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The Irrationals: A Story of the Numbers You Can't Count On
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.
312 pages, Paperback
First published June 19, 2012
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Displaying 1 - 22 of 22 reviews
May 14, 2016
A thorough examination of the history of thought about irrational numbers. Many fascinating and surprising properties of the irrationals are proven; the book also presents some questions about the irrationals that are surprisingly still unanswered. For example, though e and pi were both proven to be transcendental in the 1800s, it is still unknown whether e+pi is irrational, let alone transcendental!
This book seems to be pitched to the layman, but there are equations and detailed proofs on nearly every page, so I can't imagine reading this without having taken a few college math courses. For the most part I found the proofs followable, with some effort, but there were more than a few where my eyes glazed over and I ended up skipping to the punchline. And sometimes the thing being proved (at great length) wasn't actually all that interesting, or just a subtle variation on something that had already been proved. I also had some problems on my Android Kindle reader where no matter how large I made the text font, the equation font was still tiny. Still, I'd rather have a book be too technical than not technical enough, which is a problem I've had with many popular physics books.
This book seems to be pitched to the layman, but there are equations and detailed proofs on nearly every page, so I can't imagine reading this without having taken a few college math courses. For the most part I found the proofs followable, with some effort, but there were more than a few where my eyes glazed over and I ended up skipping to the punchline. And sometimes the thing being proved (at great length) wasn't actually all that interesting, or just a subtle variation on something that had already been proved. I also had some problems on my Android Kindle reader where no matter how large I made the text font, the equation font was still tiny. Still, I'd rather have a book be too technical than not technical enough, which is a problem I've had with many popular physics books.
March 2, 2021
Starts out gentle with high school algebra and geometry and eases into some delightful college math (where the math gets interesting and fun if not hallucinogenic). It is fun and you get a taste of the infinite without too much of a strain by route of the irrational numbers. Delightful stuff.
March 1, 2024
Totally fun romp through many diverse aspects of the irrationals. Includes lots of examples and proofs. It’s nearly guaranteed you will learn something new here, even mathematicians themselves. Highly recommended.
November 4, 2017
Overall the book was interesting and enjoyable. I liked learning about some ideas I had never considered during graduate school, and it was nice to sit down and finally read and understand proofs that e and pi are irrational and transcendental. The book follows the development of the mathematics of irrational numbers from their discovery by the Greeks to proofs of the transcendentality of e and pi and some 20th century results.
In spite of the book's claim to be accessible to anybody with a background in calculus, I doubt that anybody who hasn't taken an undergraduate analysis class will really understand most of it. That said, as a math PhD I enjoyed that the ideas were generally spelled out clearly instead of using the weasel words one generally finds in graduate math texts.
The book has a fair number of proofreading errors. Some sentences don't make any sense. There are some formatting errors in the equations. There are some occasional minor errors in the proofs, but they are geneally easy to correct. The proof that pi is transcendental contains a serious error, and it took some work for me to modify it to something correct.
In spite of the book's claim to be accessible to anybody with a background in calculus, I doubt that anybody who hasn't taken an undergraduate analysis class will really understand most of it. That said, as a math PhD I enjoyed that the ideas were generally spelled out clearly instead of using the weasel words one generally finds in graduate math texts.
The book has a fair number of proofreading errors. Some sentences don't make any sense. There are some formatting errors in the equations. There are some occasional minor errors in the proofs, but they are geneally easy to correct. The proof that pi is transcendental contains a serious error, and it took some work for me to modify it to something correct.
November 8, 2013
Do not buy the kindle edition of this book. This a mixed review, where I'm guessing that the paper book would be around a 4-star volume, but the kindle edition would need some generosity to be called a 2. The formatting is really that bad.
The first sign of trouble is when the a book about irrationals could not express the square root of two with switching from the regular font a what looked like a low resolution screen shot about two and a half times the size of the surrounding text. Worse yet, when radicals appeared in formulae the text would sometimes jump back and forth between the two modes, which led to symbols not lining up vertically. Instead of concentrating on the text material I found myself trying to reason backwards about which symbols were part of a numerator rather than a denominator. Some symbols were obviously cut in half, and others were silently absent. The captions for figures would appear interjected into a seemingly random part of the main text, often a couple screen/pages away from the figures themselves.
Formatting issues aside, the book is a decent history of a progression of mathematical thought. It is written for people who have done math, not the popular audience. It is accessible enough that physics majors and engineers should have no trouble working through the terminology and proofs, but a true layman would likely be entirely lost for large sections.
A minor peeve is that Mr. Havil seems to project a modern academic motivation to the ancient mystics who make up the early portion of the story. I imagine that the strong reaction they had to people expounding on the nature of incomesurability was not, as Mr. Havil assert, that a devastating blow had been delivered to one of their theories, but instead outrage that someone had been revealing their sacred secrets. After all, "occult" derives from a root having to do with being hidden.
All in all, this is a good math book for math guys. It doesn't meet the standard of a Hoffstader volume, but when you run out of those, this is a decent next choice.
Just not on your Kindle.
The first sign of trouble is when the a book about irrationals could not express the square root of two with switching from the regular font a what looked like a low resolution screen shot about two and a half times the size of the surrounding text. Worse yet, when radicals appeared in formulae the text would sometimes jump back and forth between the two modes, which led to symbols not lining up vertically. Instead of concentrating on the text material I found myself trying to reason backwards about which symbols were part of a numerator rather than a denominator. Some symbols were obviously cut in half, and others were silently absent. The captions for figures would appear interjected into a seemingly random part of the main text, often a couple screen/pages away from the figures themselves.
Formatting issues aside, the book is a decent history of a progression of mathematical thought. It is written for people who have done math, not the popular audience. It is accessible enough that physics majors and engineers should have no trouble working through the terminology and proofs, but a true layman would likely be entirely lost for large sections.
A minor peeve is that Mr. Havil seems to project a modern academic motivation to the ancient mystics who make up the early portion of the story. I imagine that the strong reaction they had to people expounding on the nature of incomesurability was not, as Mr. Havil assert, that a devastating blow had been delivered to one of their theories, but instead outrage that someone had been revealing their sacred secrets. After all, "occult" derives from a root having to do with being hidden.
All in all, this is a good math book for math guys. It doesn't meet the standard of a Hoffstader volume, but when you run out of those, this is a decent next choice.
Just not on your Kindle.
January 4, 2015
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 and this review appears on Amazon
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 and this review appears on Amazon
July 19, 2015
This book is more of a reference book than a book to read for pleasure. I skimmed through most of the proofs, but enjoyed much of the discussion. I especially liked the details about the three definitions of real numbers: the set of all infinite decimal expansions, the set of all equivalence classes of Cauchy sequences, and the set of Dedekind cuts.
December 5, 2023
Everyone will appreciate the historical journey. The layperson may get lost in the proofs but that's OK. Skip them and enjoy the story of discovery and exploration. Some numbers are irrational. Some numbers are even transcendental!
March 23, 2024
There are some excellent nuggets of mathematical concepts in this book that I enjoyed learning, so I can give a qualified recommendation for this book. There are things one is likely to learn from this book even if one has read many other rec math books. However, there are also large tracts of the book that are mostly equations with little or no exposition. Hence, those parts are more like a difficult math textbook with no good professor helping to explain the concepts that the equations characterize. In a number of other places where there is more exposition, the explanations tend to be somewhat muddled or incomplete. There are several of these that I know from better explanations in other books, so I can tell for certain that those explanations are not satisfactory in this book. A rec math book should read like the combination of the math text and the good prof, rather than often having to already know the topic being discussed from somewhere else to be able to make sense of the given author’s explanations. For a comparison, check out the usually much clearer explanations in The Golden Ratio by Mario Livio. That being said, this book went a lot deeper than Livio’s, and I don’t regret reading this book because I value the several extra learnings I got from it.
November 1, 2023
The Irrationals: A Story of the Numbers you can't Count on by Julian Havil is a book about the history of math, specifically irrational numbers. It spans from thousands of years ago to the present, and not only discusses the history of irrational numbers, but also gets into the philosophy of why we even care about irrational numbers. The thing this book did well is it shares an interesting and fascinating story without shying away from math itself. It still feels like reading a historical narrative, but in the process of reading it I learned many new mathematical concepts. From my experience, this is something many casual, non-textbook math books fail to do. However, there were still a few sentences I had to read multiple times, but that is often the case with books that deal with more complicated ideas. As someone who has a love for math, I really enjoyed reading this book, as each day this book would give me new thoughts to puzzle on. My only word of warning to anyone planning to read this book is that it is dense, with certain sections getting very technical, but other than that, it is a great book that I'd recommend to anyone who is interested in math.
November 2, 2018
A fascinating study of the search for irrational numbers in mathematics. The history of the quest begins, as usual, with the ancient Greeks but it is still ongoing in this century. The equations used left me in the dust almost from page one. I firmly believe that God, in His infinite wisdom, did not bless me with an understanding of mathematics. If He had done so, the result would have been too powerful a creature for its own good. I thank Him every day for denying me the opportunity to be puffed up with unmanageable pride and then destroying me for the sake of the universe. But I digress. The story of the mathematicians struggling to understand the irrationals, bit by bit, is a tribute to the human spirit.
September 11, 2020
The book can be read two ways - as a math or a history tome. I did the latter for the most part, only diving into proofs once in a while. Whilst an entertaining and informative narrative on the whole, the book is really only suitable for mathematicians.
December 7, 2024
Quite a fun history of irrational numbers, the proofs of the irrationality and transcendental-ness (?) of various famous constants, and why irrationality and continued partial fractions matter in various contexts.
April 19, 2024
A good read, but at least basic knowledge of number theory is recommended to get true value from this book. I did enjoy it, but much was just over my head.
May 14, 2018
Irrationality in math is not the same as in other arenas. In fact, if the old definition of insanity is doing the same thing over and over again while expecting different results, mathematical irrationality is its opposite. Irrational numbers are ratios that can't be expressed as a fraction of integers -- they never repeat, no matter how many times the division in the ratio is carried out. Which is one wrinkle with them already -- how do we know a decimal never ends and never repeats? Simple math alone can't get us there; irrationality requires a mathematical proof of its existence.
Some irrationals are common and famous -- π as the ratio of a circle's circumference to its diameter is one. Some showed up when mathematicians got curious about what might happen when they played this or that game with numbers or equations, like the mathematical constant called e. In his 2017 book The Irrationals, mathematician Julian Havil offers some of the history of irrationals, first discovered by ancient Greek and some Hindu mathematicians. He explains how some of the better-known were first discovered and how new ones appear even in math today. The ability of computers and their ability to calculate immense strings of digits mean mathematicians are less sure than they used to be about the non-repeating aspect of irrationals -- they probably don't repeat, but there may be some wiggle room.
As in some of his other books Havil is not shy about using mathematical formulas and equations, many of which are blank space to people who didn't progress much beyond pre-calculus and have forgotten large swaths of that. It may be unavoidable but it's an unfortunate feature of what is a really interesting set of ideas about our weird ol' universe.
Original available here.
Some irrationals are common and famous -- π as the ratio of a circle's circumference to its diameter is one. Some showed up when mathematicians got curious about what might happen when they played this or that game with numbers or equations, like the mathematical constant called e. In his 2017 book The Irrationals, mathematician Julian Havil offers some of the history of irrationals, first discovered by ancient Greek and some Hindu mathematicians. He explains how some of the better-known were first discovered and how new ones appear even in math today. The ability of computers and their ability to calculate immense strings of digits mean mathematicians are less sure than they used to be about the non-repeating aspect of irrationals -- they probably don't repeat, but there may be some wiggle room.
As in some of his other books Havil is not shy about using mathematical formulas and equations, many of which are blank space to people who didn't progress much beyond pre-calculus and have forgotten large swaths of that. It may be unavoidable but it's an unfortunate feature of what is a really interesting set of ideas about our weird ol' universe.
Original available here.
August 30, 2014
Great book for deep dive into the mathematics and proofs of irrationals. I skipped the proofs and only got half way through the book before calling it quits. You have to have a lot of time on your hands as well as some good background to dig through all of this. On the other hand, I loved his historical treatment of the ideas that form the structure from which calculus could arise.
August 3, 2014
This is one of the best books I can remember ever reading. A fascinating blend of historical anecdote, number theory, human interest, and cultural commentary, with just enough math to provide a challenge and not dumb down the subject.
January 9, 2015
This was a bit more intense than I had expected, but I enjoyed it very much. I can now prove that the square root of 2 is irrational--a simple and beautiful proof. I also learned that phi is, in a certain sense, the most irrational number of all.
December 19, 2012
Not for the mathematical novice, but very rewarding for those with some ability. Irrationals are more surprising than I expected.
June 7, 2015
Absolutely fascinating!
October 7, 2012
A hard read..
February 28, 2013
I found this book a little bit more dense than I was expecting, but otherwise I liked it.
Displaying 1 - 22 of 22 reviews