Number Theory Books

A Beginner's Guide to Constructing the Universe: Mathematical Archetypes of Nature, Art, and Science

Michael S. Schneider

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Total reviews: 51 Average rating: 4.5 of 5

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The Universe May Be a Mystery,
But It's No Secret

Michael Schneider leads us on a spectacular, lavishly illustrated journey along the numbers one through ten to explore the mathematical principles made visible in flowers, shells, crystals, plants, and the human body, expressed in the symbolic language of folk sayings and fairy tales, myth and religion, art and architecture. This is a new view of mathematics, not the one we learned at school but a comprehensive guide to the patterns that recur through the universe and underlie human affairs. A Beginner's Guide to Constructing, the Universe shows you:

Why cans, pizza, and manhole covers are round.

Why one and two weren't considered numbers by the ancient Greeks.

Why squares show up so often in goddess art and board games.

What property makes the spiral the most widespread shape in nature, from embryos and hair curls to hurricanes and galaxies.

How the human body shares the design of a bean plant and the solar system.

How a snowflake is like Stonehenge, and a beehive like a calendar.

How our ten fingers hold the secrets of both a lobster and a cathedral.

And much more.

Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

John Derbyshire

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Total reviews: 83 Average rating: 4.5 of 5

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Bernhard Riemann was an underdog of sorts, a malnourished son of a parson who grew up to be the author of one of mathematics' greatest problems. In Prime Obsession, John Derbyshire deals brilliantly with both Riemann's life and that problem: proof of the conjecture, "All non-trivial zeros of the zeta function have real part one-half." Though the statement itself passes as nonsense to anyone but a mathematician, Derbyshire walks readers through the decades of reasoning that led to the Riemann Hypothesis in such a way as to clear it up perfectly. Riemann himself never proved the statement, and it remains unsolved to this day. Prime Obsession offers alternating chapters of step-by-step math and a history of 19th-century European intellectual life, letting readers take a breather between chunks of well-written information. Derbyshire's style is accessible but not dumbed-down, thorough but not heavy-handed. This is among the best popular treatments of an obscure mathematical idea, inviting readers to explore the theory without insisting on page after page of formulae.

In 2000, the Clay Mathematics Institute offered a one-million-dollar prize to anyone who could prove the Riemann Hypothesis, but luminaries like David Hilbert, G.H. Hardy, Alan Turing, André Weil, and Freeman Dyson have all tried before. Will the Riemann Hypothesis ever be proved? "One day we shall know," writes Derbyshire, and he makes the effort seem very worthwhile. --Therese Littleton

e: The Story of a Number

Eli Maor

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Total reviews: 55 Average rating: 4.5 of 5

Amazing minds! 5 out of 5 stars.
1 of 1 people found this review helpful.

This was a good book for someone who likes math and is willing to work a little. You should have had (and enjoyed at some point) a little algebra, geometry, and calculus. Even if your math is rusty like mine, you will be able to follow this book well enough. I was surprised how much of it came back to me. (I wouldn't want to be tested on it though!)

The most fascinating thing to me was the brainpower that thought this stuff up! How they could have pumped so much out of the natural logarithm (e) was simply amazing to me, things such as the elegant infinite series of fractions and continued fractions, continued exponentials, sometimes with factorials. Perhaps the most amazing thing was the totally unintuitive formula e raised to the power of the product of i and pi = -1; imagine e, i, and pi contained in one short,neat, little formula! This book is also about the history of math, how calculus was invented, and how imaginary numbers found their place in math. Fortunately for me, Eli Maor goes slow enough and skips enough of the details and the proofs to make this book readable. He also gives neat short biographies of the main characters in the history of mathematics to break the hard math up. The one that was most fascinating to me was an 18th century mathametician named Leonhard Euler (who came up with e raised to the product of pi and i = 1), whom Eli Maor called "unquestionably the Mozart of math". He is relatively unknown simply because he was bracketed in time between Newton and Galileo. I do, however, have to confess I got a bit lost near the end of the book with his dissertation on complex variables (imaginary and real). The math there was a bit too dense for me (or maybe I was too dense).

I can't figure out how e raised to the power of the product of pi and i can come out to a real number (-1) since it is about a real number raised to an imaginary power. How is that even possible? How in the world did Euler come up with the formula! Maor says he'll leave it to the reader to decide if this remarkable formula is a part of "the Creator's grand scheme".

It was also a relief to read a math book without having to be graded. That was a first for me.

Editorial Review:

Until about 1975, logarithms were every scientist's best friend. They were the basis of the slide rule that was the totemic wand of the trade, listed in huge books consulted in every library. Then hand-held calculators arrived, and within a few years slide rules were museum pieces.

But e remains, the center of the natural logarithmic function and of calculus. Eli Maor's book is the only more or less popular account of the history of this universal constant. Maor gives human faces to fundamental mathematics, as in his fantasia of a meeting between Johann Bernoulli and J.S. Bach. e: The Story of a Number would be an excellent choice for a high school or college student of trigonometry or calculus. --Mary Ellen Curtin

Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Law for Unity in Physical Law

Peter Woit

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Total reviews: 37 Average rating: 4.0 of 5

The emperor has no clothes 5 out of 5 stars.
9 of 11 people found this review helpful.

I am writing this blurb in reply to Lubos Motl's (you guess it, another String practitioner whose livelihood is being threatened by this book) comment.

The fact that Peter Woit runs a blog critical of String only shows that he is consistent with his opinions. It should shock any impartial observer that in the eye of String practitioners--Lubos Motl being by no means exceptional among String people--this somehow automatically qualifies Woit and his readers as "crackpots". But what is even more shocking is the comparison to William Dembski's ID (Intelligent Design) blog, because String itself actually provides the perfect analogy to ID.

Let me elaborate. At the height of the recent Pennsylvania ID trial, many education experts came forward to defend the theory of Evolution. One of the hotly debated issues was the definition of a "theory" in science. The ID people, including Mr. George W the Decider, have argued repeatedly that, since Evolution is just a theory, other theories such as ID deserve at least some mention in the classrooms. The defenders of Evolution rely on the rebuttal that a theory in science actually means something that provides the best framework to explain a multitude of independent observations or experimental results and therefore has been accepted by scientific community at large. This, of course, is a lie. String has been paraded as a "theory" in every physics department for over two decades, and yet not only has it produced no testable prediction but it will almost certainly never be able to do so. This puts String squarely in the company of ID. Furthermore, just like ID has morphed constantly, free of the constraint of experimental confirmation, String has changed constantly ever since its inception. In other words, not only does String produce untestable predictions, but these predictions also change from year to year, often dramatically and in a mutually contradictive way. (Have they settled on the dimensionality of space-time yet? Is it 26, 10, 11, or all of the above?)

Other similarities exist, chief among them the religious ferver driving both groups to influence popular opinions and police dissenting thoughts, as exemplified by Lubos Motl's criticism of this book. But there is one big difference between String and ID. The String practitioners are insiders of the science community. They hire their own and make sure String-related papers get published. When a group dominates the peer-review process, it controls the career of every physicist in related fields. It then gains the power to influence, bribe, coerce and intimidate.

Of the five purported String supporters in general physics cited by Motl, I have not worked with Gell-Mann or Hawking, so I don't know what their excuses are (or if they actually need one, since String people tend to misrepresent general enthusiasm for high-energy theories as specific support for String). Prof. Weinberg is the head of a large String group even though he did not publish many real String papers himself. I would certainly count him in the String camp. Dr. Randall and Dr. Arkani-Hamid both took advantage of String referee's eagerness to find supporting work and padded their publication counts with a series of well-publicized "String-Phenomenology" (an oxymoron) papers, thus can hardly be called disinterested third parties either.

The rest of Motl's criticism can be condensed into two simple arguments. The first is that the author does not know details in String as well as he does. This is probably correct technically, particularly in light of the freely changing nature of String's "conclusions". But the core message (as well as the title) of the book is not that String gets the details wrong but that String has no details that can be proven wrong (or right). In this sense, Motl's straw-man argument is irrelevant.

The second point Motl made is, "The problematic statement that string theory makes no prediction is repeated hundreds of times, and in many particular contexts, such a statement becomes not only boring but also patently false." But Motl did not offer any justification why that statement is false. He simply proceeded to throw out more buzzwords and correct more details. One has to conclude that these buzzwords are nothing more than smoke screens meant to obscure the fact no justification could possibly be offered for his claim.

String is a unique phenomenon. It is the most "successful" pseudo science in modern history. Its catalyst is the unprecedented absence of real experimental input in high-energy physics during the past 30 years. Like a cancer, it turns otherwise worthy members of a formerly proud body into the self-promoting endlessly-propagating automatons sucking all resources dry. Later generations of historians of science will surely make their careers studying this bizarre event. Yet it has not begun. This book is not perfect, but as the first formal effort to recognize and document this historical phenomenon, it is a must read.

Editorial Review:

When does physics depart the realm of testable hypothesis and come to resemble theology? Peter Woit argues that string theory isn't just going in the wrong direction, it's not even science. Not Even Wrong shows that what many physicists call superstring "theory" is not a theory at all. It makes no predictions, not even wrong ones, and this very lack of falsifiability is what has allowed the subject to survive and flourish. Peter Woit explains why the mathematical conditions for progress in physics are entirely absent from superstring theory today, offering the other side of the story.

Elementary Number Theory (5th Edition)

Kenneth H. Rosen

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Total reviews: 11 Average rating: 4.0 of 5

Excellent introduction to a vast, fascinating subject 5 out of 5 stars.
8 of 10 people found this review helpful.

This book is awesome. Tons of material covered, but at a decent pace and with good examples. All the reader needs is a good working knowledge of how to read and construct proofs, and the time and patience to get through the material. The exercises range from the computational to the theoretical, from the routine to the extremely challenging. Many of the examples and exercises are intriguing, well-known results. The author touches on a large amount of subject matter and has many references for those interested in further reading. He makes no use of any of the methods that come from the main branches of Mathematics, namely Algebra, Analysis, etc. (though he mentions a few famous results), but he also mentions that he will not be using these methods in the beginning. It starts off a bit easy but gets moderately challenging. This book will never leave my shelf, and has sparked an interest inside of me that shows no signs of burning out. Again, an excellent book.

Editorial Review:

Elementary Number Theory and Its Applications is noted for its outstanding exercise sets, including basic exercises, exercises designed to help students explore key concepts, and challenging exercises. Computational exercises and computer projects are also provided. In addition to years of use and professor feedback, the fifth edition of this text has been thoroughly checked to ensure the quality and accuracy of the mathematical content and the exercises. The blending of classical theory with modern applications is a hallmark feature of the text. The Fifth Edition builds on this strength with new examples and exercises, additional applications and increased cryptology coverage. The author devotes a great deal of attention to making this new edition up-to-date, incorporating new results and discoveries in number theory made in the past few years.

Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills

Paul J. Nahin

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Total reviews: 16 Average rating: 4.0 of 5

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I used to think math was no fun
'Cause I couldn't see how it was done
Now Euler's my hero
For I now see why zero
Equals e^{[pi] i}+1
--Paul Nahin, electrical engineer

In the mid-eighteenth century, Swiss-born mathematician Leonhard Euler developed a formula so innovative and complex that it continues to inspire research, discussion, and even the occasional limerick. Dr. Euler's Fabulous Formula shares the fascinating story of this groundbreaking formula--long regarded as the gold standard for mathematical beauty--and shows why it still lies at the heart of complex number theory.

This book is the sequel to Paul Nahin's An Imaginary Tale: The Story of I [the square root of -1], which chronicled the events leading up to the discovery of one of mathematics' most elusive numbers, the square root of minus one. Unlike the earlier book, which devoted a significant amount of space to the historical development of complex numbers, Dr. Euler begins with discussions of many sophisticated applications of complex numbers in pure and applied mathematics, and to electronic technology. The topics covered span a huge range, from a never-before-told tale of an encounter between the famous mathematician G. H. Hardy and the physicist Arthur Schuster, to a discussion of the theoretical basis for single-sideband AM radio, to the design of chase-and-escape problems.

The book is accessible to any reader with the equivalent of the first two years of college mathematics (calculus and differential equations), and it promises to inspire new applications for years to come. Or as Nahin writes in the book's preface: To mathematicians ten thousand years hence, "Euler's formula will still be beautiful and stunning and untarnished by time."

The Fabulous Fibonacci Numbers

Alfred S. Posamentier, Ingmar Lehmann

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Total reviews: 6 Average rating: 4.0 of 5

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The most ubiquitous, and perhaps the most intriguing, number pattern in mathematics is the Fibonacci sequence. In this simple pattern beginning with two ones, each succeeding number is the sum of the two numbers immediately preceding it (1, 1, 2, 3, 5, 8, 13, 21, ad infinitum). Far from being just a curiosity, this sequence recurs in structures found throughout nature—from the arrangement of whorls on a pinecone to the branches of certain plant stems. All of which is astounding evidence for the deep mathematical basis of the natural world.

With admirable clarity, math educators Alfred Posamentier and Ingmar Lehmann take us on a fascinating tour of the many ramifications of the Fibonacci numbers. The authors begin with a brief history of their distinguished Italian discoverer, who, among other accomplishments, was responsible for popularizing the use of Arabic numerals in the West. Turning to botany, the authors demonstrate, through illustrative diagrams, the unbelievable connections between Fibonacci numbers and natural forms (pineapples, sunflowers, and daisies are just a few examples). In art, architecture, the stock market, and other areas of society and culture, they point out numerous examples of the Fibonacci sequence as well as its derivative, the "golden ratio." And of course in mathematics, as the authors amply demonstrate, there are almost boundless applications in probability, number theory, geometry, algebra, and Pascal’s triangle, to name a few. Accessible and appealing to even the most math-phobic individual, this fun and enlightening book allows the reader to appreciate the elegance of mathematics and its amazing applications in both natural and cultural settings.

Elementary Number Theory

Gareth A. Jones, Josephine M. Jones

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Total reviews: 10 Average rating: 5.0 of 5

Editorial Review:

This book gives an elementary undergraduate-level introduction to Number Theory, with the emphasis on carefully explained proofs and worked examples. Exercises, with solutions, are integrated into the text as part of the learning process. The first few chapters, covering divisibility, prime numbers and modular arithmetic, assume only basic school algebra, and are therefore suitable for first or second year students as an introduction to the methods of pure mathematics. Elementary ideas about groups and rings (summarised in an appendix) are then used to study groups of units, quadratic residues and arithmetic functions with applications to enumeration and cryptography. The final part, suitable for third year students, uses ideas from algebra, analysis, calculus and geometry to study more advanced topics such as Dirichlet series and sums of squares. The last chapter gives a concise account of Fermat's Last Theorem, from its origins in the ancient Babylonian and Greek study of Pythagorean triples to its recent proof by Andrew Wiles.

Numerical Methods for Scientists and Engineers

Richard Hamming

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Total reviews: 6 Average rating: 5.0 of 5

The Purpose of Computing is Insight, Not Numbers 5 out of 5 stars.
117 of 117 people found this review helpful.

Throughout the book, that motto is repeated.

By reading and absorbing the material in this book, the reader is left with the tools and the insights necessary to derive their own numerical methods.

No longer will numerical methods be memorized as textbook formulas -- now the reader can adapt and derive a formula to solve a specific problem, instead of trying to fit one of a small number of textbook formulas to a problem.

The distinction is made between numerical analysis and numerical methods, with emphasis on the latter.

The book is roughly divided into two parts. The first part covers classical numerical methods, using classical error analysis (truncation error, roundoff error). The second part reexamines these methods under the frequency domain, analyzing how numerical methods affect various frequencies (the "transfer function" approach).

Numerical methods are derived under an information theory model, such as by finding a quadrature formula of the highest polynomial degree of accuracy, given limited information about the function and its derivatives.

Matrices and linear systems are not discussed as much as one might expect, although one chapter convincingly leads the reader to question some classical methods.

The content is well-rounded, introducing many readers to topics such as random number generators, difference equations and summation formulas, digital filters and quantization, discrete fourier transforms and the FFT, and orthogonal polynomials. A background in calculus is all that is needed.

Many real-world examples and anecdotes are cited, but without too much detail or too many illustrations given.

This book encourages the reader to ask: "What information is available about the problem? How can it be used to solve the problem? What are the limits of this information?" The approach is practical, not merely analytical.

This book teaches what most other numerical books fail to teach: How to derive your own formulas, and thus your own solutions to problems. And that is perhaps the most important lesson of all.

Editorial Review:

For this inexpensive paperback edition of a groundbreaking classic, the author has extensively rearranged, rewritten and enlarged the material. Book is unique in its emphasis on the frequency approach and its use in the solution of problems. Contents include: Fundamentals and Algorithms; Polynomial Approximation— Classical Theory; Fourier Approximation—Modern Therory; Exponential Approximation.

A History of Pi

Petr Beckmann

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Total reviews: 55 Average rating: 3.5 of 5

Great Book, Where It Sticks To The Topic 3 out of 5 stars.
4 of 4 people found this review helpful.

First, let me say that this book is a good overview of a persistent mathematical problem; in this case, deriving the value of "pi," or the ratio between the diameter and circumference of a circle. The author begins at the beginning, by going over Stone Age "mathematics" and showing how and when it occurred to early humanity that this ratio existed. Second, Beckmann is very good when it comes to explaining the mathematics of pi and how it was analyzed historically in mathematical fashion. He also has a good handle on the primary-source material (i.e. historical treatments of pi), and can explain them in modern terms. All told, this makes it useful to someone who is new to the history of mathematics and wants to learn about one of its foremost problems.

Having said that, Beckmann clearly has some faults:

1. He frequently diverges into anti-communist rhetoric, not only tangentially, but at times when it's completely irrelevant and superfluous.

2. He views the past anachronistically; specifically his hatred of the Romans, and his contempt for Aristotle, are obvious.

He is technically correct on many of these scores; the Romans were, in fact, brutes compared to the Greeks, Carthaginians, etc. Aristotle also was also overrated. He is also correct in that, even in recent times, the Romans and Aristotle are given too much credit for things. None of that is in doubt.

What is troubling is that he arrives at these conclusions anachronistically. He see Aristotle as overrated, simply because Aristotle did not emphasize quantitative analysis over qualitative. But as a scientist he should realize that qualitative analysis has its place in the long process of learning. That Aristotle did not do what Beckmann personally (as a 20th century scientist) wished he had done, does not mean Aristotle contributed nothing useful to human knowledge.

Beckmann similarly laughs at other historical figures, implying that their lack of (modern-day) mathematical ability makes them contemptible. In his forward he even mentions that people took him to task for this, and appears amused at this critique. I'm not sure he understands the problem with anachronistic thinking, however. He clearly sees himself as "more clever" than figures of the past, as well as his colleagues who think he went too far in condemning the Romans and Aristotle; this shows a certain amount of hubris which is probably not penetrable.

His anti-communist rhetoric is, perhaps, more understandable, since he lived and studied under a communist regime, and later escaped from it. Even so, much of what he says about communism has no place at all in this book ... it's more or less irrelevant to the topic. Beckmann clearly could have written an "insider's" account of the faults and dangers of communist ideology, and perhaps he should have done so; but not here, in the guise of a treatise on pi.

To sum up: This is a valuable read, but only if you filter out Beckmann's anachronistic, personal biases.

Editorial Review:

The history of pi, says the author, though a small part of the history of mathematics, is nevertheless a mirror of the history of man. Petr Beckmann holds up this mirror, giving the background of the times when pi made progress -- and also when it did not, because science was being stifled by militarism or religious fanaticism.