One to Nine: The Inner Life of Numbers

by Andrew Hodges

Inspired by millennia of human attempts to figure things out, 'One to Nine' takes a witty, hands-on approach to such various topics as musical harmony, the chemistry of sunflowers, and the logic of the game Paper, Scissors, Stone. Hodges shows us that numbers are part of us - they are everywhere. With simple elegance, he encourages us to explore the latest questions of climate change and cosmology, unveiling a universal language which has its roots in antiquity, and yet could enable us to communicate with aliens.

Genres: Nonfiction, Mathematics, Science, Physics, History

328 pages, Hardcover

First published September 6, 2007

About the author

Andrew Hodges is a British mathematician, author and an activist in the gay liberation movement of the 1970s.
Since the early 1970s, Hodges has worked on twistor theory which is the approach to the problems of fundamental physics pioneered by Roger Penrose.
He is a Tutorial Fellow in mathematics at Wadham College, Oxford University.

Reviews

[Kara Babcock · Rating 2 out of 5]

This is perhaps the first time I have condemned a book for its concept but applauded it for its content. Writing a book that examines the integers 1 to 9 on a per-chapter basis is just silly. It's also impossible; the properties of these numbers are inextricably bound up in the properties of all other numbers. Andrew Hodges knows this, and indeed makes no attempt to conceal the fact that the structure of this book is a lie. In each chapter, Hodges gleefully digresses into topics that have only the most tenuous of connections to their titular number.

Some of these topics are very interesting and worthy of entire books in their own right. Hodges covers electromagnetism, quantum chromodynamics, Fermat, Fibonacci, and some cryptography too. But One to Nine's incredible breadth is, somewhat predictably, also a weakness. Hodges provides able summaries of these topics, but in his whirlwind tour of the first nine positive integers, he can't cover the topics in much depth. Although Hodges' explanations of some fairly complicated mathematical concepts are accessible, I don't think people would find them very helpful.

It doesn't help that Hodges jumps from topic to topic, and even from thought to thought, with the pace of a frenzied beaver on speed. And while I've never been high, reading parts of this book made me feel like I imagine being high would feel: "to consider Two-ness is to confront broken symmetries in a world crammed with them." Um . . . OK, sure. "Two-ness?" Really? Hodges' writing is a bizarre mix of airy and wistful. His attempts to come off as jaunty are merely jarring, owing to his constant transitions from one topic to another. And, oh my, I have never seen so many rhetorical questions in my life. Excess and superfluous much?

As a result, Hodges undermines the very effect essential to a popular mathematics text: the sense of wonder that mathematics evokes. He tries for it, and once in a while gets close. But before the monotonous tour guide can say, "And we're moving. . ." Hodges has gone off on a parallel track, and you're forced to catch up rather than stick around and appreciate the beauty of what's just been covered. One to Nine did little to augment my admiration of number (and much diminished any remaining shred of interest in Sudoku).

Hodges also makes sporadic references to Desperate Housewives, climate change, and Alan Turing. Alan Turing, I can understand, because his contributions to cryptography, computer science, and mathematics are pretty important. That doesn't quite justify Hodges' obvious hero worship, however. While mathematics has a role to play in answering questions about climate change, Hodges never actually addresses that role. Climate change just gets mentioned, much like Desperate Housewives (allusions to which I am at a loss to explain at all).

Then in the last chapter, the book abruptly shifts from its focus on the properties of number into a polemic about mathematics in education and how Alan Turing met an untimely end. In both cases, Hodges makes some good points. Yet I don't appreciate being ambushed by such arguments at the end of the last chapter of the book.

All of this makes for a rather dense barrier to the main event. It's clear that Hodges knows what he's talking about. He just didn't convince me of his ability to talk about it well. Nevertheless, despite his meandering through the mathematics, Hodges does make it possible to eke a semblance of erudition from this book. I'm not sure it's worth the effort. One to Nine is by no means a bad book, but it does not excite me or delight me in the way that a book like A Short History of Nearly Everything does.

Maybe it needed more Two-ness.

[Monte · Rating 3 out of 5]

A frank acknowledgment that anything I wrote was bound to resemble Constance Reid's seminal From Zero to Infinity doesn't stop mathematician and biographer Hodges (Alan Turing: The Enigma) from boldly launching into his own rather disjointed explanation of the place of the numbers one through nine in mathematics and (primarily Western) culture. Pop culture references and political topics such as global warming, presumably meant to make terms like quantum of existence a little less scary to the novice, appear alongside subjects of more interest to math nerds (the author debunks the common assumption that mathematicians are male, overweight and perennially single). Some knowledge of mathematical vocabulary and history is necessary to fully appreciate Hodges's merry skipping from one subject to another—a single page mentions Vonnegut's fiction... Plato's aesthetics, Euclid's pentagons, Fibonacci's rabbits [and:] the inspiration of Islamic art and its parallels in Kepler—but even the most halfhearted former math major will find a lot of familiar topics, like Schrödinger's cat and the equivalence of 1 with 0.99999.... The result is not entirely satisfying to either numerophobes or numerophiles.

[Daniel Messer · Rating 2 out of 5]

I was so bummed about this one. It looked really good and I love the history of mathematics. But it seemed so haphazardly written, like facts thrown on a page. There were times I thought I was missing something, so I'd go back and re-read a paragraph to find that no, I hadn't missed anything. There were times that the ideas were disjointed and there didn't seem to be much connectivity.

Too bad. It looked so good.

[Luke Paulsen · Rating 1 out of 5]

In Neal Stephenson's wonderful novel Anathem, there's a memorable description of a Book that contains a compilation of gratingly shoddy, unsound, or not-quite-accurate works of scholarship. Studying this Book, the theory goes, is the intellectual's ultimate punishment: to have to understand and internalize such inconsistencies is a kind of torture to the scholarly mind.

This is roughly how I felt when reading One to Nine.

It's not that any one point in the book is wrong, exactly. Most of the points made were (I happen to know) accurate enough, and a couple were even interesting news to me. What made One to Nine painful reading was the lack of any discernible structure. Or of any thesis. Or of any intended audience. Or, indeed, of anything much resembling an argument at all.

In part, the book's structure is to blame. To any serious mathematician-- and even to some unserious ones like myself-- the idea of organizing a math book around the digits one through nine is almost laughably incoherent. And Hodges utterly fails to rescue the book from this perception. In content, the text is one part mathematical trivia, one part poorly-explained math fundamentals, and one part opinionated sermonizing. What's more, the content is mixed together with no apparent regard for rational order. Some of the chapters profess to have hazy themes supposedly correlated with their numbers, but Hodges fails to treat this standard with any kind of rigor. Even material on the same topic will appear in several different chapters, and within any given chapter many different topics are hopelessly mashed up with the aid of credulity-straining transitions.

Hodges's prose doesn't help much either. A rose-tinted review on the cover describes it as "lively", but even adjectives like "breezy" or "blithe" would be too generous. The text has a certain stream-of-consciousness feel to it. Nowhere is there much concerted effort at argument or explanation. Hodges simply leaps from one assertion to the next. And as often as not, what he asserts isn't even mathematics; it's history, sociology, or outright politics. On these topics Hodges shows no qualms about vomiting the most cliched conventional wisdom of mid-2000's British intelligensia onto the page. Most often these come as pot-shot asides for which one might forgive the complete lack of justification or self-awareness. But there are yet more blatant examples: Hodges harps on global warming in every single chapter, apparently out of some vague belief that everyone would agree with him in only they understood the numbers One to Nine. Which, after reading, they still won't. Hodges's line of thought, such as it is, is too technical for non-mathematicians to follow and too flimsy for mathematicians to enjoy. It's merely a professor's undisciplined holding-forth, with no audience in mind except himself.

I exaggerate, of course. Slightly. But I am in complete earnest that this book is not worth reading. For readers wanting an engaging, not-overly-technical introduction to the historical, aesthetic, and cultural side of mathematics, I can heartily recommend Journey through Genius: The Great Theorems of Mathematics by William Dunham as a far superior alternative.

[Sean Maguire · Rating 4 out of 5]

A great romp through the integers, with plenty of side trips and quizzes available for those even more mathematically inclined than I. And who knew that Queen guitarist Brian May (who even knew the name of a Queen guitarist) was an astrophysicist who interrupted his studies to play in a band and recently completed his PhD? And took over from Cherie Blaire as chancellor of some UK university. But I digress. This was a great refresher for anyone who has temporarily misplaced their regard for the depth and wonder of numbers. And it makes me want to read Constance Reid's 'From Zero to Infinity' (1955).

[M · Rating 3 out of 5]

One to Nine is a stream of consciousness about everything relating to numbers: the numbers themselves, the history of their discovery, computers, algorithms, the universe, theoretical physics, science, climate change, and the relevance of mathematical thought and learning to modern society. Especially with that last point, it isn't strictly about the numbers one to nine, but it's interesting. I found a lot of misprints, even with the math problems.

[Sarah Sammis · Rating 2 out of 5]

I remember hearing about One to Nine: The Inner Life of Numbers by Andrew Hodges. I want to say I heard it on KQED, my local public radio station but I'm having trouble verifying my memory. I know it was an extended review that I heard somewhere was the reason behind me adding it to my wishlist. As I am married to a math professor and have a calculus teacher mother in law and, frankly, like math, I had to read the book.

The book has a chapter devoted to a different number, the first nine natural numbers. So chapter one tries to cover everything numerically interesting about one. Then the book repeats the process with two, three, and so forth, all the way through nine.

I hoped I might pick up something new, a numeric tidbit I didn't know. Or maybe I'd learn a little history about the numbers. What I got instead was an encyclopedia of numerology presented as prose. It was dense, disjointed reading.

What I also discovered is that at least for a layman's book on numbers, I had heard of nearly everything presented in the book. So it wasn't the new and exciting look at numbers I was hoping for. That said, it is still a great reference on things one to nine.

[Heron · Rating 2 out of 5]

Ugh. Couldn't finish it. I love math and I love books - why would I hate a book about math? Oh yeah, cause it's not about math - it's about the pop science and pseudo-history of numbers. Leave it.


UPDATE: So I accidentally kept reading this book (it was on my bedside table) and I'm amending my review. It got a LOT more mathy as I read further. Too mathy. Like complicated imaginary number equation mathy. Still bad, but not one-star bad.

[Dennis Littrell · Rating 2 out of 5]

Not for the mathematically challenged

Oxford Fellow Andrew Hodges, who wrote the very well received biography, Alan Turing: The Enigma (1992), uses--rather quixotically I might say--the one to nine format to delve into the world of mathematics. His emphasis is on number theory, mathematics as applied to physics, and mathematics as applied to cryptology. The text is difficult, and the puzzles strewn throughout, whether labeled, EASY, GENTLE, TOUGH, HARD, TRICKY or DEADLY, proved mostly too difficult for this non-mathematician.

For those readers versed in number theory, that branch of mathematics in which numbers are explored purely for their own sake without even the dream of a practical application, this book is probably a delight. And for cryptologists it is probably a double delight since Hodges explores in some considerable depth the delicious irony of how pure mathematics became contaminated, as it were, when it was noticed some years ago that the encryption of messages could be facilitated by using very large numbers with unique divisors. While it is easy to multiply two even very large numbers and get a unique result it is enormously difficult to find the unique factors that make up a very large number.

At any rate that is my understanding. And if I have gotten it wrong it is only because I am not much of a mathematician. Which brings me to the central criticism of this book. To put it bluntly I don't think anyone but a mathematician can fully appreciate Andrew Hodges' text. It's that difficult. Additionally, Hodges, who is a physicist as well as a mathematician, brings string and twistor theory into the fray further multiplying the difficulties for the general reader.

But even more off-putting (and this explains some of the negative reviews this book has garnered) is the fact that the book is more than a bit self-indulgent. Hodges's political views are a bit too obvious and gratuitous (although not necessarily disagreeable). He digresses often, sometimes whimsically, sometimes unaccountably. He employs naked jargon, insider allusions, and unexplained references. His subject matter spills over and jumps around from one chapter to other making the "One to Nine" structure seem artificial what with matter pertaining to the number six, for example, appearing in the chapter on the number seven and vice-versa.

I think it's obvious that the sort of book that Hodges has written here must needs another sort of structure, perhaps in three parts, one dealing with encryption, the second with pure number theory, and the third with mathematical physics. He is following to some extent (as he acknowledges) the structure that Constance Reid used so successfully in "From Zero to Infinity" (1956, new edition 2006) in which the chapters were entitled "Zero," "One," "Two,"…"Nine," and then "e" and "Aleph Zero." It's too bad that Hodges didn't emulate Reid's reader friendly prose--and he's a good enough writer to do it--instead of her structure.

Finally I didn't like the fact that the reader has to go to a Website to get the answers to the puzzles!

--Dennis Littrell, author of “The World Is Not as We Think It Is”

[Steven · Rating 2 out of 5]

I can't really add more than what's been elucidated in other reviews: the topics are interesting, the writing/approach is haphazard, and the overall effect is going to both overwhelm lay persons and underwhelm those with expertise in the field.

FWIW, I've done my share of differential equations and linear algebra during my studies, but I haven't revisited those topics in earnest in years, and I found the explanations and trivia frustratingly inaccessible most times. I would've appreciated more explanations of concepts like 'modulo' and 'congruence,' even though this isn't a textbook, as such 'bridges' might've helped more readers at least attempt the posed questions.

[RGD · Rating 3 out of 5]

“It is the eye of mathematics, seeing with insight”

A kind of good books to be read many times. It needs a strong mathematical background to go through.

[Craig · Rating 4 out of 5]

Interesting snippets and puzzles on the first few natural numbers.

[Shane Campbell · Rating 1 out of 5]

Weak sauce

[Old Man Aries · Rating 1 out of 5]

Nella mia carriera di lettore c'è sempre stato un punto quasi totalmente fermo.
Per quanto potessi non gradire un libro, per quanto potesse stufarmi, non sono mai arrivato al punto di abbandonarlo, salvo forse una volta: è sempre stata una questione di principio, unita al desiderio di farmi l'idea completa del libro senza potermi domandare se magari in seguito migliorasse.
Orbene, tutte le regole hanno un'eccezione e la mia è giunta proprio in questi giorni: "Il curioso dei numeri", di Andrew Hodges, è forse uno dei peggiori libri mi si capitato tra le mani e sicuramente uno dei peggio scritti.
L'idea originale era assolutamente intrigante: prendere i numeri più comuni (dall'1 al 9) e raccontarne misteri, aspetti affascinanti, coincidenze, potenzialità, ecc...
Il problema è il modo in cui l'autore ha messo in pratica l'idea di base: abbiamo un'accozzaglia di frasi spesso slegate tra di loro, argomenti che saltano di palo in frasca, disserzioni che non è ben chiaro cosa abbiano in comune col numero cui si riferiscono, conclusioni tratte senza fornire le basi per condividerle o meno, e via dicendo in questo modo.
Non dubito che l'autore sia esperto nel suo campo, tutt'altro, ma questo non lo rende certo un bravo divulgatore (e di libri di divulgazione ne ho letti parecchi); il risultato è che non è chiaro per chi sia stato scritto questo libro: se per i neofiti, allora fallisce in pieno l'intento, dato che dopo due o tre paragrafi allontanerebbe anche l'umanista più volenteroso, se per matematici, beh, allora non si capisce per quale motivo uno dovrebbe andare a leggersi un libro del genere.
Deludente, veramente deludente.

[Maurizio Codogno · Rating 4 out of 5]

(se vuoi una mia recensione più seria di questo libro, va' su Galileo, http://www.galileonet.it/recensioni/1... !)
Un Vero Matematico non deve necessariamente usare i numeri irrazionali, immaginali o surreali (anche se sono divertenti, una volta che uno riesce a capire come si formano). Kronecker asserì che "i numeri interi provengono da Dio, tutto il resto è opera dell'uomo": ma anche solo con i numeri da uno a nove c'è già materiale assolutamente a sufficienza, come Andrew Hodges mostra in questo libro che in originale ha appunto il titolo molto più icastico "One to Nine". Ispirandosi a una tradizione che parte da G.H.Hardy che tuonò contro il "marxista pratico" Lancelot Hogben che scriveva dell'utilità della matematica per giungere a Constance Reid e il suo "Da zero a infinito", Hodges - noto per la sua biografia di Alan Turing - racconta un po' di matematica ma non solo in nove capitoli, ciascuno dedicato ad alcuni aspetti. Il filo conduttore, oltre ai numeri stessi, è dato dal... sudoku, che evidentemente Hodges apprezza molto: nel libro ci sono escursioni nella fisica antica e moderna, ma anche una tirata contro il modo in cui si insegna la matematica nelle scuole inglesi (la sua proposta è "farne magari di meno, ma scegliere cose utili"). Una lettura piacevole, anche per lo stile di scrittura caustico ma leggero ben reso nella traduzione; solo in qualche caso verso metà del libro l'autore forse esagera con le formule matematiche, che potrebbero rimanere indigeste al lettore casuale.

[St Fu · Rating 5 out of 5]

OK, it's unfair of me to be rating and reviewing this book which I've just started but seeing all the negative reviews on this list disturbed me and forced my hand. Yes, it's sometimes more free-associative (in the psychoanalytic sense, not some mathematical sense involving associative laws) than is typical for a non-fiction ouvre, but this isn't a negative from my perspective. I find it freeing and exciting. It doesn't talk down to the reader, bringing up deep philosophical considerations (yes, and often rushing through them when they are worth a book of their own) and making references to pop and not so pop cultural phenomena some of which I admit I could not identify. But it doesn't obfuscate either. So far, it's like a smarter Gödel, Escher, Bach.

OK, I'll be back to say more when I finish.

[Ron · Rating 3 out of 5]

Each chapter was supposed to be about a specific number. However, it wasn't always easy to keep in mind how the number related to the subject. For example, RSA encryption is covered in chapter 8, which is dependant on Prime Numbers, and obviously eight is not prime.

However, I did enjoy the breadth and depth of this book in covering the math and physics and computer science details covered in the book. But I would have preferred more common history for each number, like the brief sentance in chapter 7 covering how seven has these famous celebrities: Vices, Virtues, League Boots, Year Itches and Dwarves. I would have liked these celebrities discussed more.

[Aaron Humphrey · Rating 3 out of 5]

I was expecting a bit of a light-hearted romp through the numbers. While it does wander a bit, and makes a bizarre number of Pet Shop Boys references (and mentions of climate change), it does settle down once a chapter or so for a not-completely-superficial examination of a topic free-associated to by the number of that chapter. Which topics include particle physics, probability theory, number theory (overlapping with, and even referencing, the Marcus du Sautoy book I read the week before), and more that I've forgotten. None of it was too far over my head, but then let's not forget that my major was in physics, and I took a few math options as well.

[Caitlin · Rating 1 out of 5]

The author knows a little about a lot. This clever-esque book devotes a chapter to each number 1- 9, but I didn't learn a thing about anything. Other than, that is, how smug the author was about dropping references to some obscure mathematical/physics/philosophical topic with every new sentence.

This book would be a lot of fun if you had the exact same knowledge base as the author. But, for everyone else, the references aren't explained for the most part, and the book quickly becomes a non-illuminating slog.

Bah!

Oh, and I finished this book months ago, but felt uninspired to write about it, so avoided this site. Bought time, neh?

[Riccardo · Rating 2 out of 5]

��Adori un libro quando, a venti pagine dalla fine, decidi che non ne puoi proprio pi�� e lo butti nel cestino.��

In realt�� non �� spazzatura perch�� contiene alcuni spunti interessanti. Tipo, cinque frasi in tutto il libro. Per�� per il resto �� il classico libro divulgativo scritto da chi invece non sa divulgare: capitoli che dovrebbero essere dedicati a un numero (e di roba da dire ce n'�� un sacco) che invece divagano sul nulla, perch�� quando si cita qualcosa senza spiegarlo di nulla si tratta, solamente perch�� l'autore possa rileggere le proprie parole e dire ��CAZZO MA QUANTO NE SO?��

Da evitare.

[Antonio Plastini · Rating 3 out of 5]

Che dire... troppa carne al fuoco.
Mi aspettavo qualcosa di più "semplice", anche perché ritrovarsi
al centro della teoria dei twistori è un po' sentirsi naufraghi in un mare in tempesta.
Spicca ogni tanto qualche stilettata al sistema scolastico inglese (vedesse
quello italiano!) per non parlare delle ridicolaggini dei media che, in Inghilterra
come in Italia, hanno il simpatico vizio di parlare di cose che non comprendono.
In definitiva il libro lo si può definire "godibile", anche se ogni tanto bisogna
aggrapparsi a qualcosa di solido per non sprofondare negli abissi della matematica di frontiera.

[John]

This book never seemed to know its audience. There were countless references to Desperate Housewives, and Pet Shop Boys, which seemed to be a nod to the average layperson who doesn't know much about math, but then a lot of the math was completely over my head. Which is understandable because I don't know much about math, but often times the mathematics was not clearly explained.
Also, there were more than a few typos and such, which makes me wonder about the accuracy of the math. Not that I'd know either way.

[Liz De Coster]

I stuck with this book for a while, but into the Third section I realized that the constant use of Sudoku as a mathematical comparison was not going to fade with the book, and I bailed. I just didn't find the book interesting - most sections (within the chapter) lasted a page or two at most, so it was hard to keep track of what was going on. There wasn't really anything that pulled me in or grabbed me, nor did I find what I read very informative.

[Lane Wilkinson · Rating 3 out of 5]

A fairly interesting meditation on the integers 1 through 9. Though, by 'meditation' I mean an unstructured, free-association between many disparate thoughts rather than a concerted effort. For those who are interested in the trivia, oddities, and bizarre connections among the numbers, the various Wikipedia pages are more informative. But, for a reasonably entertaining read, One to Nine is all right.

[Steven Feeney · Rating 1 out of 5]

A rambling collection of chapters which refers to constant in-jokes about random subject matter barely related to numbers or maths.

The style of prose can border on condescending but it smashes together haphazard references to Chess, Pet Shop Boys, Bach and countless other items.

Poorly written, poorly executed but a good concept.

[Konrad]

Getallen beheersen ons leven. Ze verwijzen naar verschillende dingen. Maar wat zijn ze eigenlijk? En wat vertellen ze over ons en over ons wereld?



Ik ben net begonnen te proberen te ontdekken wat ik tot nu toe op moeilijke wijze begrepen heb.

[Jennifer · Rating 2 out of 5]

This book has some very interesting parts, but is so disjointed that I'm finding it really difficult to get through. Not at all what I expected from a book about math. If it were called "A Whole Bunch of Observations about Math and the World," I might have gotten past the first chapter.

[Dave Guia · Rating 2 out of 5]

I like mathematics, but not enough to be swooped up into the laborious formulas and theories that the author puts us through. Way too advanced for a guy like me. On a scale of one to nine...I'd give it a 22/7...

[Karen · Rating 3 out of 5]

You really have to be a math geek to enjoy this, I think. I am, I did.

[C · Rating 5 out of 5]

"Interesting number relationships"