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41 of 45 people found the following review helpful
5.0 out of 5 stars Takes some effort on the readers part but the payoff is great.
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...
Published on July 20, 2012 by Peter D. McLoughlin

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18 of 28 people found the following review helpful
2.0 out of 5 stars For the mathematically inclined
I have read tons of popularized accounts on mathematics, number theory and related topics. This book gives the impression that laypeople can read it, stating: "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." However, the book is clearly...
Published on January 4, 2013 by tormodg


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41 of 45 people found the following review helpful
5.0 out of 5 stars Takes some effort on the readers part but the payoff is great., July 20, 2012
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This review is from: The Irrationals: A Story of the Numbers You Can't Count On (Hardcover)
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.
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46 of 53 people found the following review helpful
5.0 out of 5 stars Why do irrationals exist at all? It is great to be given a glimpse of it., July 17, 2012
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This review is from: The Irrationals: A Story of the Numbers You Can't Count On (Hardcover)
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
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18 of 19 people found the following review helpful
5.0 out of 5 stars Language Skills are Astonishing, July 28, 2012
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This review is from: The Irrationals: A Story of the Numbers You Can't Count On (Hardcover)
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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18 of 20 people found the following review helpful
5.0 out of 5 stars "FASCINATING, MYSTICAL, EDUCATIONAL!", July 22, 2012
This review is from: The Irrationals: A Story of the Numbers You Can't Count On (Hardcover)
Julian Havil, a Master in math for decades delivers a very interesting story about numbers we cannot count on. He presents illuminating research on the counting theory, number theory, and a fun-filled historical background on the hidden patterns of numbers. The author includes historical facts from Ancient Greece, India, Arabia, and Europe. In Addition, historical facts are provided as to where it all began to the present day. This amazing book is entertaining from beginning to end as it makes the mind curious in historical research. There's never a dull moment in the information presented on the Irrationals. Highly recommended for math lovers and teachers!
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13 of 16 people found the following review helpful
5.0 out of 5 stars Mathematic on a high level, but very detailed and clearly explained, July 30, 2012
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This review is from: The Irrationals: A Story of the Numbers You Can't Count On (Hardcover)
I've have now finished reading the book and must say that it's a mathematical book containing all the details in the explanations and in the proofs. And thereby makes it possible to be understood without seeking in other books, or asking persons, if there should be something not learned about in high school or university. Mathematic is what I through out my life have liked most of all, next comes programming, of course for by in this using the mathematic, and then comes history.

Especially I liked the more which I in the book learned about Nicole Oresme, a person who was much in ahead of the other back there 700 hundred years ago. And about whom I in the most by me read mathematical books only have found little. And also I in the this book am getting some more details concerning the work by Cantor, his explanations and proofs, which I don't have in the books I owe especially concerning Cantor. But particularly I liked the many sides concerning continued fractions, where there especially come some interesting interrupted continued fractions which I never by myself had been thinking about going on for construction, that is that there in these cases should be something especially beautiful.

But here follow a few historical comments. Concerning Galois Evariste we by a remark, correctly, are told that he was in a mixture of politic and love. And then caused by political reasons were killed as result of a duel. But nearly on the same lines we learn about Abel, and also close to this have Gauss, but we are not told that Gauss newer read the paper from Abel, or meat Abel. And Abel had to return home to Norway, after the travel sponsored by the citizens, for going to Germany universities, and Gauss, and then died poor (born as a "Dane", and died as a "Swede", after Denmark went bankrupt as result of the Napoleon war). Think about which mathematical works we might have got if Gauss had taken care of Abel, like later Hardy to care of Ramanujan, after reading the letter from him. Besides why do we always in the books read that Cantor is a German, even though he actually was born and lived the first 11 years of his life in Russia? Besides his family originally moved from Denmark to Russia, but that was around 38 year before he was born. And this was a result of English bombing of Copenhagen, in 1807, and thereby having destroyed their house. And concerning Kronecker who are mentioned, he actually was the person who counteracted that Cantor ever came to the university where he by himself was professor, and which was where Cantor most highly wanted to be a professor (the mathematical center point in Europe), and this is maybe the reason that Cantor died in a mental hospital. But again these lines are only containing some side marks.

But finally, again, an excellent book which I in the future will be rereading for relaxing.
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3 of 3 people found the following review helpful
4.0 out of 5 stars Great Book but Wrong Format, January 17, 2014
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I loved reading this book. It's a fascinating history of irrational numbers and full of interesting stories about the people and times along the way (question: Why is the "Dunce Cap" a cone??). My frustration is with the electronic format and is the same I've had in the past. The book cries out for you to flip around to look at illustrations and proofs. I read it on my Kindle and it just does not work well. I also like to mark sections with small post-it notes (you should see "The History of Mathematics" by Boyer!) so I can flip to sections I want to share with my classes, etc. I will look for this in book form. But with all that said, this is a fun book to read and shows the struggle we had even admitting that irrational numbers are numbers (and how absolutely clever our predecessors were)! Working my way through their proofs was a real treat. I wonder what we refuse to see, today, even though it's sitting right in front of us?
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4 of 5 people found the following review helpful
5.0 out of 5 stars A wonderful book with a collection of interesting facts appealing to those with an interest in numbers and their history, August 25, 2012
By 
Didaskalex "Eusebius Alexandrinus" (Kellia on Calvary, Carolinas, USA) - See all my reviews
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This review is from: The Irrationals: A Story of the Numbers You Can't Count On (Hardcover)
*****
"One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beautiful theories." --Philip J. Davies

An irrational number is defined by the Wolfram MathWorld as, "a number that cannot be expressed as a fraction for any integers and . Irrational numbers have decimal expansions that neither terminate nor become periodic. Every transcendental number is irrational." There is no standard notation for the set of irrational numbers, but the notations Q, R-Q or R\Q, where the bar, minus sign, or backslash indicates the set complement of the rational numbers Q over the reals R, could all be used. The most famous irrational number is sq. root of 2, sometimes called Pythagoras's constant. Legend has it that the Pythagorean philosopher Hippasus used geometric methods to demonstrate the irrationality of sq root of 2 while at sea and, upon notifying his comrades of his great discovery, was immediately thrown overboard by the fanatic Pythagoreans. Other examples include sq. root of 3, e or pi(my Amazon icon).

In the captivating and intellectually informative book, "The Irrationals," Julian Havil, utilizes the talents of the 'Mathematical troubadour' to articulate the story of irrational numbers and the stories of mathematicians who have pioneered to tackle the irrationals fascinating world, explaining why they are illusive to define, leaving numerable questions still to be tackled. But, through "The Irrationals" Havil portray them as real complex numbers, which arouse the curiosity with a fascinating history, from Euclid's sq. rt. of 2 to Roger Apéry's Zeta. The talented author explores and expounds others from the irrationality of e and pi, to the discrimination between the irrationals and transcendentals. An appealing question for mathematical minds is the nature of the decimal expansion of irrationals.

I loved the Discovery educational mathematics appetizer on irrationals, that may enhance your curiosity on the Irrationals, please check it on youtube.
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1 of 1 people found the following review helpful
4.0 out of 5 stars Book Review, September 8, 2013
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This review is from: The Irrationals: A Story of the Numbers You Can't Count On (Hardcover)
Requires a bit more knowledge of math to really appreciate the book than what I believe the average reader would have. But, if this aspect of math appeals to you, it is a nice book to read and have on the shelf.
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5.0 out of 5 stars Wickedly clever!, June 11, 2014
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I read this book on the recommendation of a mathematician friend after our many philosophical debates. I'm a physiker and cranky to boot. At the very edge of our understanding, there exists a mysterious set of constants. They have been increasingly observed in nature, socialized as precise with one additional decimal place as better instrumentation and computers will do. Then again, there is a problem ... these constants may not be numbers at all as we imagine numbers. At issue is plugging constants into formulas and performing algebraic manipulations to imagine that we might someday describe everything in an algorithm. I defer to Feynman to describe one of these constants, the fine structure constant that holds everything together ...

"It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the "hand of God" wrote that number, and "we don't know how He pushed his pencil." We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out, without putting it in secretly!"

The fine structure constant is not in this book, however, a whole bunch of more basic, taken for granted, mathematical processes are. The P versus NP problem is also not discussed but the solution will determine if our "explanation of everything" might even be theoretically possible in polynomial time.

Havil takes the reader to more basic mathematical curiosities using killer arguments that make you reimagine "numbers". Is `Pi' really a "number"? The Euler number? The sqrt of -1? Havel deep dives dozens of these constructs and mathematical manipulations that we have discovered and used well to help us along in the world. Havel runs down the concept that the proof of a proof, is a proof only after we socialize it for a couple millennia or centuries. Here is fun stuff guaranteed to make you think.

While some reviewers have been critical regarding the amount of math here, I think Havel is simply peeling the onion for the reader to the 3rd or 4th level of detail and that will require experience to understand. However, at the tier one or even tier two level discussions, most any reader that might first finds this book and then is interested enough in history and the philosophy of mathematics will get it. Apply the math you know to enjoy this one. Don't get scared off.
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5.0 out of 5 stars A sound description of how the irrationals went from the bizarre to routine high school material, November 20, 2013
This review is from: The Irrationals: A Story of the Numbers You Can't Count On (Hardcover)
When ranking the level of difficulty of a mathematical textbook, the phrase "mathematical maturity" is often used. This refers to that general increase in mathematical ability that one expects students to achieve as they study more and more mathematics. The phrase can also be used to describe the mathematical community as a whole as it develops, assimilates and then refines new concepts until they often reach the level of the routine.
One sees this thread throughout mathematics, in this book the maturity of the mathematical community in discovering, developing and refining the theorems of irrational numbers is covered. Although there is scholarly debate on the severity of the reaction to the initial knowledge of the existence of the irrationals, there is no question that it was significant. Havil does an excellent job in describing this collective mathematical process and spares no equation in the process. He captures the spirit and difficulties as generations of mathematics toiled for hundreds of years in order to develop a sound definition of the irrational numbers as well as the logical mechanisms to work with them.
There is no question that this ongoing process was a major success, I do not remember the precise time in my education where I was first exposed to irrational numbers, but believe that it was in the ninth grade. Now, irrational numbers are routinely discussed, manipulated and occasionally cussed in the high schools, which is certainly the definition of a topic that is "mature." In no way a popular book on mathematics, this book is a sound description of this trek from the "bizarre" to the routine.

Published in Journal of Recreational Mathematics, reprinted with permission
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The Irrationals: A Story of the Numbers You Can't Count On
The Irrationals: A Story of the Numbers You Can't Count On by Julian Havil (Hardcover - July 22, 2012)
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