Teaching Old Numbers New Tricks

by Nicholas Laurie
June 14, 2013

Nicholas Laurie '14 is a Royce Fellow working on revealing symmetries and patterns that form the foundation of quantitative thought. 

“And see what we’ve accomplished in 5 minutes of conversation?”

Like the height of his shoulders, laughter eases down in the raised cheeks as he pauses to take a look at his watch.

“It’s been more like forty-five.”

We return to laughing.

This little excerpt is exactly how most of my conversations go when discussing mathematics--it’s easy to get absorbed.

This summer I have embarked on a project that can only be described as uniquely ironic. I’ve never been good with numbers, and while our relationship has been rather rocky over the years, suffice it to say that we have had our differences. I only ever began to become interested in math when numbers started to leave the room; yet here I find myself today, trying to create more of them. Well, maybe not trying to create more, but at least teaching them to be a bit better behaved.

If you take the time to think about it, we really have a lot of different kinds of numbers floating around. We have numbers to count things, we have numbers to tell time, we have numbers to measure distance, and so on. Moreover, these numbers all know different trick

s. They can add, they can multiply, and, if you’re really nice to some of them, they can even do calculus. But the numbers on a clock are different from those counting people. Namely, if you’re on clock at one and you add twelve hours, you’re back again to where you started. Yet if you count a person and add twelve more, you’ve got thirteen.

It turns out there are many examples like this, all sorts of different numbers that people have tamed and taught to do different tricks over the years; different kinds of additions, and multiplications, and exponentiations. In my course of study, I stumbled upon an interesting way to describe addition and multiplication, a kind of iterative process that I can then use to create a new operation, and several beyond that. The trick is that each preserves the structure of the previous, and my hope is to explore these new behaviors adding layer upon layer of complexity to these number systems, like a vast meshwork of cogs and wheels all whirring in unison, each reliant upon the others as they churn out the infinite rhythms of their orchestration.

A lofty goal, and frightening too. Perhaps a little ill-fitted to the picture of a man pouring over a library book and sipping a warm cup of tea; but alas, I’ve already warned of the unique irony of my endeavor. Just don’t ever let anyone tell you, you can’t teach an old number new tricks.