How many folds of a sheet of paper would it take to reach the moon? And how many more to the edge of the known universe? Clue: it’s less than a hundred. Isn’t the idea of solving a mathematical conundrum by the simple act of inventing Imaginary numbers taking the piss? What is Euler’s Identity? Why is π not necessarily all it’s cracked up to be and what the hell do : τ, ℑ, ∫, ∑, ∝ all mean? Are all Cretans liars?
So many questions for which I don’t have an answer but here’s one chap who does: Alex Bellos. Alex’s Adventures in Numberland is a pop-sci book that blurs the line between accessibility and brain-acher – at least for a chump like me who’s not had to try and decipher impenetrable equations for the best part of 15 years – but well worth the challenge nonetheless.
Working in a library I have become well used to spotting many varied books, both fiction and non-fiction that come in leaving me thinking ” I must read that”, “there’s another for the list” and “ooh that’s something different, why not?” The list is presently a pretty modest two pages long so far, hopefully not too unhealthy after a year “in the service”, covering crime capers to classics, science books to politics, history, nature and philosophy. This most recent one covers a hitherto under-read realm in my reading sphere.
As I am always acutely aware that there are so many subject areas that I know too little about, when a book pops up the list in one of those areas that I’ve allowed to go untouched for too long I welcome the opportunity to renew acquaintances. And so MATHS! I remember at least some classes at school being not completely awful and as maths is in everything, everywhere according to all the science documentaries, why not find out more? At the very least I was able to go into this one with the surefire guarantee there would no exam waiting at the end of this book.
This is Bellos’ second book – a fact I discovered upon reading the first pages – so I would assume he’s accustomed to judging the level required for the lay reader – and one certainly cannot be a lazy reader if you want to enjoy what he has to offer. Across ten chapters the author draws out a thematic exploration of some of the key discoveries in mathematics history, roughly keeping it to a chronological narrative where possible – though there’s plenty of room for back and forth – it’s no surprise to see the Ancient Greeks regularly re-appear with each new chapter. Each section comes in roughly three acts, designed to help you keep up with the fast moving flow of the argument : i) an accessible introduction with an anecdote or two to help establish the context, ii) a fleshed out middle bit that goes into specific details, often using graphs and pictorial assistance to convey the significance claimed initially, before moving swiftly into iii); a conclusion that assumes you’ve taken all the information in in one nice digested mouthful and are ready to move on to the next – connected – discovery.
If you’ll excuse the consumption metaphor here a moment – certain chapters go down more easily than others, the opener is a pleasant aperitif that considers how societies have various relationships and superstitions with certain numbers – in Japan there is a particular reverence for 3, 5 and 7 – but don’t like 4 . The reason being the word for 4 – shi – is also the word for death. Likewise the number 9’s homophone ku also means torture. I can’t help but feel this was an avoidable confusion, but then linguistic evolution everywhere can be quite mad.
There are surprising statements that can be drawn on so simple an issue as odd and even numbers – historically and culturally there has been a gendered split in association – the number one and odd numbers in general are considered more masculine, whereas even numbers have feminine links – or so suggest certain studies cited in the chapter. We – in its most general sense – also are rather fond of even numbers because of their ease of understanding – a point that can be intuitively accepted quite easily. We generally “like” seeing the number 50 more than we would 23 for example. Understandable, I can’t stand 23. This observation has not gone unnoticed by researchers for Advertising companies. However much we would consciously protest we would not be duped by such superficial differences, we are far more likely to buy a product called Solus 36 than one called Solus 37 – so say the statistics at least!
Thus, the opening chapter is an enjoyably breezy and informative overview of some of the ways people respond to and interact with numbers – interesting, accessible and digestible – the ham salad of starters. It isn’t necessarily a gradual linear progression from smooth mouthfuls to calling for Dr. Heimlich, but there are – for the more rustier mathematicians approaching this – certain parts that will require a clear head, a pitcher of coffee and more than just a one hour lunch window of reading time to get stuck in.
In chapter 7, the abstraction of mathematical concepts has come to the fore with the question of Imaginary numbers – a concept that was invented to deal with the somewhat vexatious issue of negative numbers. We remember ( well so claims Alex) from school that a negative times a positive is a negative, and a negative times a negative is a positive right? So what’s the square root of -1? An apt response comes from the 16th century Italian mathematician Girolamo Cardono who revealed that thinking about these things “gave him mental tortures.“
It is a position that I had some sympathy with at the trickier parts of the book. For the lay reader who’s not touched equations in a long time, there will be times where you may, like me, having to re-read words like sinesoid and parabolica several times before you understand what is going on. However, once you do get there, there is a satisfying sense of at least being able to appreciate at a basic level why those more “in the know” are in awe of certain breakthroughs. The development of understanding of the triangle, conical shapes, and angles, and it’s practical application in the worlds they emerge in is fascinating. For me at least, the word triangulation now actually means something! Being able to calculate far off distances for travelling ships or the position of the stars in the sky using the same simple principle is fascinating and rewarding to read once you’ve settled your mind into a mathematical approach.
In fairness Bellos is upfront about the varying level of technical language used to explain these concepts and one has the sense that there is a limit below which an explanation behind the notation and terminology for Imaginary numbers, quadernicons, and infinitesimals would be too simplistic to make it worthwhile. Still, there is enough to be able to enjoy to compensate for the parts where you may find your brain curled up in the fetal position demanding a more straight forward read (I duly followed up with some PG Wodehouse after finishing this). I tried to take notes on most of the chapters I read with a little bit of information for each one, and I succeeded mostly – although Chapter 8 in my notebook simply reads: “Calculus – hmm, something Newton, 1800s, infinitesimal, help.”
On the other hand, and looping back to my introduction the nature and spectacular size of doubling explained through folding paper is a strikingly succinct way of conveying the exponential swiftness of growth. A 0.1mm fold of paper, folded 6 times has the thickness of a small book, a further 6 folds ( doubling ) gets you close to a metre in size. So far no quibble from the brain. But then it begins to take off rather quickly. Another 6 folds leads to a height equivalent to that of the Arc de Triomphe. Now we’re at 18 folds. It takes just a further 22 doubling folds to reach the Moon. It makes straightforward sense when you see the numbers written down, and multiplied by 2, but this visualistion lends a resounding sense of “wow” about this very simple idea.
The final chapter that discusses proofs is a welcome relief after the more abstract concepts that have preceded it, with Euclid’s Elements invoked to remind us how savvy those Ancient Greeks were, and discussion of the cellular computation behind the mathematical plaything The Game of Life neatly brings old and new together to sign off a challenging but engaging book about numbers and maths. After an appropriate break I’ll happily try out Bellos’ first book in this area, and I do recommend it for those with an interest in numbers and popular science books – it is a worthwhile addition to the bookshelf, but for now, where are you Jeeves?