10 + 10 = 100

“Any questions?”

I asked this at the end of class today*, and one of my students thought he’d get smart and ask “Why is two plus two equal to four?”

Luckily for the student, the bell was about to ring, and I did not get a chance to work out my modeling thing on him and his smart-alecky question. (That’s what tomorrow’s class is for ;) My only response was “Because we said so.”

Here’s what I’m thinking… the words two and four, and the symbols 2 and 4 are just models we use to represent some quantities. Think back before we had symbols to when Og, the recently ex-hunter/gatherer wants to marry Uga who is also a recent farmer/herder. Og says, “Me have this many egg-layers,” pointing to his two chickens. “You have same,” he continues, pointing to Uga’s chickens, “so together us have this many.” Og does not have symbols for the number of chickens, so he actually refers to them. Uga, being slightly more advanced, and not particularly fond of Og, says “You marry Moga, and you get this many egg-layers.” She holds up five fingers, showing that she has created a new model, representing the chickens with her fingers. Mistaking Uga’s model for something entirely different, Og clubs her on the head, and drags her and her chickens off to his cave.

Fast forward to a civilization with bartering and without my corny dialogue. Now, traders know they cannot bring their chickens with them everywhere, so they must have some way of representing the chickens. Maybe the trader carried with him bagful of small pebbles, a model for the hen pen he has back home. Or, better yet, he has learned how to write – a very powerful and in some cases magical model – and represents his chickens with a bunch of hash marks. Now, we start seeing verbal and symbolic rather than concrete models for quantities.

So, to ancient Romans, and much of Europe in the Middle Ages, there was no 2+2=4. There was only II+II=IV, only without the + or = signs; I’m not sure how they represented addition. The Arab world at this time was much more savvy about their models, and could write something more like 2+2=4.

Now, here’s where we see the power of a good model. While Albertus is trying to represent the population of his town with the numeric model MMMDXXXVII, al-Khwarizmi can use 3087 for the same number, and can also add and subtract his accounts much more quickly than Albertus. A merchant named Leonardo of Pisa (also known as Fibonacci) is one of the people to figure out that Arabic numerals are so much nicer than the Roman ones, and starts using them, being one of the guys who helps start the Renaissance in Italy.

Now a good model is only fun if we can mess with it, and see where it leads, right? So we use the Arabic numerals to represent our accounts, and as we pay our bills, we find that we have to subtract the amount we owe (325 lira) for the gilding on our Renaissance chair from the amount we have in our account (268 lira) and we suddenly realize that we still owe 57 lira! If we tweak the model ever so slightly, we have negative numbers – no longer are we restricted to our model representing some physical quantity! We can now model a more abstract concept, like money owed.

If we keep tweaking our model, we eventually end up with lots of algebraic symbols, like the superscript for raising to a power (which Rene DesCartes and Francois Viete invented, since they decided that xx was too cumbersome and x^2 was much prettier**), which led to all sorts of properties of exponents.

Or, even more fun, we start using the square root symbol (which is actually just a check-mark shape; the horizontal bar part we always write is actually a grouping symbol called a vinculum – it’s the same grouping symbol we use when writing fractions or repeating decimals). So we merrily use our symbols to model the square root of 9 (3) and the square root of 2 (approximately 2.414, which irritated the Greeks because it turned out to be irrational), and then some smart-alecky student asks “What’s the square root of negative nine?” So we tweak the model a little more, and “EUREKA!” we have complex numbers!

So, why is 2+2=4? Because this is our current model for Og’s chickens. If you don’t like it, use a different model, like base2, in which 10+10=100.



* I actually think this is a pretty lame way to end a class, especially when the bell is about to ring. I’m not really expecting any questions, and even if there are any, I won’t really have time to answer them adequately. It just kind of slipped out today, but I’m glad it did, since it gave me a side track for my train of thought.<\teacher mode>

** Do you see my problem with algebraic notation here? x^2 just isn’t as pretty as if I were actually able to figure out how to get superscripts and other math symbols.