering, to name just a few of its applications. Infinity may be irrational; transcendentals may be irrational and unseeable; but the imaginary number i is not just irrational, and unseeable, it is entirely incomprehensible. And yet it too is shown by Euler to be a fundamental constituent of the world.
Benjamin Peirce, The 19th century American mathematician (and father of the first great American philosopher, Charles Sanders Peirce) summed up the import of the Euler equation rather succinctly. “It is absolutely paradoxical,” Stipp quotes from Peirce’s published lectures, “We cannot understand it, and we don’t know what it means, but we have proved it, and therefore we know it must be the truth.” I have never encountered a better or more inspiring claim to divine revelation.
Postscript: for a further literary interpretation of the Euler formula, see: https://www.goodreads.com/review/show...
Further postscript: https://www.wired.co.uk/article/googl...
February 19, 2018
Aside from E = mc2, there is no other mathematical formula that has had more books dedicated to it than Euler's equation, eiπ +1 = 0. In some ways it's not surprising - like Einstein's equation, Euler's is simple, yet combines essential quantities in a way that surprises and has interesting uses.
Not long ago I read Robin Wilson' Euler's Pioneering Equation, which started really well with some good history of maths on the main components of the equation, but then became too complex for the typical non-mathematician. I'm pleased to say that David Stipp in A Most Elegant Equation doesn't fall for this same trap. This book remains easily readable throughout.
Stipp also takes us through a little of the background to the main components of the equation (though in a far more summary fashion for 1 and 0). It would have been nice to have had a little more history of maths to round out these introductions - as it was, what you get is plenty to understand e or i, for example, but a little more context would have been pleasing. One thing he does do well here, though, is give us some nice biographical detail on Euler himself.
Where Stipp triumphs, though, is continuing to make the whole process accessible as we discover where the equation comes from and what it (and the more generalised version of the equation) is capable of doing for oscillating values such as waves. Stipp takes us through step by step from the basics of definitions of i and sines and cosines, through the use of radians and the way that the complex plane combined with the rotational interpretation of sine and cosine make this approach so powerful. (If any of that doesn't make sense, it will after you've read this book).
If anything, the approach taken is almost too simplified - taking the time, for instance, step by step to explain even something as simple as why x+1=0 is the same as x=-1. But those who remember their high school maths can skip a little. My only other small complaint is that Stipp tries relentlessly to be funny. Sometimes this works fine, but at other times it can feel a little laboured. Having said that, I can forgive a lot of someone whose longest footnote is an exposition of why Mr Spock's emotional responses are more nuanced than they seem. (Yes, there is a reason for this.)
To get a picture of why so many mathematicians and physicists think this equation is beautiful, what the equation does and what lies beneath it, A Most Elegant Equation is hard to beat. There's even a section looking at why a few rebels think the equation is rather boring... and what's wrong with their assertion. I'm not saying this book will make every maths-hater suddenly decide the topic is fascinating. But for anyone who is puzzled by mathematicians talking about beauty, or who knows enough to be surprised at the way these disparate quantities come together without having the mathematical background to explain it, this will make a short but sweet read.
Not long ago I read Robin Wilson' Euler's Pioneering Equation, which started really well with some good history of maths on the main components of the equation, but then became too complex for the typical non-mathematician. I'm pleased to say that David Stipp in A Most Elegant Equation doesn't fall for this same trap. This book remains easily readable throughout.
Stipp also takes us through a little of the background to the main components of the equation (though in a far more summary fashion for 1 and 0). It would have been nice to have had a little more history of maths to round out these introductions - as it was, what you get is plenty to understand e or i, for example, but a little more context would have been pleasing. One thing he does do well here, though, is give us some nice biographical detail on Euler himself.
Where Stipp triumphs, though, is continuing to make the whole process accessible as we discover where the equation comes from and what it (and the more generalised version of the equation) is capable of doing for oscillating values such as waves. Stipp takes us through step by step from the basics of definitions of i and sines and cosines, through the use of radians and the way that the complex plane combined with the rotational interpretation of sine and cosine make this approach so powerful. (If any of that doesn't make sense, it will after you've read this book).
If anything, the approach taken is almost too simplified - taking the time, for instance, step by step to explain even something as simple as why x+1=0 is the same as x=-1. But those who remember their high school maths can skip a little. My only other small complaint is that Stipp tries relentlessly to be funny. Sometimes this works fine, but at other times it can feel a little laboured. Having said that, I can forgive a lot of someone whose longest footnote is an exposition of why Mr Spock's emotional responses are more nuanced than they seem. (Yes, there is a reason for this.)
To get a picture of why so many mathematicians and physicists think this equation is beautiful, what the equation does and what lies beneath it, A Most Elegant Equation is hard to beat. There's even a section looking at why a few rebels think the equation is rather boring... and what's wrong with their assertion. I'm not saying this book will make every maths-hater suddenly decide the topic is fascinating. But for anyone who is puzzled by mathematicians talking about beauty, or who knows enough to be surprised at the way these disparate quantities come together without having the mathematical background to explain it, this will make a short but sweet read.
January 31, 2018
The book did exactly what the author set out for it to do. The author explains how this equation connects several seemingly disparate branches of mathematics. David Stipp does this with unconcealed wonder and glee. He also takes us through the steps that Euler underwent in order to derive the equation. David does this in such a simple way that only an elementary understanding of Mathematics is necessary to understand. This book is not a poetic work of art or "high literature" so the 5 star rating is merely based on the effectiveness of the author in conveying a deep understanding of the equation.
December 27, 2017
Relatively short book. Once the rotating vector form of e^i theta comes into the book, the casual reader is probably lost. I'm and EE/Physics major so I liked this and wish more detail would have been given in the book overall. Non-math/tech people would have stopped reading very early in the book, so I would have taken a much stronger approach to the presentation. I recall a couple key statements in the book I liked: pg 140:
"Today, Euler's formula is a tool as basic to electrical engineers and physicists as the spatula is to short-order cooks. It's arguable that the formula's ability to simplify the design and analysis of circuits contributed to the accelerating pace of electrical innovation during the twentieth century."
Then, pg 149:
"The most inspiring teachers I've known possessed the gift of infectious enthusiasm - they communicated intellectual excitement about their subjects by seeming to regard them with the fresh eyes of impassioned novices. ... The killjoys, I must confess, I wouldn't want them as my kids' math teachers."
"Today, Euler's formula is a tool as basic to electrical engineers and physicists as the spatula is to short-order cooks. It's arguable that the formula's ability to simplify the design and analysis of circuits contributed to the accelerating pace of electrical innovation during the twentieth century."
Then, pg 149:
"The most inspiring teachers I've known possessed the gift of infectious enthusiasm - they communicated intellectual excitement about their subjects by seeming to regard them with the fresh eyes of impassioned novices. ... The killjoys, I must confess, I wouldn't want them as my kids' math teachers."
March 30, 2018
Not bad for what it is: a pre-analysis approach to the famous equation. He tosses in enough historical background to keep it from getting too mathy. (I was a math major, so I’m trying to put myself as a reader into the mind of someone who isn’t afraid of math yet who doesn’t want to overwhelmed.)
October 28, 2021
This is a four-star book, but with a couple of caveats.
First, what made it good?
Stipp does a relatively easy explanation of how Euler derived his original version of the formula as well as how it's normally phrased today. He starts by explaining e, then pi, which sets us up for transitioning from degrees-based trigonometry to radian-based when he gets to i. The illustrations help with explaining i, pi, and how e fits in on values of sines and cosines, sines being the only thing of concern for the equation. At the end, he explains how i to the i power is a real number, not an imaginary or a complex one, based off the radian geometry based version of Euler's equation.
He concludes with a dicussion of mathematical Platonism vs non-Platonism. I would consider myself a semi-Platonist. Certainly, integers are "real" and basically universal. That said, pi as a ratio is certainly universal itself. However, I'm not a mathematics Platonist because of all that Platonism implies, including that there would be a Platonic ideal of pi in addition to the actual, and of course, that's not true.
Caveats?
First, it's got a few grammatical issues and related.
Second, the biggie? A science error, or better, per philosopher Gilbert Ryle, a category mistake right on page 1.
Stipp claims, in talking about Euler in St. Petersburg, Russia, that fall has not yet started in mid-September.
In reality? Fall starts there in late August. Let's be serious.
Where I live, in north Texas, fall doesn't start until October.
The category mistake, of course, is claiming that the astronomical equinoxes and solstices define the starts of the four climatological seasons when they do no such thing.
I still give four stars as, from what I've seen here and Amazon, two other books about Euler's formula are more mathematically dense and have other problems.
An error like this, though, means one should read other books of Stipp's carefully.
First, what made it good?
Stipp does a relatively easy explanation of how Euler derived his original version of the formula as well as how it's normally phrased today. He starts by explaining e, then pi, which sets us up for transitioning from degrees-based trigonometry to radian-based when he gets to i. The illustrations help with explaining i, pi, and how e fits in on values of sines and cosines, sines being the only thing of concern for the equation. At the end, he explains how i to the i power is a real number, not an imaginary or a complex one, based off the radian geometry based version of Euler's equation.
He concludes with a dicussion of mathematical Platonism vs non-Platonism. I would consider myself a semi-Platonist. Certainly, integers are "real" and basically universal. That said, pi as a ratio is certainly universal itself. However, I'm not a mathematics Platonist because of all that Platonism implies, including that there would be a Platonic ideal of pi in addition to the actual, and of course, that's not true.
Caveats?
First, it's got a few grammatical issues and related.
Second, the biggie? A science error, or better, per philosopher Gilbert Ryle, a category mistake right on page 1.
Stipp claims, in talking about Euler in St. Petersburg, Russia, that fall has not yet started in mid-September.
In reality? Fall starts there in late August. Let's be serious.
Where I live, in north Texas, fall doesn't start until October.
The category mistake, of course, is claiming that the astronomical equinoxes and solstices define the starts of the four climatological seasons when they do no such thing.
I still give four stars as, from what I've seen here and Amazon, two other books about Euler's formula are more mathematically dense and have other problems.
An error like this, though, means one should read other books of Stipp's carefully.
April 11, 2018
When the educational merits of rote memorization come up in discussion, I have this annoying habit of telling my friends that math is a lot like a good poem: if you’re young enough when you memorize it, it gets burnt to the “hard disk” part of your brain, from where you can retrieve it when you’re mature enough to understand, or at least explore, its meaning.
I was fifteen when I became familiar with the equation on the cover (don’t ask!) and sure enough I recall it fully, so I had high hopes for this book. Perhaps it would take me to a more intuitive level of understanding.
It resolutely didn’t.
In the interest of perhaps walking math-shy people through the equation (a task that I’ll bet good money it fails to achieve) this is far far far too dumbed down. It explains totally evident stuff. And it leaves the proof to a short appendix. I was expecting that appendix to be 150 pages rather than ten pages. Instead, you get a biography of Euler, a description of pi and waffle about beauty, aesthetics etc.
Regardless, I’m a sucker, so I enjoyed this. I enjoyed the definition of what e is at the very beginning (it’s what you get if you infinitely compound a 100% return over a year: if you compound semiannually, you end up with 2.25 instead of 2, if you compound every quarter you’re at 2.44, if you compound every day you get 2.72, which is e) I also enjoyed the discussion regarding Platonic versus man-made math (not that it moved my views one iota) and nobody had ever shown me what i raised to the i is, so I liked that too.
But the reading time I won’t get back. And I’m none the wiser about any deeper meanings of an equation that to me is the equivalent of a slide rule. Oops, did I say that?
I was fifteen when I became familiar with the equation on the cover (don’t ask!) and sure enough I recall it fully, so I had high hopes for this book. Perhaps it would take me to a more intuitive level of understanding.
It resolutely didn’t.
In the interest of perhaps walking math-shy people through the equation (a task that I’ll bet good money it fails to achieve) this is far far far too dumbed down. It explains totally evident stuff. And it leaves the proof to a short appendix. I was expecting that appendix to be 150 pages rather than ten pages. Instead, you get a biography of Euler, a description of pi and waffle about beauty, aesthetics etc.
Regardless, I’m a sucker, so I enjoyed this. I enjoyed the definition of what e is at the very beginning (it’s what you get if you infinitely compound a 100% return over a year: if you compound semiannually, you end up with 2.25 instead of 2, if you compound every quarter you’re at 2.44, if you compound every day you get 2.72, which is e) I also enjoyed the discussion regarding Platonic versus man-made math (not that it moved my views one iota) and nobody had ever shown me what i raised to the i is, so I liked that too.
But the reading time I won’t get back. And I’m none the wiser about any deeper meanings of an equation that to me is the equivalent of a slide rule. Oops, did I say that?
December 11, 2017
Read the full review at my site Digital Amrit
Mathematics textbooks call it Euler’s formula. But some people feel that that name is too mundane for what is arguably math’s most magnetic truth, as well as one of its most startling, and so they all it God’s Equation
What is the book about?
A Most Elegant Equation: Euler’s Formula and the Beauty of Mathematics is written by David Stipp.
This book explores the history of this equation as well as the beauty of mathematics. This is a book for us to sit back and enjoy the brilliance behind this profundity which might have escaped us when we were in school.
What does this book cover?
A Most Elegant Equation has eleven chapters.
The book starts of by giving us a look at Euler. Then it breaks down the component parts of Euler’s equation – e, i, pi and 1 and 0. We then come back to see more of Euler followed by the impact of the equation on Mathematics. There are a couple of appendices which deal with the original derivation of the equation as well as the “real” nature of i.
...
Read the full review at my site Digital Amrit
Mathematics textbooks call it Euler’s formula. But some people feel that that name is too mundane for what is arguably math’s most magnetic truth, as well as one of its most startling, and so they all it God’s Equation
What is the book about?
A Most Elegant Equation: Euler’s Formula and the Beauty of Mathematics is written by David Stipp.
This book explores the history of this equation as well as the beauty of mathematics. This is a book for us to sit back and enjoy the brilliance behind this profundity which might have escaped us when we were in school.
What does this book cover?
A Most Elegant Equation has eleven chapters.
The book starts of by giving us a look at Euler. Then it breaks down the component parts of Euler’s equation – e, i, pi and 1 and 0. We then come back to see more of Euler followed by the impact of the equation on Mathematics. There are a couple of appendices which deal with the original derivation of the equation as well as the “real” nature of i.
...
Read the full review at my site Digital Amrit
May 28, 2018
Decent look at Euler's formula. Stipp clearly has a high regard for Euler (and rightfully so!) and his formula. He does a good job of describing the formula, especially with the trigonometry and the imaginary numbers. However, he does a lousy job at describing the impact and importance of the formula. He hints at it constantly but never says much more than "people studying electric currents used this formula." Seriously. What a let-down.
The last chapter was a bit of a switch. He discusses if math is "out there," waiting to be discovered, objectively true regardless of man, or if it is merely a human invention (in other words, would aliens from Mars do math the same way we do?). He arrives at a hybrid conclusion that leans more toward the former (I would put myself entirely in the former camp). However, I was struck in his discussion by the realization of how sad the atheistic worldview is. He clearly wants to laud someone or something for the beauty of mathematics, but he has nowhere to turn. He is stuck. He doesn't even seem to know what do with Euler's deep piety and faith (he has no idea how to connect it to Euler's love for math). How much more straightforward to recognize that math reflects the beauty, wisdom, and order of a transcendent being, one who has made us in his image and thus able to "think his thoughts after him."
The last chapter was a bit of a switch. He discusses if math is "out there," waiting to be discovered, objectively true regardless of man, or if it is merely a human invention (in other words, would aliens from Mars do math the same way we do?). He arrives at a hybrid conclusion that leans more toward the former (I would put myself entirely in the former camp). However, I was struck in his discussion by the realization of how sad the atheistic worldview is. He clearly wants to laud someone or something for the beauty of mathematics, but he has nowhere to turn. He is stuck. He doesn't even seem to know what do with Euler's deep piety and faith (he has no idea how to connect it to Euler's love for math). How much more straightforward to recognize that math reflects the beauty, wisdom, and order of a transcendent being, one who has made us in his image and thus able to "think his thoughts after him."
April 19, 2020
A fun book. When my wife asked me why I liked it so, I actually quoted one of our fellow Goodreads members, Jimmy Ele (https://www.goodreads.com/review/show...), referring to the author's "unconcealed wonder and glee" in guiding the reader through the usually technical and dry subject matter. I wish I had had Mr. Stipp as a mathematics professor. I would have cried less and smiled more!! I recommend this book with equally strong, "wonder and glee".
April 26, 2020
It’s math. And it’s history. And philosophy. But mostly math.
I found the math descriptions well simplified. I thought I would get a lot of math knowledge out of this book. Or at least I would be reminded of a lot of math from my courses decades ago. There was a little faint recognition. I appreciate the job the author did in jogging my memory on e. But I found the history, especially the story of Euler, more interesting here. When I reflect on this book a week after reading, I most remember not the math, but the Euler stories, especially about his family. I noted while listening to this audiobook that the last quarter or so was more philosophical, about the meaning of beauty. This discussion was a bit overdone and didn’t add to the story as much as I would have hoped. Go for the math, stay for the history.
I found the math descriptions well simplified. I thought I would get a lot of math knowledge out of this book. Or at least I would be reminded of a lot of math from my courses decades ago. There was a little faint recognition. I appreciate the job the author did in jogging my memory on e. But I found the history, especially the story of Euler, more interesting here. When I reflect on this book a week after reading, I most remember not the math, but the Euler stories, especially about his family. I noted while listening to this audiobook that the last quarter or so was more philosophical, about the meaning of beauty. This discussion was a bit overdone and didn’t add to the story as much as I would have hoped. Go for the math, stay for the history.
October 10, 2020
I cheated.
My knowledge of math exceeds that of the book's intended audience, and I already know the derivation of Euler's formula. But I read the book anyway because I needed an easy read to get me back to the habit of focused extended reading. I needed to get away from endless doom-scrolling through the news. Alas, I failed: still cycling through CNN, NBC, NY Times, etc etc
I really don't know how to rate this book. If a reader is fluent in calculus then he/she would find this book a bit of a slog. This type of reader could watch any of a large number of videos on YouTube and get the same result shown in this book in about 30 minutes. On the other hand if a reader's knowledge of math matches the level that the author intends then I wonder if he/she would make the leap to the higher maths that Euler's formula requires even though the book explains it very well.
Let's not forget that the true aim of the book is to share the sense of awe and beauty that Euler's formula inspires. I was lucky to receive that gift, that sense of wondrous beauty. I was watching MIT Prof. Gilbert Strang's lecture on the exponential function. He approaches it like no one I've seen before (and I went to MIT). I immediately thought, "So cool! He constructed the exponential function, and from there it was a short skip and a jump to e^ix = cos(x) + isin(x). At this point I was, "Wow!" And then a simple substitution x = pi, and BAM! that was the derivation of Euler's formula which up to that point I so desperately had wanted to understand. "Holy cow! Oh My God!" I literally jumped for joy! I was filled with joy, and wonder, and awe, and suffused with beauty. Thank you, Prof. Strang. Your teaching is genius.
My knowledge of math exceeds that of the book's intended audience, and I already know the derivation of Euler's formula. But I read the book anyway because I needed an easy read to get me back to the habit of focused extended reading. I needed to get away from endless doom-scrolling through the news. Alas, I failed: still cycling through CNN, NBC, NY Times, etc etc
I really don't know how to rate this book. If a reader is fluent in calculus then he/she would find this book a bit of a slog. This type of reader could watch any of a large number of videos on YouTube and get the same result shown in this book in about 30 minutes. On the other hand if a reader's knowledge of math matches the level that the author intends then I wonder if he/she would make the leap to the higher maths that Euler's formula requires even though the book explains it very well.
Let's not forget that the true aim of the book is to share the sense of awe and beauty that Euler's formula inspires. I was lucky to receive that gift, that sense of wondrous beauty. I was watching MIT Prof. Gilbert Strang's lecture on the exponential function. He approaches it like no one I've seen before (and I went to MIT). I immediately thought, "So cool! He constructed the exponential function, and from there it was a short skip and a jump to e^ix = cos(x) + isin(x). At this point I was, "Wow!" And then a simple substitution x = pi, and BAM! that was the derivation of Euler's formula which up to that point I so desperately had wanted to understand. "Holy cow! Oh My God!" I literally jumped for joy! I was filled with joy, and wonder, and awe, and suffused with beauty. Thank you, Prof. Strang. Your teaching is genius.
April 15, 2018
Very well done. A well researched, and deftly put forth argument in favor of the titular equation being worthy of high esteem (as equations go, which is far in the author's case). He makes his case, and then proceeds to give historical context all the while pointing out landmarks of great mathematical beauty. I am not a "math person" although it intrigues me and I something feel I could be one.
This book had the perfect amount of hand-holding for the lay person, and I took great relief when near the beginning the author states that even if the math is not grasped, this book will pay off in ways highlighting the equation. I think I understand about 80% of what was presented as far the workings of the equation, and that is more than I hoped for. Also, I had an unfortunate break in time between beginning the audiobook and the final hour of it which probably accounts for most of the missing 20%.
Overall, very entertaining, and I understand a lot more about how calculus of the type here works and is applied.
This book had the perfect amount of hand-holding for the lay person, and I took great relief when near the beginning the author states that even if the math is not grasped, this book will pay off in ways highlighting the equation. I think I understand about 80% of what was presented as far the workings of the equation, and that is more than I hoped for. Also, I had an unfortunate break in time between beginning the audiobook and the final hour of it which probably accounts for most of the missing 20%.
Overall, very entertaining, and I understand a lot more about how calculus of the type here works and is applied.
December 23, 2018
We are in 2017, David noticed that the prettiest equation was not in his textbooks. And you probably never saw it at school either. What?! So he decided to write this book to fix this ugly mistake. Of course bright people (say autoditats thrilled with mathematics) find Euler’s identity early in life but we (hungry-minded laypersons) never see the sexy stuff. Good news. Here comes an accessible compilation of history (including a bit of Euler's impressive biography), fundamental concepts and a gentle/incremental deduction of this elegant relation. Or flabbergasting, beautiful, intriguing, provocative… formula putting together the top five celebrity numbers (e, i, pi, 1 and 0) using just three arithmetic operations (+, = and ^).
November 15, 2017
A neat little book, good for amateurs like myself. I've long wondered what Euler's equation or formula means, and now my curiosity has been satisfied. The most interesting chapter was the last one pertaining to the nature of mathematics, although I believe the author could have made good use of the writings of Immanuel Kant regarding the subject. All in all, informative for the layperson with interesting bits of philosophy and history scattered in between.
November 25, 2018
A strong start, but the momentum fizzles by the last 3 chapters or so. The author’s prose is strong in parts where equations are actually made unnecessary! Yet the constant inclusion of equations served to break the book’s pace and snap me out of the word-based journey through Euler’s formula.
(For what it’s worth: this is coming from a STEM professional for whom numeric notation sometimes feels more native than English)
(For what it’s worth: this is coming from a STEM professional for whom numeric notation sometimes feels more native than English)
February 8, 2022
3.5 Stars. I enjoyed this book but it is for true math nerds (which) I am. I listened to the book and it didn't translate great to audio, there were far too many equations for that. Fortunately, I knew most of the math so I was able to follow along and I really appreciated his explanations. It really is a beautiful thing. I also really enjoyed his inclusion of the mathematicians and their characters as part of the story.
January 4, 2023
This book provides an excellent guide for understanding Euler’s Formula. I thought that it built on concepts well and was quite easy to understand. It did go over quite a bit of math that I already knew, especially in the first few chapters, but I stuck with it because the historical context was very interesting.
March 8, 2024
Ok it starts off really good and I looked it but at a certain point it felt both repetitive and less comprehensible. The last two chapters were a bit over my head, but it’s hard to tell how much of it I didn’t understand and how much of it was just boring and I didn’t care to put the time in to learn it properly. 3.1 stars
August 18, 2022
Good book in the history of mathematics. Contains some assertions about the nature of mathematics and its philosophy that are contested, but does note that they are the philosophical opinion of the author. Overall a good read.
February 9, 2019
An excellent surprise of a book. The author has written this book as a layman survey of the Euler formula for the special case where x = pi, e^i*pi + 1 = 0, but eventually expands it to the general form of e^i*thera = cos theta + i*sin theta, and makes fairly elementary, but intricate, demonstrations like DeMoivres relation, with a brief interlude into finite series, then finally constructs the relation using a LHS, RHS argument with a finite series expansion of cos theta and sin theta, I *think* using DeMoivres relation.
Though the above is the roadmap to the "destination", the true value of the book is the insightful commentary the author has peppered between the simple mathematical construction, on a range of topics like the nature of elegance in math, discussion on abstraction and concreteness in math, and how the mathematical literati which value the former helps retard the advancement of general mathematical education for the public, skewing the subject to an esoterica appreciated by a few, instead of a joint human activity for all, historical tidbits on Leibniz, Wessel (who was the first to visualize the Euler formula as a spherical coordinate), and Euler himself, and gives the reader a glimpse of the multitude of applications Euler's formula has made possible in engineering. There should have been something on Fourier transform, that application alone has touched probably every field of computing, engineering, science, and even parts of social science. Nothing on that, or the associated notion of convolutions, another deep construct enabled ultimately by the Eurler relation.
The biggest miss in this domain, and should be included in version 2 of the book is the Bloch Sphere construct in QM. Given the Born interpretation of the square of the parameters of the ket-states of a superposition, one can normalize these series of square parameters to equal 1, and thus map them into a spherical coordinate system, using the Euler relation. Thus, constructing properties of states of qubits by contextualizing them as rotations across 2 axes.
This application of Euler's formula may be the most profound and impactful yet of all applications in the history of science. There are many other applications of the formula which could triple or quadruple the page count. As I was reading the book I believe that the layman may realize that although they may start off viewing Euler's formula similarly as other famous equations, like E=MC2, as the author demonstrates, this isn't true. Whereas the former is a sort of deep discovered fact about nature, Euler's formula is more of a swiss army knife that allows one to generate facts about nature. If the general readers come away with that sort of insight, the book has definitely done its job of expanding minds.
To students of math or pros, I think this books commentary on the subject and history is sufficient and keen enough to justify a reading, especially for those who may be dusting off their mathematical chops after a few years of atrophy from non-use. A great book all around, recommended
Though the above is the roadmap to the "destination", the true value of the book is the insightful commentary the author has peppered between the simple mathematical construction, on a range of topics like the nature of elegance in math, discussion on abstraction and concreteness in math, and how the mathematical literati which value the former helps retard the advancement of general mathematical education for the public, skewing the subject to an esoterica appreciated by a few, instead of a joint human activity for all, historical tidbits on Leibniz, Wessel (who was the first to visualize the Euler formula as a spherical coordinate), and Euler himself, and gives the reader a glimpse of the multitude of applications Euler's formula has made possible in engineering. There should have been something on Fourier transform, that application alone has touched probably every field of computing, engineering, science, and even parts of social science. Nothing on that, or the associated notion of convolutions, another deep construct enabled ultimately by the Eurler relation.
The biggest miss in this domain, and should be included in version 2 of the book is the Bloch Sphere construct in QM. Given the Born interpretation of the square of the parameters of the ket-states of a superposition, one can normalize these series of square parameters to equal 1, and thus map them into a spherical coordinate system, using the Euler relation. Thus, constructing properties of states of qubits by contextualizing them as rotations across 2 axes.
This application of Euler's formula may be the most profound and impactful yet of all applications in the history of science. There are many other applications of the formula which could triple or quadruple the page count. As I was reading the book I believe that the layman may realize that although they may start off viewing Euler's formula similarly as other famous equations, like E=MC2, as the author demonstrates, this isn't true. Whereas the former is a sort of deep discovered fact about nature, Euler's formula is more of a swiss army knife that allows one to generate facts about nature. If the general readers come away with that sort of insight, the book has definitely done its job of expanding minds.
To students of math or pros, I think this books commentary on the subject and history is sufficient and keen enough to justify a reading, especially for those who may be dusting off their mathematical chops after a few years of atrophy from non-use. A great book all around, recommended
June 3, 2019
Seduced by the Sirens of Math
David Stipp loves math. He doesn't just like math, wants to settle down with math, maybe have a little family, or at least a really great roll in the hay. He is a drooling idiot who thinks the girl loves him. Of course, she does not. She is, in fact, one of the biggest flirts in human history and has used and tossed aside greater minds than his. Well, I am unimpressed. Her tattoo is a little much and her elegance seems stale and practiced. Ask Pedro Domingos The Master Algorithm: How the Quest for the Ultimate Learning Machine Will Remake Our World, about the tattoo on his grad student. It seems over the top to me, but maybe that's what turns these guys on. She's in.
The idea that math is beautiful has always confused me. I think Euler's equation is contrived. I appreciate the identity. But to set up the equation to contain "all the important numbers" so that you can write a book and talk about zero, pi, e, i, and -1 all in one equation reminds me of the violin performances where the composer broke one string after another only to complete the piece on the last string remaining. I know he broke the strings on purpose and I can only say "nice job, but you made all that shit up".
This book is good, but it is propaganda. It promotes the idea that aesthetics should be important in the math that we use - especially in the math we use to describe nature. I don't think so, and neither does Sabine Hossenfelder. She has written a nice book:Lost in Math: How Beauty Leads Physics Astray in the spirit of The Trouble with Physics: The Rise of String Theory, the Fall of a Science and What Comes Next and Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Law. I have to agree with Sabine despite also being sure that she is batshit crazy: https://www.youtube.com/watch?v=m47mJ... Physics is off the rail chasing "elegance". And yes Sabine, it's fucking with my brain too.
I'm not seduced, like Peter Godfrey-Smith I see no reason why 'elegant', 'natural' or 'beautiful' (whatever the fuck-all any of that means), should be any justification for choosing one theory over any other (see: Theory and Reality: An Introduction to the Philosophy of Science.)
So, I've read Sabine's book, and Peter's book and David's book, and I think they are all pointing to the same idea: just because you think something is art (Duchamp: I'm looking at you), doesn't mean it's pretty, or right. Nature owes you nothing.
David Stipp loves math. He doesn't just like math, wants to settle down with math, maybe have a little family, or at least a really great roll in the hay. He is a drooling idiot who thinks the girl loves him. Of course, she does not. She is, in fact, one of the biggest flirts in human history and has used and tossed aside greater minds than his. Well, I am unimpressed. Her tattoo is a little much and her elegance seems stale and practiced. Ask Pedro Domingos The Master Algorithm: How the Quest for the Ultimate Learning Machine Will Remake Our World, about the tattoo on his grad student. It seems over the top to me, but maybe that's what turns these guys on. She's in.
The idea that math is beautiful has always confused me. I think Euler's equation is contrived. I appreciate the identity. But to set up the equation to contain "all the important numbers" so that you can write a book and talk about zero, pi, e, i, and -1 all in one equation reminds me of the violin performances where the composer broke one string after another only to complete the piece on the last string remaining. I know he broke the strings on purpose and I can only say "nice job, but you made all that shit up".
This book is good, but it is propaganda. It promotes the idea that aesthetics should be important in the math that we use - especially in the math we use to describe nature. I don't think so, and neither does Sabine Hossenfelder. She has written a nice book:Lost in Math: How Beauty Leads Physics Astray in the spirit of The Trouble with Physics: The Rise of String Theory, the Fall of a Science and What Comes Next and Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Law. I have to agree with Sabine despite also being sure that she is batshit crazy: https://www.youtube.com/watch?v=m47mJ... Physics is off the rail chasing "elegance". And yes Sabine, it's fucking with my brain too.
I'm not seduced, like Peter Godfrey-Smith I see no reason why 'elegant', 'natural' or 'beautiful' (whatever the fuck-all any of that means), should be any justification for choosing one theory over any other (see: Theory and Reality: An Introduction to the Philosophy of Science.)
So, I've read Sabine's book, and Peter's book and David's book, and I think they are all pointing to the same idea: just because you think something is art (Duchamp: I'm looking at you), doesn't mean it's pretty, or right. Nature owes you nothing.
August 15, 2022
Difficult to understand in places, but overall very enjoyable. Euler's formula is e raised to the iPi power+1=0. One of my physics professors went through a quick derivation of the formula when I was in college. When he arrived at the end result, he suggested the simplicity and elegance of how these various constants interrelate was evidence of the existence of the hand of God in our universe. Consequently, I'm adding "God's Equation: Einstein, Relativity, and the Expanding Universe" to my reading wish list. Fortunately, at my age, I don't have to worry about understanding all of it for an exam!
January 27, 2019
A great Maths book, well-written so it is appropriate for most who read it.
Fermat's last theorem - from Simon Singh is the one to beat though.
Fermat's last theorem - from Simon Singh is the one to beat though.
February 2, 2022
Easy to follow demonstration of Euler’s formula. What may have made the book more interesting would have been a longer exposition on the ubiquitous nature of e and pi in the real world. That these two seemingly unrelated numbers are actually related as Euler demonstrated, is incredible to me. I’m convinced that in spite of understanding the proof, I can only scratch the surface of what a mathematician sees. But, I CAN feel the sense of “This is really weird”. I guess I’ll just have to be happy with that.
December 8, 2017
I love author's passion for this topic - the man Euler and his math. I haven't thought about radians and imaginary numbers for many decades, but author kept me interested and engaged. Nice to be challenged by a book. The connections between branches of math (and how the world applies them) are a wonderful revelation.
January 20, 2018
A very interesting book that explains, in concrete and easy to follow concepts and examples, one of the most beautiful mathematical expressions of all time in all its dept and, by many standards, in pretty much all of its complexity. It reminds us that out there, wether we know it or not, the cosmos keeps speaking its eerie and enticing mathematical language, for us to understand and admire.
May 6, 2018
Very interesting history of euler’s number and the equation e^i(pi)-1=0
I recommend getting the actual book as I had some trouble with the equations being read. But interestingly Euler probably wouldn’t have any trouble with non-visual mathematics.
I recommend getting the actual book as I had some trouble with the equations being read. But interestingly Euler probably wouldn’t have any trouble with non-visual mathematics.
July 13, 2018
Mai capito i numeri complessi.
Sempre stato una capra in matematica, da un certo livello in su. Effettivamente anche oggi, un poco lo sono.
Però il saggio è decisamente interessante e ben scritto, abbastanza comprensibile anche per me.
Quindi 3 stelle e mezza li merita.
Sempre stato una capra in matematica, da un certo livello in su. Effettivamente anche oggi, un poco lo sono.
Però il saggio è decisamente interessante e ben scritto, abbastanza comprensibile anche per me.
Quindi 3 stelle e mezza li merita.
August 2, 2018
An informative read, beginning with the basics of the transcendental number e and going into Euler’s equation. There’s another book about e which also touches upon Euler’s formula, called ��The Story of e”, it’s a bit more technical, so I recommend this book by David Stripp first.
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