The Irrationals: A Story of the Numbers You Can’t Count On

*Julian Havil*

Princeton University Press sends me lots of cool books to review! Here’s one. Remember the irrational numbers, which can’t be expressed as a ratio of integers ? I bet you even know a few cool facts about them, like how is irrational, or how there are actually more irrational numbers than rational numbers. Well, here is a whole book full of in-depth history and analysis of the irrationals. It’s really quite astounding how much Havil manages to pack into this book. Do you know what continued fractions have to do with irrationals? (Quite a lot, it turns out.) Or did you know that the golden ratio, , is the *most irrational* number? And there are transcendental numbers, and the Riemann zeta function… in my opinion the theory of irrational and transcendental numbers is a really beautiful and astonishing branch of number theory, and Havil does a great job showing it. **Be warned**, however: this is not a book for casual reading, and it makes no attempt to dumb things down or gloss over details! To fully follow all the details (which few will actually have the patience for) requires excellent algebra skills and occasionally some calculus. It presents the full details of even some relatively recent advances in our understanding of irrational numbers. But for all that it remains quite engaging, and there’s lots to get out of it even if you can’t make it through the whole book.

I’m curious—in what sense is the “most” irrational number? I was recently introduced to some of the results regarding continued fractions and rational approximations in pursuit of my own research, and it seems that one possible way of discussing how “irrational” a number is might be in terms of how slowly or quickly the continued fraction convergents converge. In such a case, wouldn’t it make more sense to say that or Liouville’s constant is more irrational (depending on whether we consider slow or fast convergence to signify greater irrationality)? And what about any of the uncountable number of numbers that aren’t even representable? Wouldn’t it be fair to say that these are “more” irrational that ?

I ask not to be a pain in the ass—I am genuinely curious. I suppose that I will have to get a copy of Havils book. ;)

Whoops! That was a typo. I meant to say, indeed, that the golden ratio is the most irrational number, in the sense that it is the “hardest” to approximate by rational numbers (we don’t get “lucky” with rational numbers that are very close to but have relatively small denominators). Thanks for the catch!

Groovy. Thanks for the clarification.

Julian Havil is one of my favourite math writers. If you liked The Irrationals then you’ll probably also like Gamma, which covers Euler’s constant. (I enjoyed The Irrationals, but I think Gamma is even better.) He’s also written Impossible? and Nonplussed!, which are more “popular” in format — with each chapter covering a different puzzle or curiosity — but without holding back on the mathematical details.