The Clock Notation

I’m reading the book Prime Obsession, by John Derbyshire, which tries to explain the Riemann Hypothesis to non-mathematicians. I got the book early in my undergraduate but never got around to reading it and then came across it in my book shelf the other day. I’m about half-way through and it has been a delightful read. Derbyshire manages to explain a lot of difficult mathematics without getting too technical and I would recommend the book to anyone starting an undergraduate in mathematics.

When I turned page 268 I was a bit surprised to see Derbyshire’s notation for the quotient rings $\mathbb{Z}/n \mathbb{Z}$ ; he uses $\mathbb{CLOCK}_n$ . The reason for his choice is that arithmetic modulo $n$ is just like adding numbers (or positions) on a (analog) clock. Let’s work this out for $n = 12$ . We begin by replacing the $12$ on our clock by ${0}$ . Then we can start adding things. Say it’s $11$ and we’re starting to drive home from work which will take $2$ hours (damn commute). What will the time be when we arrive at our doorstep (late for dinner)? Well it’s $1$ of course, $11 + 2 = 1$ on the clock face. This is exactly the same as addition modulo $12$ .

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Jonathan Lubin Says:

February 18, 2010 at 8:16 pm

Actually, this is a fairly common way to introduce the concept to elementary-school kids. Problems: multiplication makes no sense (except to consider $\ZZ/12\ZZ$ as a $\ZZ%-module, which I believe they don’t do), and *nothing* is done with the concept later on in kids’ education. Hmmm, actually, I think there’s a problem prior to these: what’s actually happening is that the numbers on the face of the clock are a phs over the cyclic group, so it doesn’t really make sense to add four o’clock to nine o’clock.

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The Clock Notation

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