The Square Root of 2: A Dialogue Concerning a Number and a Sequence

A review in two voices

Note:  This book is reviewed by Doug Ensley, a math professor, and John Ensley, his teenage son currently taking precalculus in high school.  In the spirit of the book, they have written this review in dialogue form.

What did you think of David Flannery’s book on the square root of two?

At first I thought, “Can there really be 250 pages of stuff to say about the square root of two?”  I also thought it had a cool cover.

So you judged a book by its cover?

Yeah, I guess I did at first.  I also wanted to find out how anyone could possibly write an entire book about the square root of two.

In all fairness, the book is more about some mathematics pertaining to the square root of two among other things.   It is only as long as it is because of the playful informal dialogue used throughout the entire book.

Isn’t it kind of weird to write a math book in dialogue form?  The boldface voice made me think of God on a mountain speaking from a burning chalkboard.

The dialogue form between Master and Pupil is a tradition that goes back to Greek writers like Plato.

I still thing it is weird to write in dialogue form, but I’ll admit that it does make the math pretty easy to understand.

Did you catch that the main focus of the book is on the Pell sequence?

The what now?

The driving force behind what the author of the book calls the “fundamental sequence.” Specifically the Pell sequence consists of the whole numbers that appear in the denominators of the successive approximations to the square root of two given early in the book.

Oh, you mean like 1/1, 3/2, 7/5, 17/12, … I remember now that the book did mention the “Pell sequence.” Why didn’t the author just call it that the whole time?

I think the author wanted to avoid traditional sequence notation, and he also wanted to emphasize that this sequence of numbers is natural in the context of approximating the square root of two.

I thought it was pretty cool to see that you could go from one fraction in that sequence to the next number using the rule  .

It seems like most of the book was about just that rule.

You are right.  Everything is centered on that partially because the algebra in that rule is relatively easy to follow.  The author proves the irrationality of the square root of two from that rule, derives the form of the continued fraction for square root of two from that rule, and shows how Heron’s rule can be applied to his “fundamental sequence” to give an acceleration of approximations to the square root of two.  In the final chapter the author even talks a little bit about problems solved by Ramanujan and Gauss.  And all the while the algebra never gets more difficult than multiplying polynomials and simplifying rational expressions!

Speaking of algebra, I was confused a few times because the author would explain something easy in painful detail and then jump right past some more complicated stuff.

Really?  I thought the pace was pretty uniform so that anyone could follow along.  However, I found a few of the marginal notes puzzling.

Like what?

Did you notice the marginal note on page 83 that explains what the symbol “<” means?

Less than?   No, I didn’t see that one.  Isn’t the “less than symbol” used earlier than page 83?

I looked back and found it many times before, the first of which is on page 8.  The same sort of thing happens with a marginal explanation of FOIL well after its first use.

So other than the weird marginal notes, did you like the book?

I did, but I found myself skipping ahead quite a bit. Frankly, I thought the dialogue style got dull after a while, so I found myself just skimming the text looking for the next math point to be made.

I admit that even I did some of that.  I wonder who this book is written for anyway.

I think the author is trying to walk a fine line.  I am not sure of the size of the intersection of “those who are interested” and “those who are at the level at which the book is written.”  I would recommend it to good high school students, but I would warn them to be prepared to stop and check algebra from time to time as well as to skim ahead from time to time.  I also think it would be a wonderful topic for a colloquium presentation for undergraduate students.

Well, I think the book is easy to understand and interesting as long as you like math.  Since it is easy to flip ahead when the dialogue form gets old, I guess I would recommend it to other kids in algebra II or precalculus as well, but only to those who really like math.

Doug Ensley is Professor of Mathematics at Shippensburg University.