In this informal and engaging history, Eli Maor portrays the curious characters and the elegant mathematics that lie behind the number. Designed for a reader. The story of [pi] has been told many times, both in scholarly works and in popular books. But its close relative, the number e, has fared less well: despite the. e, The Story of a Number - Eli soundofheaven.info Uploaded by . 3 ² @ / / Û I Ü 1 ' Ú ä ç è é ç è å æ ê Ž Œ Š ‹ ò ÿ õ ð ï ñ ô ó ì ë í ö ÷ ø ú û þ û î ù ü ù ù ù ü ý í ø þ ü ù!.
|Language:||English, Spanish, Japanese|
|ePub File Size:||20.43 MB|
|PDF File Size:||8.59 MB|
|Distribution:||Free* [*Regsitration Required]|
e: the story of a number I Eli Maor. p. em. Includes bibliographical references and index. ISBN I. e (The number). QAM 'dc 1. "e": The Story of a Number Read Online · Download PDF; Save; Cite this Item . Leonhard Euler proved that the expression xxxxx, as the number of exponents grows to infinity, The principal value of these (the value for k=0) is e−(π/2). of Hazrat Mawlana Jalaluddin Rumi's work, regardless of who they are and Immersing oneself in the ocean of love and co.
I loved his "Infinity" book. Jul 05, Jeffrey rated it really liked it. And it is an excellent companion to a course in calculus. But its close relative, the number e, has fared less well: That was the merit of several mathematicians among which Gauss. What happens if n tends to infinity?
What happens if n tends to infinity? The result, as we know is e r , but it took a very long time to come to this result. It needs binomial powers, the concept of infinity and of a limit.
This came only after calculus was introduced by Newton and Leibniz, Jacob Bernoulli linked compound interest and the exponential, and it became only fully explored by Euler, the master of them all. Of course there are many precursors before one arrived at calculus. There are Zeno's paradoxes that relate to the concept of infinity. And the Greek approximated the circle by an n -gon with n indefinitely increasing, which lead to this other magic number we know as p.
This polygon approximation allowed to approximate the circumference but also the area of the circle, and similar techniques applied to other conic sections. But the proper machinery to compute this integral as an anti-derivative was only provided by Newton and Leibniz. And there appears the number e like magic. Once the exponential and logarithmic functions are known, they show up in all kinds of applications like solutions of differential equations, music scales, spirals, catenary and other curves, hyperbolic functions, and of course the most magic formula showing a family picture of the most famous actors of our number system: Of course the latter directly relates to complex numbers too.
The spreading of Leibniz's ideas throughout Europe was mainly due to the Bernoulli family. Leibniz himself died at the age of seventy almost completely forgotten. Jacob was particularly fond of the logarithmic spiral spira mirabilis and he had it carved on his tomb. Euler was the first to considered the exponential function in its own right, next to the logarithm, and not as just as an inverse by-product.
The acceptance of negative and complex numbers is another interesting story that Maor takes the opportunity to tell. Mathematics had developed well with only positive numbers basically only rationals.
Negative numbers were known to Hindus, but were neglected by Europeans. It was only when Bombelli used a number line to represent numbers that a meaning could be given to negative numbers.
Similarly, it was the geometric interpretation of complex numbers as points in the complex plane that finally began to make sense. That was the merit of several mathematicians among which Gauss. Only later Hamilton gave a formal definition as couples of real numbers with appropriate addition and multiplication, which he later generalized to quaternions.
Maor even discusses complex functions and complex calculus of course including the complex exponential and logarithm. Another topic that could not have been missed is in the trailing chapter about the transcendence of e proved by Hermite in Since the original publication in , J. Havil has compiled a biography on Napier: John Napier: Life, Logarithms, and Legacy Princeton University Press, and there were of course also several books that had chapters on the logarithm or the exponential, which emerged in a period where mathematics experienced a boost in Europe.
This story has e as the central star, but e has many strings attached to it Thus many other issues of mathematics are also wonderfully told by the author, much to the liking of the public who made it a bestseller.
It reads smoothly, is well illustrated, with some more technical material moved to appendices although the main text is not avoiding formulas, it remains quite accessible for a general interested reader.
Also the brief interleaving sections on several topics e. The book begins with an introduction to logarithms, highlighting the relationship between the arithmetic and geometric progressions contained therein. Then we learn how the enigmatic number e was already slyly peeking out at us, way back in the day, in the realm of compound interest. Next we have a fairly decent discussion of limits and infinity. Then, after some binomial formula gymnastics, which are aided by an obliging infinite series, e: A slightly more rigorous proof is fortunately included in the appendices, which, among other things, also offer a gorgeous proof of the irrationality of e.
Elegant, elegant math. This brings us to e: The area in question is expressed by the logarithmic function with base e, or the natural log. We soon discover, through differential calculus, that e is pretty badass. And get this: Then a few rather run-of-the-mill physical applications of the function are perfunctorily trotted out.
Now we investigate the natural log function, or the inverse of the exponential function. The logarithmic spiral, or "spira mirabilis," plotted in polar coordinates, is really quite pretty check out that equiangular property in action! Next we examine e as it relates to hyperbolic trig functions, and then we get to some good stuff: Sweet Jesus, how do they think of these things??
Euler was a legend. The rest of the book deals with mapping complex functions, complex analysis, polar representations of complex functions, etc. The big star of this section is how "the imaginary becomes real. Also, it is pointed out that hyperbolic functions are, like, super well-behaved when they play with purely imaginary variables. Well, that was a long-ass summary!
Here are some pros and cons I believe are worth mentioning: Also, the math in question, while detailed, can still be grasped by those who have studied a bit of calculus at the university level. The book did not really explore the more interesting real-life applications of e. Also, I had some issues with the organization of the book. For instance, the last chapter looked at different types of numbers integers, rational versus irrational numbers, and algebraic versus transcendental numbers.
It seems to me that this material ought by rights to have been presented in the beginning of the goddamn book! Seriously, I have no idea why he jammed this shit into the conclusion.
Yuck city. Bottom line: Look at this sexy girl: View 2 comments. Oct 05, Katia N rated it it was amazing. Eli Maor wrote quite a few books about the history of Mathematics. They are wonderful in combining interesting historical insights with the maths per se, but on the level of a school program. I loved his "Infinity" book. This is as well extremely erudite and fascinating. Sounds daunting, but one can think of this number as a basis for measuring rate of change in many processes involving so called exponential growth the rate o Eli Maor wrote quite a few books about the history of Mathematics.
Sounds daunting, but one can think of this number as a basis for measuring rate of change in many processes involving so called exponential growth the rate of growth is proportional to the current state of the system. It is very relevant nowadays while almost everything what matters grows exponentially including information, population, pollution etc. The book is not very technical at all. It explains underlying maths.
But also talks about fascinating historical characters and anecdotes. To illustrate, I will just mention here one episode: As the story goes, the Calculus were "discovered" in the 17th simultaneously by Newton in England and Leibniz on the continent.
Apparently, it was huge unresolved argument who hold the priority over this discovery. The majority of academy in England claimed it was Newton and that Leibniz has stollen his ideas after seeing some of the Newton's papers.
This was severely rejected by the Leibniz's defenders on the continent. Nevertheless, Leibnitz's system of notation was more intuitive and easier to understand and apply. In England, where it originated, the calculus fared less well.
Newton's towering figure discouraged British mathematicians from pursuing the subject with vigor. Worse, by siding completely with Newton in the priority dispute, they cut themselves off from developments on the Continent.
The stubbornly stuck to Newton's dot notation, failing to see the advantages of Leibniz's differential notation.
As a result, over the next hundred years, while mathematics flourished in Europe as never before, England did not produce a single first-rate mathematician. Jul 13, Bill Ward rated it really liked it Recommends it for: Valerie Neer.
The constant e is just as important if not more so, but never managed to break its way into popular culture because it's a little hard to understand just what makes it so special.
This book makes a valiant effort to redress that shortcoming, by explaining the history of logarithms and calculus and how the last years of mathematics developed, empowered largely by this mysterious number which, before the invention of computers and calculators, was critical in doing any kind of serious arithmetic. Nowadays they don't even teach how logarithms are used to do multiplication - I'm 40 years old, and it was not taught when I was a kid either - but for hundreds of years the only realistic way to do it was to look up the numbers in a log table, add them up, look the sum up in another table, and get your result.
This book talks about the lives of mathematicians and their discoveries, and how those built on each other to produce the knowledge we now have about the amazing world of numbers. But books like this tend to have a fatal flaw, either dumbing down the math so much that it becomes basically just biography and handwaving, or having so much math that you need an advanced math degree to understand it. This one strikes a very careful balance between those extremes.
There were definitely parts where I had to stretch my brain back 20 years to high school and college calculus classes, but each of the formulas was pretty well explained, and I'd like to think you could come away from this book with some understanding even if you'd never taken any advanced math.
View 1 comment. Dec 22, Elijah Oyekunle rated it it was amazing Shelves: I love this concise history of one of Mathematics' most interesting numbers. Calculus was required to explain and understand it, which brought the Bernoullis, Leibnitz, Newton, Euler and a lot of other scientific geniuses to tackle it.
Unlike pi, which has been known for thousands of years, and which was foundational to geometry, one of Mathematics' oldest branches, e has been around for a shorter I love this concise history of one of Mathematics' most interesting numbers. Unlike pi, which has been known for thousands of years, and which was foundational to geometry, one of Mathematics' oldest branches, e has been around for a shorter period of time about years , and deals with a bunch of things like irrationality, infinity and stuff that ancient mathematicians never liked to think much about.
I always find interesting the story of Hippasus , a Pythagorean who is famous for getting drowned by other Pythagoreans for his threat to expose irrationality. Although a lot of stuff in the book was over my head and I steadily refused the urge to read the Appendices, I still think this book is a good work of mathematical history.
Oct 09, Andy rated it it was ok. Here he departs from his straight-laced account to describe, at length, an imagined conversation between J.
Bach and Johann Bernoulli. That perfectly fits my love for orderly sequences of numbers. But there is a problem. A scale constructed from these ratios consists of three basic intervals: The first two are nearly identical, and each is called a whole tone, or a second… But the same ratios should hold regardless of which note we start from.
Every major scale consists of the same sequence of intervals. I can see the confusion… This bizarre interlude aside, Maor has a difficult time keeping to the project he outlines in his introduction.
After a brief and entertaining history of logarithms, as Maor begins his approach to the subject at hand, his text quickly becomes mired in equations—limits, infinite series, and calculus notation. Jul 05, Jeffrey rated it really liked it.
OK, so books on math, not going to become national best sellers by any stretch of the imagination. But any story in the field of math be it zero, 'e,' Phi, PI tells us more about that mystical, insightful language that can tell us so much about the why's and what's of our surroundings, as well as provide the more practical to suit our human needs.
Math is interesting in the sense that it dictates to the mathematician not the mathematician to it to determine outcome. So, to the book.
Maor has done a great job giving us some background on 'e' and its beginnings in logarithmic use. And even though 'e's use can be found in diverse places--"the interest earned in a bank account, the arrangement of seeds in a sunflower, and the shape of the Gateway Arch in St. Louis"--its significance, only second to PI in importance, as a number is greatly and clearly expressed by Maor. It's written for the non-mathematician, no great depth of understanding needed to get the points here.
Some anecdotes and diversions to bring home points made. Good effort. Nov 26, Ben Pace rated it it was ok Shelves: Enjoyable skim through the basics of logarithms, conic sections, calculus, and various other areas of mathematics relating to e. Not a textbook, so don't read this to learn those subjects, only to glance at them.
The historical aspects add a narrative element, and of course the writing is far more pleasant than a textbook too. The background given, and also the original explanations, helped me to understand some of the concepts better, so I am glad that I read it. I will only be giving it a curs Enjoyable skim through the basics of logarithms, conic sections, calculus, and various other areas of mathematics relating to e. I will only be giving it a cursory glance though. Subject to edit on completion Jan 18, Stanley Xue rated it really liked it.
Great book to explore mathematics from a different perspective recreational rather than traditional mathematics education. Even suitable if you haven't touched and been learning more maths for a while. Many of the explanations were built from first principles. Although there was a lot of overlap initially with mathematics covered in high school cirricula e. Similarly to An Imaginary Tale, this subject matter was approached in a chronological manner, with stories about the characters and mathematicians involved in the story of e.
This improved the ease of the read and helped maintan my interest in combination with other general trivial facts and case studies e. There was ample new subject matter especially in the latter half of the book for me stuff that wasn't covered in high school, wasn't proven and accepted as fact, or just forgotten by me: It was especially satisfying to read about the relationship between e and pi e.
Would recommend reading in conjunction with An Imaginary Tale a book about the imaginary number: Having only read that book a month ago, I seem to have forgotten some of the theorems and proofs within. As such, this book was great to remind me of those theorems and the beauty of their proofs.
Euler's thoerem, Laplace's equations, hyperbolic functions. Maybe I wish that there was more maths in this book. Some proofs seem to be glossed over and "outside the scope of this book". Some of the explanations seem less clear than those within An Imaginary Tale. Maybe this is why I rate this book 4 starts instead of 5. As well since the novelty of a book that is not a textbook fiddling with a lot of maths has been attenuated for me. Although I'm not sure which one I retain better since it seems that a lot of the proofs and examples in An Imaginary Tale have already been lost on me.
As said by others - picked this up wanting to understand a complex mathematical topic, got this and also an awesome historical overview of the development of the calculus and more over hundreds of years. I was hoping this would be more like The Golden Ratio: For one thing, this book has differential equations.
A lot of them. What really helped get me through the book were the historical anecdotes, and the parts of the book I was able to follow well were also well-done.
Feb 25, Tim rated it really liked it. Maor's account of the place of e, the base of the natural logarithms, in the history of mathematics provides a peek inside a mathematician's brain.
More connected by mathematical ideas than by chronology or the usual social, cultural, economic, or political themes taken up by historians, Maor's book opened vistas in the calculus I did not see when I first ploddingly confronted derivatives and integrals some decades ago. He thoroughly covers the differing views of Newton and Leibniz as they devel Maor's account of the place of e, the base of the natural logarithms, in the history of mathematics provides a peek inside a mathematician's brain.
He thoroughly covers the differing views of Newton and Leibniz as they developed the calculus. He discusses some of the special characteristics of e revealed in the fact that the exponential function is its own derivative.