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The Irrationals: A Story of the Numbers You Can't Count On, by Julian Havil

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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"Mathematicians and serious students of mathematics will find much to admire in this book. . . . Every mathematician and student of mathematics with appropriate background will find [it] to be a valuable resource."---Pamela Gorkin, Mathematical Intelligencer"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

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From the Inside Flap

"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 andElliptic Tales: Curves, Counting, and Number Theory

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Product details

Hardcover: 312 pages

Publisher: Princeton University Press; First Edition edition (July 22, 2012)

Language: English

ISBN-10: 0691143420

ISBN-13: 978-0691143422

Product Dimensions:

6.8 x 1 x 9.8 inches

Shipping Weight: 1.3 pounds (View shipping rates and policies)

Average Customer Review:

3.9 out of 5 stars

26 customer reviews

Amazon Best Sellers Rank:

#1,145,133 in Books (See Top 100 in Books)

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.

It fit the bill.

Interesting

For mathematicians, teachers or just math lovers, very good. It can be read several times, each with a deeperlevel of understanding (specially the demonstrations, some of which are quite elaborated).

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.

Loved it. Sent it on to my Son.

Great!

Great book, no problems with delivery

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