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The Irrationals: A Story of the Numbers You Can't Count On, by Julian Havil
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The ancient Greeks discovered them, but it wasn't until the nineteenth century that irrational numbers were properly understood and rigorously defined, and even today not all their mysteries have been revealed. In The Irrationals, the first popular and comprehensive book on the subject, Julian Havil tells the story of irrational numbers and the mathematicians who have tackled their challenges, from antiquity to the twenty-first century. Along the way, he explains why irrational numbers are surprisingly difficult to define--and why so many questions still surround them. Fascinating and illuminating, this is a book for everyone who loves math and the history behind it.
- Sales Rank: #873951 in Books
- Brand: Havil, Julian
- Published on: 2012-07-22
- Original language: English
- Number of items: 1
- Dimensions: 9.40" h x 1.02" w x 6.41" l, 1.29 pounds
- Binding: Hardcover
- 312 pages
Review
"The insides of this book are as clever and compelling as the subtitle on the cover. Havil, a retired former master at Winchester College in England, where he taught math for decades, takes readers on a history of irrational numbers--numbers, like v2 or p, whose decimal expansion 'is neither finite nor recurring.' We start in ancient Greece with Pythagoras, whose thinking most likely helped to set the path toward the discovery of irrational numbers, and continue to the present day, pausing to ponder such questions as, 'Is the decimal expansion of an irrational number random?'"--Anna Kuchment, Scientific American
"The Irrationals is a true mathematician's and historian's delight."--Robert Schaefer, New York Journal of Books
"From its lively introduction straight through to a rousing finish this is a book which can be browsed for its collection of interesting facts or studied carefully by anyone with an interest in numbers and their history. . . . This is a wonderful book which should appeal to a broad audience. Its level of difficulty ranges nicely from ideas accessible to high school students to some very deep mathematics. Highly recommended!"--Richard Wilders, MAA Reviews
"To follow the mathematical sections of the book, the reader should have at least a second-year undergraduate mathematical background, as the author does not shrink from providing some detailed arguments. However, the presentation of historical material is given in modern mathematical form. Many readers will encounter unfamiliar and surprising material in this field in which much remains to be explored."--E. J. Barbeau, Mathematical Reviews Clippings
"[I]t is a book that can be warmly recommended to any mathematician or any reader who is generally interested in mathematics. One should be prepared to read some of the proofs. Skipping all the proofs would do injustice to the concept, leaving just a skinny skeleton, but skipping some of the most advanced ones is acceptable. The style, the well documented historical context and quotations mixed with references to modern situations make it a wonderful read."--A. Bultheel, European Mathematical Society
"This is a well-written book to which senior high school students who do not intend to study mathematics at university should be exposed in their last two years at school. The ideas are challenging and provocative, with numerous clear diagrams. The topics are presented with numerous examples, and unobtrusive humour which renders the exposition even more palatable. The book would be an ideal source of ideas in a mathematics course within a liberal arts college because it links not only with the historical source of mathematics problems, but also with some of the great ideas of philosophy."--A. G. Shannon, Notes on Number Theory and Discrete Mathematics
From the Back Cover
"Readers will be swept away by Havil's command of the subject and his wonderful writing style. The Irrationals is a lot of fun."--Robert Gross, coauthor of Fearless Symmetry: Exposing the Hidden Patterns of Numbers and Elliptic Tales: Curves, Counting, and Number Theory
About the Author
Julian Havil is the author of "Gamma: Exploring Euler s Constant", "Nonplussed!: Mathematical Proof of Implausible Ideas", "Impossible?: Surprising Solutions to Counterintuitive Conundrums", and "John Napier: Life, Logarithms, and Legacy" (all Princeton). He is a retired former master at Winchester College, England, where he taught mathematics for more than three decades.
Most helpful customer reviews
45 of 49 people found the following review helpful.
Takes some effort on the readers part but the payoff is great.
By Peter D. McLoughlin
I had to read this book twice. The first time I skimmed it and shyed away from the proofs. That was a mistake. If one takes one's time and tries to get a gist of what the proofs are trying to show, the reader will get a glimpse into the mysteries of irrational numbers. I would recommend the readers have some familiarity with college level mathematics when approaching this book. The reader will come away from this book with a better understanding of how mathematicians struggled with the irrationals over history and expanded understanding of this pandora's box opened by the legendary Hippias who was a pythagorean who shared the secret of their existence to the world and as the legend goes was thrown overboard a ship by his angry brethren.You will learn about Greek geometric proofs of incommeasurables. Next you will be introduced to surd arithmetic in India and Islamic civilization,then Medieval Europeans then pick up the thread. The proofs of pi and e irrationality are worth a closer look and Algebraic and Trancendentals are discussed along with proofs of e and pi as transcendental numbers. The later chapters cover the 19th century rigorists in Germany as the hammer out definitions of the irrationals and the real numbers. The last chapter covers some of the applications of the study of irrationals.One chapter I really like has a good discussion of the randomness with regard to irrationals. If you willing to put in some effort an it is an illuminating book.
49 of 56 people found the following review helpful.
Why do irrationals exist at all? It is great to be given a glimpse of it.
By Malcolm Cameron
The Irrationals
A Story of the Numbers You Can't Count On
by Julian Havil
Irrational numbers are defined by what they are not.
They are numbers which "cannot be expressed as the ratio of two integers" or numbers with a "decimal expansion which is neither finite nor recurring". As author Julian Havil goes on to explain, these are definitions in terms of one characteristic quality, not as entities in their own right. How do we use such definitions to define equality between, or arithmetic operations on, two irrational numbers? Although familiar, convenient, harmless definitions: Who is to say that irrationals exits at all?
That is the problem and fascination. We have all heard of pi. We may have heard the quote of Leopold Kronecker "God made the whole numbers, all the rest is the work of man". For a reader fascinated with a universe containing irrational numbers - not the Pythagorean "All things are (whole) number" numbers - this is a book to increase ones mystification and spread one's lack of understanding. It is still a field beset with unanswered questions.
For instance, "every rational number is equidistant from two other rational numbers" or "there is no rational number such that it is a different distance from all other rational numbers" BUT "all irrational numbers have different distances from all rational numbers". And there is not one rational number that lies on the circle the circle x2 + y2 = 3. But x2 + y2 = 5 has an infinite number of points with at least one coordinate irrational but also an infinite number of rational coordinates too. Mathematics starts clearly enough with the integers: how did all this arise the next instant? Read and sympathize with the Pythagoreans with their star pentagram symbol and the Golden Ratio (1 + sqrt5)/2 the "most irrational number" of the nineteenth century.
Then the warnings: see the proofs that pi and e are irrational. Understand them if you can. And read the quote "The book to read is not the one that thinks for you but the one that makes you think". It is as clear as "Messieur Descartes is right and Messieur Fermat is not wrong". But there are surprising things. Continued fractions seemed (to me) to be just mathematical play-things. Yet it is through these "constructions" that both pi and e were proved to be irrational, so the proofs are well worth seeing. A simple continued fraction form of a rational number is finite and that of an irrational number unending.
This is history from Ancient Greece, India and Arabia, medieval Europe nineteenth-century Germany to the twentieth century. No mathematical equation is spared or so it seems. Augustinian monk Micheal Stifel (1487-1567) stated: "It is rightly disputed whether irrational numbers are true numbers or false. Since in studying geometrical figures, where rational numbers fail us irrational numbers take their place and prove exactly those things that rational numbers could not prove ... On the other hand, other considerations compel us to deny that irrational numbers exist at all ... hidden in a kind of infinity".
The story concludes with one of the most remarkable achievements of twentieth century mathematics when Roger Apéry proved the Zeta Function S(3) irrational. Again, we may not fully understand but it is great to be given a glimpse of it. It was important enough for Apéry's 1994 tomb in a Parisian cemetery to be inscribed:
1 + 1/8 + 1/27 + 1/64 + ... =/= p/q
and moreover for the author to include detailed instructions to find it!
Malcolm Cameron
18 July 2012
19 of 20 people found the following review helpful.
Language Skills are Astonishing
By Let's Compare Options Preptorial
Other reviewers point out the amazing depth and breadth of Havil's mathematical journey, and they're right. But another wonderful facet: Havel's command of English, and his continual use of subtle metaphor is delightful, humorous and, frankly, shocking! This is an awesomely right/left brain balanced talent whose intellect is as much fun to watch as the proofs and historical adventure. "We hope with sufficient conviction for hand-waving to be a positive signal" he quips, as an example of his continual tongue in cheek about ponderous scientific method and epistemology. He is genius at making sure we don't take "axioms" too seriously, and what better secret and oyster pearl grain irritant is there than the irrationals? This isn't just subtle-- Havel often points out limits in many senses of the word, from the essence of defining irrationals to quotes like "It is terrifying to think how much research is needed to determine the truth of even the most unimportant fact" (Stendhal) and "One can measure the importance of a scientific work by the number of earlier publications rendered superflous by it" (Hilbert). (Wonder how this would apply to the succession of the planet's Prophets??).
In passing, Havel mentions the recent color reprint of Euclid's Elements as a "novelty" the reader might enjoy. That book:Six Books of Euclid is one of the most astonishing accomplishments in math, printing, education, graphics... and bankrupted the printer in the mid 1800's (originals now go for over $25,000). Pick it up-- it will also be a collector's item!
Havel himself states that if you're not comfortable with real variable calculus and limits (as well as series), you'll get lost, however, with work, you can put the puzzle together with some missing pieces. There are several millions of dollars of worldwide unclaimed prizes in irrational proofs, so this might be a small stretch, but if you review calc II, you can, with effort, get this whole book. You don't need partial derivatives or imaginary numbers.
Like many others, 5 stars! The diagrams and illustrations are not overwhelming, but if you add the graphics you'll find in The Six Books edition above, you'll have both an artist's and a mathematician's dream pair.
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