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The Square Root of 2: A Dialogue Concerning a Number and a Sequence
An elegantly dramatized and illustrated dialog on the square root of two and the whole concept of irrational numbers.
- GenresMathematicsNonfiction
272 pages, Hardcover
First published December 8, 2005
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Displaying 1 - 6 of 6 reviews
May 18, 2020
Recommended by GR acquaintance Robert - looks fun and maybe easy enough for me.
Ok, here's the background. In school I only got as far as 'pre-Trigonometry' four decades ago. A few years after that I explored a little bit of statistics, topography, and non-Euclidean geometry. I remember the statistics best because the topic comes up in the science books that I read, but I still wish that I had a better understanding.
So now I'm trying to read this. I have to be wide awake, and everyone else in the house has to be engrossed in their own stuff so as not to distract me... so I'm not very far. But I have already learned one thing I'm grateful for--'rational' and 'irrational' numbers refer to the fact that former can be expressed as a *ratio* and the latter can't. Oh if only that were taught in public schools! I think that many math teachers, at least in the US, aren't particularly adept in math (or science)... because mainly they have to be adept in so much else like classroom management, political negotiation, etc.
Anyway, wish me luck... here I go to try some more!
---
Nearly half done. I admit I read, but didn't study, most of the algebra chapter, because it was all about proving things. And I'm satisfied, at least for now, with the proofs, without working them all out myself.
The thing that would most have improved the book would be more sensitive publishing choices. Wiser page breaks, a more clear font, and a larger typeface would have made the book much more readable.
The second thing that would have improved it would have been if different symbols than m/n and p/q were chosen by the author. I know they're traditional. My point is that not only are they hard to make out without the best font (and eyes) ever, but also that they are too familiar to the author, and unfamiliar to the reader. If the author had chosen, say, a crescent & a star, a trefoil & a box, he would have been in the same disadvantaged position as the reader, and would have been more likely to slow down and take more care making sure everything was worked out in full detail with no 'typos.' (Not that there were typos that I could see... but then how would any of his beta readers know and how would I?)
---
Done. Too many proofs, too much about the Pell sequence (whatever). I skimmed from p. 150 to 231, at which point it got very interesting again. p. 233, for example, reveals that the fraction of 1/root 2 = 1/2. Seeing how that works, and how it doesn't for other numbers, is nifty. And the Golden Flower section is very cool.
I'm still looking for a math book that charms me the way the ones that I read four decades did. But maybe I need one written as a textbook, with exercises, and/or for younger audiences, that goes more slowly. Most importantly I need one that is simply easier on the eyes!
Ok, here's the background. In school I only got as far as 'pre-Trigonometry' four decades ago. A few years after that I explored a little bit of statistics, topography, and non-Euclidean geometry. I remember the statistics best because the topic comes up in the science books that I read, but I still wish that I had a better understanding.
So now I'm trying to read this. I have to be wide awake, and everyone else in the house has to be engrossed in their own stuff so as not to distract me... so I'm not very far. But I have already learned one thing I'm grateful for--'rational' and 'irrational' numbers refer to the fact that former can be expressed as a *ratio* and the latter can't. Oh if only that were taught in public schools! I think that many math teachers, at least in the US, aren't particularly adept in math (or science)... because mainly they have to be adept in so much else like classroom management, political negotiation, etc.
Anyway, wish me luck... here I go to try some more!
---
Nearly half done. I admit I read, but didn't study, most of the algebra chapter, because it was all about proving things. And I'm satisfied, at least for now, with the proofs, without working them all out myself.
The thing that would most have improved the book would be more sensitive publishing choices. Wiser page breaks, a more clear font, and a larger typeface would have made the book much more readable.
The second thing that would have improved it would have been if different symbols than m/n and p/q were chosen by the author. I know they're traditional. My point is that not only are they hard to make out without the best font (and eyes) ever, but also that they are too familiar to the author, and unfamiliar to the reader. If the author had chosen, say, a crescent & a star, a trefoil & a box, he would have been in the same disadvantaged position as the reader, and would have been more likely to slow down and take more care making sure everything was worked out in full detail with no 'typos.' (Not that there were typos that I could see... but then how would any of his beta readers know and how would I?)
---
Done. Too many proofs, too much about the Pell sequence (whatever). I skimmed from p. 150 to 231, at which point it got very interesting again. p. 233, for example, reveals that the fraction of 1/root 2 = 1/2. Seeing how that works, and how it doesn't for other numbers, is nifty. And the Golden Flower section is very cool.
I'm still looking for a math book that charms me the way the ones that I read four decades did. But maybe I need one written as a textbook, with exercises, and/or for younger audiences, that goes more slowly. Most importantly I need one that is simply easier on the eyes!
June 9, 2022
A mathematical exploration of the irrationality proof of sqrt(2) and the remarkable properties of this number via a dialogue between a master and his student. A breezy read interspersed with mathematical gems.
March 29, 2009
An excellent book which gives good insight into how to think about math problems, and how to think like a mathematician. And, it's a fun read with plenty of thought-provoking puzzles.
October 4, 2025
This is an absolute funfest of a book! Never could I have imagined there was so much to talk about regarding that most primal of irrational numbers, a number that is often used in explaining and defining the concept of irrational numbers, a number that has its provenance going back to thousands of years, across civilizations and theories, all the way to the incandescent genius of Srinivasa Ramanujan.
Though, stepping back for a moment, one must caution that this book is going to take some patience, and will need the reader to be interested in math in general, and algebra and basic number theory in particular. This may not incite an interest, but it can certainly ignite a spark of an interest into something that is sure to last a lifetime. I wish I had a chance to read this, or even better, have a conversation like this - in my younger years!
I can't recommend this book enough. It explains dense concepts like irrational numbers, and defining the conditions of irrationality and rationality, and number theory with generous doses of conversational wit and contagious enthusiasm. The book starts off with the most basis of questions around what is the square root of the number 2, and how did the earliest mathematicians in human history go about proving what seems obvious to most of us - why is the square root of 2 (or of 3, or of 5, etc.) considered irrational. There are continuous fractions, identities, Pell sequence, even the fabled Fibonacci series makes a guest appearance. More than anything else, it is the perspicacity lucidity found all across the text that makes it a genuine pleasure to read. The author makes it really easy to follow, and in fact he had his daughter act as a proxy for all young readers out there, to call out anytime the text was becoming too difficult or complex, or veering off into dense mathematics that would diminish the appeal to all such casual readers.
Once he establishes the basic concepts and definitions of the irrational number that is the square root of 2, he subsequently reinforces that understanding in numerous ways, and along the way scatters gems of mathematical understanding that are just marvelous. Towards the end, the book tries to pick up some real world examples and usage of not just the square root of 2, but also other irrational numbers, and along the way even alludes to such stalwarts as Ramanujan, Gauss and Hardy, and even the Golden Ratio, before calling it a day.
A brilliant achievement!
Though, stepping back for a moment, one must caution that this book is going to take some patience, and will need the reader to be interested in math in general, and algebra and basic number theory in particular. This may not incite an interest, but it can certainly ignite a spark of an interest into something that is sure to last a lifetime. I wish I had a chance to read this, or even better, have a conversation like this - in my younger years!
I can't recommend this book enough. It explains dense concepts like irrational numbers, and defining the conditions of irrationality and rationality, and number theory with generous doses of conversational wit and contagious enthusiasm. The book starts off with the most basis of questions around what is the square root of the number 2, and how did the earliest mathematicians in human history go about proving what seems obvious to most of us - why is the square root of 2 (or of 3, or of 5, etc.) considered irrational. There are continuous fractions, identities, Pell sequence, even the fabled Fibonacci series makes a guest appearance. More than anything else, it is the perspicacity lucidity found all across the text that makes it a genuine pleasure to read. The author makes it really easy to follow, and in fact he had his daughter act as a proxy for all young readers out there, to call out anytime the text was becoming too difficult or complex, or veering off into dense mathematics that would diminish the appeal to all such casual readers.
Once he establishes the basic concepts and definitions of the irrational number that is the square root of 2, he subsequently reinforces that understanding in numerous ways, and along the way scatters gems of mathematical understanding that are just marvelous. Towards the end, the book tries to pick up some real world examples and usage of not just the square root of 2, but also other irrational numbers, and along the way even alludes to such stalwarts as Ramanujan, Gauss and Hardy, and even the Golden Ratio, before calling it a day.
A brilliant achievement!
April 8, 2023
I love the fact that this books is
1) about a single number, sqrt(2), that has so much to say about itself.
and
2) written like a dialogue between a pupil and a mentor. I would love to read more books written like this.
1) about a single number, sqrt(2), that has so much to say about itself.
and
2) written like a dialogue between a pupil and a mentor. I would love to read more books written like this.
Displaying 1 - 6 of 6 reviews