Monday, October 22, 2012


What's a number?

That's a surprisingly difficult question that I plan to revisit a few times. But for now, we can go with this quick definition. Numbers are how we keep track of things. We make a one-to-one relationship between particular marks or actions, and something we want to count. For instance, when keeping score, we often use tally marks to count. I scored ||||| points, you scored |||| points, so I won.

But this is very very impractical. In game 3 of this years NBA finals, the Thunder scored ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| points,
the Heat scored ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| points.

Who won?

Each month, my rent is

How do I know if I've paid the right amount? What if I'm ||||||||||||||||| short? Care to check if I've paid in full?

As I think you can see, things get ridiculously complicated.

Addition and subtraction are super easy (just delete tallies, or add more), but what about multiplication and division?

I have ||||||||||||| pieces of candy, and I want to share them between ||||| students. How many pieces does each student get? (No fair converting to our notation!)

All these problems would be infinitely easier with our modern decimal notation. But our notation didn't fall from heaven. People invented our notation, and invented the algorithms we use. (And, perhaps surprisingly to some of us, there are different algorithms, often used in different parts of the world.)

The first step was relatively natural. "Instead of writing all those lines, let's group them." The Thunder scored XIXXIIIXXIXXX (order doesn't matter in the notation I'm using here). But how do we go from something like XXIXXIIIXXIXXX or LXXXV to 85?--Historically, before  there was our modern decimal notation, that is. That was an incredibly difficult mathematical step. And now that we have this new way of writing--instead of LXXXV, we write 85--how do we do arithmetic in this new system? And how do we know that system will always work? Can you guarantee me that my books will always come out right if I use this new notation of yours, and these new algorithms? That I won't owe more (or less) that I actually pay? These are the questions Leonardo of Pisa, better known as Fibonacci, had to answer when he published his Liber Abaci which popularized this new numeration system. And he's rightly considered one of the greatest mathematicians ever, mostly due to his work with number systems.

See what we've done? Our children are in a position to relive the questions Leonardo of Pisa asked. To work through the same mathematics he did. To conjecture, and play with numbers. To work out our notation, and to come up with algorithms. And what do we do? We think they're being mathematical if they can find:


Or that somehow magically they'll learn to think like Fibonacci if they can just write 395 under the line, and not 415. (A common error.) But what is 395? And what are 537 and 142? And what is subtraction? And why do we get that particular answer, and not a different one?

It's probably impossible for us, as adults, to relive these questions. But we can perhaps get something toward the struggle if we try to work them backwards. What's DXXXVII-CXLII? Are you sure? What's XXXVII times XLIII? (No fair converting to our notation!) Can you come up with an algorithm that always works?

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